:: NDIFF_5 semantic presentation
begin
theorem Th1: :: NDIFF_5:1
for S being non trivial RealNormSpace
for R being Function of REAL,S holds
( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) ) )
proof
let S be non trivial RealNormSpace; ::_thesis: for R being Function of REAL,S holds
( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) ) )
let R be Function of REAL,S; ::_thesis: ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) ) )
A1: dom R = REAL by PARTFUN1:def_2;
A2: now__::_thesis:_(_R_is_RestFunc-like_&_ex_r_being_Real_st_
(_r_>_0_&_(_for_d_being_Real_holds_
(_not_d_>_0_or_ex_z_being_Real_st_
(_z_<>_0_&_|.z.|_<_d_&_not_(|.z.|_")_*_||.(R_/._z).||_<_r_)_)_)_)_implies_for_r_being_Real_st_r_>_0_holds_
ex_d_being_Real_st_
(_d_>_0_&_(_for_z_being_Real_st_z_<>_0_&_|.z.|_<_d_holds_
(|.z.|_")_*_||.(R_/._z).||_<_r_)_)_)
assume A3: R is RestFunc-like ; ::_thesis: ( ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & |.z.| < d & not (|.z.| ") * ||.(R /. z).|| < r ) ) ) ) implies for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) ) )
assume ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & |.z.| < d & not (|.z.| ") * ||.(R /. z).|| < r ) ) ) ) ; ::_thesis: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) )
then consider r being Real such that
A4: r > 0 and
A5: for d being Real st d > 0 holds
ex z being Real st
( z <> 0 & |.z.| < d & not (|.z.| ") * ||.(R /. z).|| < r ) ;
defpred S1[ Element of NAT , Real] means ( $2 <> 0 & |.$2.| < 1 / ($1 + 1) & not (|.$2.| ") * ||.(R /. $2).|| < r );
A6: for n being Element of NAT ex z being Real st S1[n,z]
proof
let n be Element of NAT ; ::_thesis: ex z being Real st S1[n,z]
1 / (n + 1) is Real by XREAL_0:def_1;
hence ex z being Real st S1[n,z] by A5; ::_thesis: verum
end;
consider s being Real_Sequence such that
A7: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6);
A8: now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
|.((s_._m)_-_0).|_<_p
let p be real number ; ::_thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.((s . m) - 0).| < p )
assume A9: 0 < p ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.((s . m) - 0).| < p
consider n being Element of NAT such that
A10: p " < n by SEQ_4:3;
reconsider q0 = 0 , q1 = 1 as real number ;
(p ") + q0 < n + q1 by A10, XREAL_1:8;
then A11: 1 / (n + 1) < 1 / (p ") by A9, XREAL_1:76;
take n = n; ::_thesis: for m being Element of NAT st n <= m holds
|.((s . m) - 0).| < p
let m be Element of NAT ; ::_thesis: ( n <= m implies |.((s . m) - 0).| < p )
assume n <= m ; ::_thesis: |.((s . m) - 0).| < p
then n + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (n + 1) by XREAL_1:118;
then |.((s . m) - 0).| < 1 / (n + 1) by A7, XXREAL_0:2;
hence |.((s . m) - 0).| < p by A11, XXREAL_0:2; ::_thesis: verum
end;
then A12: s is convergent by SEQ_2:def_6;
then A13: lim s = 0 by A8, SEQ_2:def_7;
s is non-zero by A7, SEQ_1:5;
then reconsider s = s as non-zero 0 -convergent Real_Sequence by A12, A13, FDIFF_1:def_1;
( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0. S ) by A3, NDIFF_3:def_1;
then consider n0 being Element of NAT such that
A14: for m being Element of NAT st n0 <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r by A4, NORMSP_1:def_7;
A15: ||.((((s ") (#) (R /* s)) . n0) - (0. S)).|| < r by A14;
A16: ||.(((s . n0) ") * (R /. (s . n0))).|| = (abs ((s . n0) ")) * ||.(R /. (s . n0)).|| by NORMSP_1:def_1
.= (|.(s . n0).| ") * ||.(R /. (s . n0)).|| by COMPLEX1:66 ;
A17: rng s c= dom R by A1;
||.((((s ") (#) (R /* s)) . n0) - (0. S)).|| = ||.(((s ") (#) (R /* s)) . n0).|| by RLVECT_1:13
.= ||.(((s ") . n0) * ((R /* s) . n0)).|| by NDIFF_1:def_2
.= ||.(((s . n0) ") * ((R /* s) . n0)).|| by VALUED_1:10
.= ||.(((s . n0) ") * (R /. (s . n0))).|| by A17, FUNCT_2:109 ;
hence for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) ) by A7, A15, A16; ::_thesis: verum
end;
now__::_thesis:_(_(_for_r_being_Real_st_r_>_0_holds_
ex_d_being_Real_st_
(_d_>_0_&_(_for_z_being_Real_st_z_<>_0_&_|.z.|_<_d_holds_
(|.z.|_")_*_||.(R_/._z).||_<_r_)_)_)_implies_R_is_RestFunc-like_)
assume A18: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) ) ; ::_thesis: R is RestFunc-like
now__::_thesis:_for_s_being_non-zero_0_-convergent_Real_Sequence_holds_
(_(s_")_(#)_(R_/*_s)_is_convergent_&_lim_((s_")_(#)_(R_/*_s))_=_0._S_)
let s be non-zero 0 -convergent Real_Sequence; ::_thesis: ( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0. S )
A19: ( s is convergent & lim s = 0 ) ;
A20: now__::_thesis:_for_r_being_Real_st_r_>_0_holds_
ex_n0_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n0_<=_m_holds_
||.((((s_")_(#)_(R_/*_s))_._m)_-_(0._S)).||_<_r
let r be Real; ::_thesis: ( r > 0 implies ex n0 being Element of NAT st
for m being Element of NAT st n0 <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r )
assume r > 0 ; ::_thesis: ex n0 being Element of NAT st
for m being Element of NAT st n0 <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r
then consider d being Real such that
A21: d > 0 and
A22: for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r by A18;
consider n0 being Element of NAT such that
A23: for m being Element of NAT st n0 <= m holds
|.((s . m) - 0).| < d by A19, A21, SEQ_2:def_7;
take n0 = n0; ::_thesis: for m being Element of NAT st n0 <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r
thus for m being Element of NAT st n0 <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r ::_thesis: verum
proof
A24: rng s c= dom R by A1;
let m be Element of NAT ; ::_thesis: ( n0 <= m implies ||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r )
assume n0 <= m ; ::_thesis: ||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r
then A25: |.((s . m) - 0).| < d by A23;
A26: s . m <> 0 by SEQ_1:5;
(|.(s . m).| ") * ||.(R /. (s . m)).|| = (abs ((s . m) ")) * ||.(R /. (s . m)).|| by COMPLEX1:66
.= ||.(((s . m) ") * (R /. (s . m))).|| by NORMSP_1:def_1
.= ||.(((s . m) ") * ((R /* s) . m)).|| by A24, FUNCT_2:109
.= ||.(((s ") . m) * ((R /* s) . m)).|| by VALUED_1:10
.= ||.(((s ") (#) (R /* s)) . m).|| by NDIFF_1:def_2
.= ||.((((s ") (#) (R /* s)) . m) - (0. S)).|| by RLVECT_1:13 ;
hence ||.((((s ") (#) (R /* s)) . m) - (0. S)).|| < r by A22, A25, A26; ::_thesis: verum
end;
end;
hence (s ") (#) (R /* s) is convergent by NORMSP_1:def_6; ::_thesis: lim ((s ") (#) (R /* s)) = 0. S
hence lim ((s ") (#) (R /* s)) = 0. S by A20, NORMSP_1:def_7; ::_thesis: verum
end;
hence R is RestFunc-like by NDIFF_3:def_1; ::_thesis: verum
end;
hence ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < r ) ) ) by A2; ::_thesis: verum
end;
theorem Th2: :: NDIFF_5:2
for S being non trivial RealNormSpace
for R being RestFunc of S st R /. 0 = 0. S holds
for e being Real st e > 0 holds
ex d being Real st
( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) )
proof
let S be non trivial RealNormSpace; ::_thesis: for R being RestFunc of S st R /. 0 = 0. S holds
for e being Real st e > 0 holds
ex d being Real st
( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) )
let R be RestFunc of S; ::_thesis: ( R /. 0 = 0. S implies for e being Real st e > 0 holds
ex d being Real st
( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) ) )
assume A1: R /. 0 = 0. S ; ::_thesis: for e being Real st e > 0 holds
ex d being Real st
( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) )
let e be Real; ::_thesis: ( e > 0 implies ex d being Real st
( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) ) )
assume A2: e > 0 ; ::_thesis: ex d being Real st
( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) )
R is total by NDIFF_3:def_1;
then consider d being Real such that
A3: d > 0 and
A4: for z being Real st z <> 0 & |.z.| < d holds
(|.z.| ") * ||.(R /. z).|| < e by A2, Th1;
take d ; ::_thesis: ( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) )
now__::_thesis:_for_h_being_Real_st_|.h.|_<_d_holds_
||.(R_/._h).||_<=_e_*_|.h.|
let h be Real; ::_thesis: ( |.h.| < d implies ||.(R /. b1).|| <= e * |.b1.| )
assume A5: |.h.| < d ; ::_thesis: ||.(R /. b1).|| <= e * |.b1.|
A6: 0 <= |.h.| by COMPLEX1:46;
percases ( h <> 0 or h = 0 ) ;
supposeA7: h <> 0 ; ::_thesis: ||.(R /. b1).|| <= e * |.b1.|
then (|.h.| ") * ||.(R /. h).|| <= e by A4, A5;
then |.h.| * ((|.h.| ") * ||.(R /. h).||) <= |.h.| * e by A6, XREAL_1:64;
then A8: (|.h.| * (|.h.| ")) * ||.(R /. h).|| <= e * |.h.| ;
|.h.| <> 0 by A7, COMPLEX1:45;
then 1 * ||.(R /. h).|| <= e * |.h.| by A8, XCMPLX_0:def_7;
hence ||.(R /. h).|| <= e * |.h.| ; ::_thesis: verum
end;
supposeA9: h = 0 ; ::_thesis: ||.(R /. b1).|| <= e * |.b1.|
reconsider p0 = 0 as Real ;
p0 * |.h.| <= e * |.h.| by A2, A6;
hence ||.(R /. h).|| <= e * |.h.| by A1, A9; ::_thesis: verum
end;
end;
end;
hence ( d > 0 & ( for h being Real st |.h.| < d holds
||.(R /. h).|| <= e * |.h.| ) ) by A3; ::_thesis: verum
end;
theorem Th3: :: NDIFF_5:3
for S, T being non trivial RealNormSpace
for R being RestFunc of S
for L being Lipschitzian LinearOperator of S,T holds L * R is RestFunc of T
proof
let S, T be non trivial RealNormSpace; ::_thesis: for R being RestFunc of S
for L being Lipschitzian LinearOperator of S,T holds L * R is RestFunc of T
let R be RestFunc of S; ::_thesis: for L being Lipschitzian LinearOperator of S,T holds L * R is RestFunc of T
let L be Lipschitzian LinearOperator of S,T; ::_thesis: L * R is RestFunc of T
consider K being Real such that
A1: 0 <= K and
A2: for z being Point of S holds ||.(L . z).|| <= K * ||.z.|| by LOPBAN_1:def_8;
dom L = the carrier of S by FUNCT_2:def_1;
then A3: rng R c= dom L ;
A4: R is total by NDIFF_3:def_1;
then A5: dom R = REAL by PARTFUN1:def_2;
now__::_thesis:_for_e_being_Real_st_e_>_0_holds_
ex_d_being_Real_st_
(_d_>_0_&_(_for_h_being_Real_st_h_<>_0_&_|.h.|_<_d_holds_
(|.h.|_")_*_||.((L_*_R)_/._h).||_<_e_)_)
let e be Real; ::_thesis: ( e > 0 implies ex d being Real st
( d > 0 & ( for h being Real st h <> 0 & |.h.| < d holds
(|.h.| ") * ||.((L * R) /. h).|| < e ) ) )
assume A6: e > 0 ; ::_thesis: ex d being Real st
( d > 0 & ( for h being Real st h <> 0 & |.h.| < d holds
(|.h.| ") * ||.((L * R) /. h).|| < e ) )
set e1 = (e / 2) / (1 + K);
consider d being Real such that
A7: 0 < d and
A8: for h being Real st h <> 0 & |.h.| < d holds
(|.h.| ") * ||.(R /. h).|| < (e / 2) / (1 + K) by A1, A4, A6, Th1;
A9: e / 2 < e by A6, XREAL_1:216;
now__::_thesis:_for_h_being_Real_st_h_<>_0_&_|.h.|_<_d_holds_
(|.h.|_")_*_||.((L_*_R)_/._h).||_<_e
let h be Real; ::_thesis: ( h <> 0 & |.h.| < d implies (|.h.| ") * ||.((L * R) /. h).|| < e )
assume A10: ( h <> 0 & |.h.| < d ) ; ::_thesis: (|.h.| ") * ||.((L * R) /. h).|| < e
then (|.h.| ") * ||.(R /. h).|| < (e / 2) / (1 + K) by A8;
then (K + 1) * ((|.h.| ") * ||.(R /. h).||) <= (K + 1) * ((e / 2) / (1 + K)) by A1, XREAL_1:64;
then A11: (K + 1) * ((|.h.| ") * ||.(R /. h).||) <= e / 2 by A1, XCMPLX_1:87;
|.h.| <> 0 by A10, COMPLEX1:45;
then A12: |.h.| > 0 by COMPLEX1:46;
reconsider p0 = 0 , p1 = 1 as Real ;
p0 + K < p1 + K by XREAL_1:8;
then A13: K * ||.(R /. h).|| <= (K + 1) * ||.(R /. h).|| by XREAL_1:64;
||.(L . (R /. h)).|| <= K * ||.(R /. h).|| by A2;
then ||.(L . (R /. h)).|| <= (K + 1) * ||.(R /. h).|| by A13, XXREAL_0:2;
then (|.h.| ") * ||.(L . (R /. h)).|| <= (|.h.| ") * ((K + 1) * ||.(R /. h).||) by A12, XREAL_1:64;
then A14: (|.h.| ") * ||.(L . (R /. h)).|| <= e / 2 by A11, XXREAL_0:2;
L . (R /. h) = L /. (R /. h) ;
then L . (R /. h) = (L * R) /. h by A5, A3, PARTFUN2:5;
hence (|.h.| ") * ||.((L * R) /. h).|| < e by A9, A14, XXREAL_0:2; ::_thesis: verum
end;
hence ex d being Real st
( d > 0 & ( for h being Real st h <> 0 & |.h.| < d holds
(|.h.| ") * ||.((L * R) /. h).|| < e ) ) by A7; ::_thesis: verum
end;
hence L * R is RestFunc of T by A4, Th1; ::_thesis: verum
end;
theorem Th4: :: NDIFF_5:4
for S, T being non trivial RealNormSpace
for R1 being RestFunc of S st R1 /. 0 = 0. S holds
for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L being LinearFunc of S holds R2 * (L + R1) is RestFunc of T
proof
let S, T be non trivial RealNormSpace; ::_thesis: for R1 being RestFunc of S st R1 /. 0 = 0. S holds
for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L being LinearFunc of S holds R2 * (L + R1) is RestFunc of T
let R1 be RestFunc of S; ::_thesis: ( R1 /. 0 = 0. S implies for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L being LinearFunc of S holds R2 * (L + R1) is RestFunc of T )
assume R1 /. 0 = 0. S ; ::_thesis: for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L being LinearFunc of S holds R2 * (L + R1) is RestFunc of T
then consider d0 being Real such that
A1: 0 < d0 and
A2: for h being Real st |.h.| < d0 holds
||.(R1 /. h).|| <= 1 * |.h.| by Th2;
let R2 be RestFunc of S,T; ::_thesis: ( R2 /. (0. S) = 0. T implies for L being LinearFunc of S holds R2 * (L + R1) is RestFunc of T )
assume A3: R2 /. (0. S) = 0. T ; ::_thesis: for L being LinearFunc of S holds R2 * (L + R1) is RestFunc of T
let L be LinearFunc of S; ::_thesis: R2 * (L + R1) is RestFunc of T
consider r being Point of S such that
A4: for h being Real holds L . h = h * r by NDIFF_3:def_2;
reconsider K = ||.r.|| as Real ;
R2 is total by NDIFF_1:def_5;
then dom R2 = the carrier of S by PARTFUN1:def_2;
then A5: rng (L + R1) c= dom R2 ;
R1 is total by NDIFF_3:def_1;
then L + R1 is total by VFUNCT_1:32;
then A6: dom (L + R1) = REAL by PARTFUN1:def_2;
then dom (R2 * (L + R1)) = REAL by A5, RELAT_1:27;
then A7: R2 * (L + R1) is total by PARTFUN1:def_2;
now__::_thesis:_for_e_being_Real_st_e_>_0_holds_
ex_dd1_being_Real_st_
(_dd1_>_0_&_(_for_h_being_Real_st_h_<>_0_&_|.h.|_<_dd1_holds_
(|.h.|_")_*_||.((R2_*_(L_+_R1))_/._h).||_<_e_)_)
let e be Real; ::_thesis: ( e > 0 implies ex dd1 being Real st
( dd1 > 0 & ( for h being Real st h <> 0 & |.h.| < dd1 holds
(|.h.| ") * ||.((R2 * (L + R1)) /. h).|| < e ) ) )
assume A8: e > 0 ; ::_thesis: ex dd1 being Real st
( dd1 > 0 & ( for h being Real st h <> 0 & |.h.| < dd1 holds
(|.h.| ") * ||.((R2 * (L + R1)) /. h).|| < e ) )
A9: e / 2 < e by A8, XREAL_1:216;
set e1 = (e / 2) / (1 + K);
consider d being Real such that
A10: 0 < d and
A11: for z being Point of S st ||.z.|| < d holds
||.(R2 /. z).|| <= ((e / 2) / (1 + K)) * ||.z.|| by A3, A8, NDIFF_2:7;
set d1 = d / (1 + K);
set dd1 = min (d0,(d / (1 + K)));
A12: ( min (d0,(d / (1 + K))) <= d / (1 + K) & min (d0,(d / (1 + K))) <= d0 ) by XXREAL_0:17;
A13: now__::_thesis:_for_h_being_Real_st_h_<>_0_&_|.h.|_<_min_(d0,(d_/_(1_+_K)))_holds_
(|.h.|_")_*_||.((R2_*_(L_+_R1))_/._h).||_<_e
let h be Real; ::_thesis: ( h <> 0 & |.h.| < min (d0,(d / (1 + K))) implies (|.h.| ") * ||.((R2 * (L + R1)) /. h).|| < e )
assume that
A14: h <> 0 and
A15: |.h.| < min (d0,(d / (1 + K))) ; ::_thesis: (|.h.| ") * ||.((R2 * (L + R1)) /. h).|| < e
|.h.| < d0 by A12, A15, XXREAL_0:2;
then A16: ||.(R1 /. h).|| <= 1 * |.h.| by A2;
reconsider p0 = 0 as Real ;
L . h = h * r by A4;
then (||.(L . h).|| - (K * |.h.|)) + (K * |.h.|) <= p0 + (K * |.h.|) by NORMSP_1:def_1;
then ( ||.((L . h) + (R1 /. h)).|| <= ||.(L . h).|| + ||.(R1 /. h).|| & ||.(L . h).|| + ||.(R1 /. h).|| <= (K * |.h.|) + (1 * |.h.|) ) by A16, NORMSP_1:def_1, XREAL_1:7;
then A17: ||.((L . h) + (R1 /. h)).|| <= (K + 1) * |.h.| by XXREAL_0:2;
then A18: ((e / 2) / (1 + K)) * ||.((L . h) + (R1 /. h)).|| <= ((e / 2) / (1 + K)) * ((K + 1) * |.h.|) by A8, XREAL_1:64;
|.h.| < d / (1 + K) by A12, A15, XXREAL_0:2;
then (K + 1) * |.h.| < (K + 1) * (d / (1 + K)) by XREAL_1:68;
then ||.((L . h) + (R1 /. h)).|| < (K + 1) * (d / (1 + K)) by A17, XXREAL_0:2;
then ||.((L . h) + (R1 /. h)).|| < d by XCMPLX_1:87;
then ||.(R2 /. ((L . h) + (R1 /. h))).|| <= ((e / 2) / (1 + K)) * ||.((L . h) + (R1 /. h)).|| by A11;
then A19: ||.(R2 /. ((L . h) + (R1 /. h))).|| <= ((e / 2) / (1 + K)) * ((K + 1) * |.h.|) by A18, XXREAL_0:2;
A20: R2 /. ((L . h) + (R1 /. h)) = R2 /. ((L /. h) + (R1 /. h))
.= R2 /. ((L + R1) /. h) by A6, VFUNCT_1:def_1
.= (R2 * (L + R1)) /. h by A6, A5, PARTFUN2:5 ;
A21: |.h.| <> 0 by A14, COMPLEX1:45;
then |.h.| > 0 by COMPLEX1:46;
then (|.h.| ") * ||.((R2 * (L + R1)) /. h).|| <= (|.h.| ") * ((((e / 2) / (1 + K)) * (K + 1)) * |.h.|) by A20, A19, XREAL_1:64;
then (|.h.| ") * ||.((R2 * (L + R1)) /. h).|| <= ((|.h.| * (|.h.| ")) * ((e / 2) / (1 + K))) * (K + 1) ;
then (|.h.| ") * ||.((R2 * (L + R1)) /. h).|| <= (1 * ((e / 2) / (1 + K))) * (K + 1) by A21, XCMPLX_0:def_7;
then (|.h.| ") * ||.((R2 * (L + R1)) /. h).|| <= e / 2 by XCMPLX_1:87;
hence (|.h.| ") * ||.((R2 * (L + R1)) /. h).|| < e by A9, XXREAL_0:2; ::_thesis: verum
end;
0 < min (d0,(d / (1 + K))) by A1, A10, XXREAL_0:15;
hence ex dd1 being Real st
( dd1 > 0 & ( for h being Real st h <> 0 & |.h.| < dd1 holds
(|.h.| ") * ||.((R2 * (L + R1)) /. h).|| < e ) ) by A13; ::_thesis: verum
end;
hence R2 * (L + R1) is RestFunc of T by A7, Th1; ::_thesis: verum
end;
theorem Th5: :: NDIFF_5:5
for S, T being non trivial RealNormSpace
for R1 being RestFunc of S st R1 /. 0 = 0. S holds
for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L1 being LinearFunc of S
for L2 being Lipschitzian LinearOperator of S,T holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T
proof
let S, T be non trivial RealNormSpace; ::_thesis: for R1 being RestFunc of S st R1 /. 0 = 0. S holds
for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L1 being LinearFunc of S
for L2 being Lipschitzian LinearOperator of S,T holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T
let R1 be RestFunc of S; ::_thesis: ( R1 /. 0 = 0. S implies for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L1 being LinearFunc of S
for L2 being Lipschitzian LinearOperator of S,T holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T )
assume A1: R1 /. 0 = 0. S ; ::_thesis: for R2 being RestFunc of S,T st R2 /. (0. S) = 0. T holds
for L1 being LinearFunc of S
for L2 being Lipschitzian LinearOperator of S,T holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T
let R2 be RestFunc of S,T; ::_thesis: ( R2 /. (0. S) = 0. T implies for L1 being LinearFunc of S
for L2 being Lipschitzian LinearOperator of S,T holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T )
assume A2: R2 /. (0. S) = 0. T ; ::_thesis: for L1 being LinearFunc of S
for L2 being Lipschitzian LinearOperator of S,T holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T
let L1 be LinearFunc of S; ::_thesis: for L2 being Lipschitzian LinearOperator of S,T holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T
let L2 be Lipschitzian LinearOperator of S,T; ::_thesis: (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T
( L2 * R1 is RestFunc of T & R2 * (L1 + R1) is RestFunc of T ) by A1, A2, Th4, Th3;
hence (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of T by NDIFF_3:7; ::_thesis: verum
end;
theorem Th6: :: NDIFF_5:6
for S, T being non trivial RealNormSpace
for x0 being Element of REAL
for g being PartFunc of REAL, the carrier of S st g is_differentiable_in x0 holds
for f being PartFunc of the carrier of S, the carrier of T st f is_differentiable_in g /. x0 holds
( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) )
proof
let S, T be non trivial RealNormSpace; ::_thesis: for x0 being Element of REAL
for g being PartFunc of REAL, the carrier of S st g is_differentiable_in x0 holds
for f being PartFunc of the carrier of S, the carrier of T st f is_differentiable_in g /. x0 holds
( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) )
let x0 be Element of REAL ; ::_thesis: for g being PartFunc of REAL, the carrier of S st g is_differentiable_in x0 holds
for f being PartFunc of the carrier of S, the carrier of T st f is_differentiable_in g /. x0 holds
( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) )
let g be PartFunc of REAL, the carrier of S; ::_thesis: ( g is_differentiable_in x0 implies for f being PartFunc of the carrier of S, the carrier of T st f is_differentiable_in g /. x0 holds
( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) ) )
assume A1: g is_differentiable_in x0 ; ::_thesis: for f being PartFunc of the carrier of S, the carrier of T st f is_differentiable_in g /. x0 holds
( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) )
consider N1 being Neighbourhood of x0 such that
A2: N1 c= dom g and
A3: ex L1 being LinearFunc of S ex R1 being RestFunc of S st
( diff (g,x0) = L1 . 1 & ( for x being Real st x in N1 holds
(g /. x) - (g /. x0) = (L1 . (x - x0)) + (R1 /. (x - x0)) ) ) by A1, NDIFF_3:def_4;
let f be PartFunc of the carrier of S, the carrier of T; ::_thesis: ( f is_differentiable_in g /. x0 implies ( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) ) )
assume f is_differentiable_in g /. x0 ; ::_thesis: ( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) )
then consider N2 being Neighbourhood of g /. x0 such that
A4: N2 c= dom f and
A5: ex R2 being RestFunc of S,T st
( R2 /. (0. S) = 0. T & R2 is_continuous_in 0. S & ( for y being Point of S st y in N2 holds
(f /. y) - (f /. (g /. x0)) = ((diff (f,(g /. x0))) . (y - (g /. x0))) + (R2 /. (y - (g /. x0))) ) ) by NDIFF_1:47;
consider R2 being RestFunc of S,T such that
A6: R2 /. (0. S) = 0. T and
A7: for y being Point of S st y in N2 holds
(f /. y) - (f /. (g /. x0)) = ((diff (f,(g /. x0))) . (y - (g /. x0))) + (R2 /. (y - (g /. x0))) by A5;
reconsider L2 = diff (f,(g /. x0)) as Lipschitzian LinearOperator of S,T by LOPBAN_1:def_9;
consider L1 being LinearFunc of S, R1 being RestFunc of S such that
A8: ( diff (g,x0) = L1 . 1 & ( for x being Real st x in N1 holds
(g /. x) - (g /. x0) = (L1 . (x - x0)) + (R1 /. (x - x0)) ) ) by A3;
consider r being Point of S such that
A9: for p being Real holds L1 . p = p * r by NDIFF_3:def_2;
reconsider p0 = 0 as Element of REAL ;
(g /. x0) - (g /. x0) = (L1 . (x0 - x0)) + (R1 /. (x0 - x0)) by A8, RCOMP_1:16;
then 0. S = (L1 . 0) + (R1 /. 0) by RLVECT_1:15;
then 0. S = (p0 * r) + (R1 /. 0) by A9;
then 0. S = (0. S) + (R1 /. 0) by RLVECT_1:10;
then R1 /. 0 = 0. S by RLVECT_1:4;
then reconsider R0 = (L2 * R1) + (R2 * (L1 + R1)) as RestFunc of T by A6, Th5;
A10: dom (L2 * L1) = REAL by FUNCT_2:def_1;
reconsider q = L2 . r as Point of T ;
now__::_thesis:_for_p_being_Real_holds_(L2_*_L1)_._p_=_p_*_q
let p be Real; ::_thesis: (L2 * L1) . p = p * q
L2 . (L1 . p) = L2 . (p * r) by A9;
then L2 . (L1 . p) = p * q by LOPBAN_1:def_5;
hence (L2 * L1) . p = p * q by A10, FUNCT_1:12; ::_thesis: verum
end;
then reconsider L0 = L2 * L1 as LinearFunc of T by NDIFF_3:def_2;
g is_continuous_in x0 by A1, NDIFF_3:22;
then consider N3 being Neighbourhood of x0 such that
A11: g .: N3 c= N2 by NFCONT_3:10;
consider N being Neighbourhood of x0 such that
A12: N c= N1 and
A13: N c= N3 by RCOMP_1:17;
now__::_thesis:_for_x_being_set_st_x_in_N_holds_
x_in_dom_(f_*_g)
let x be set ; ::_thesis: ( x in N implies x in dom (f * g) )
assume A14: x in N ; ::_thesis: x in dom (f * g)
then reconsider x9 = x as Real ;
A15: x in N1 by A12, A14;
then g . x9 in g .: N3 by A2, A13, A14, FUNCT_1:def_6;
then g . x9 in N2 by A11;
hence x in dom (f * g) by A2, A4, A15, FUNCT_1:11; ::_thesis: verum
end;
then A16: N c= dom (f * g) by TARSKI:def_3;
A17: now__::_thesis:_for_x_being_Real_st_x_in_N_holds_
((f_*_g)_/._x)_-_((f_*_g)_/._x0)_=_(L0_._(x_-_x0))_+_(R0_/._(x_-_x0))
let x be Real; ::_thesis: ( x in N implies ((f * g) /. x) - ((f * g) /. x0) = (L0 . (x - x0)) + (R0 /. (x - x0)) )
assume A18: x in N ; ::_thesis: ((f * g) /. x) - ((f * g) /. x0) = (L0 . (x - x0)) + (R0 /. (x - x0))
A19: (g /. x) - (g /. x0) = (L1 . (x - x0)) + (R1 /. (x - x0)) by A8, A12, A18;
A20: x in N1 by A12, A18;
then g . x in g .: N3 by A2, A13, A18, FUNCT_1:def_6;
then g . x in N2 by A11;
then A21: g /. x in N2 by A2, A20, PARTFUN1:def_6;
A22: x0 in N by RCOMP_1:16;
A23: R1 is total by NDIFF_3:def_1;
then A24: dom R1 = REAL by PARTFUN1:def_2;
dom L2 = the carrier of S by FUNCT_2:def_1;
then A25: rng R1 c= dom L2 ;
A26: dom (L2 * R1) = REAL by A23, PARTFUN1:def_2;
L1 + R1 is total by A23, VFUNCT_1:32;
then A27: dom (L1 + R1) = REAL by PARTFUN1:def_2;
R2 is total by NDIFF_1:def_5;
then dom R2 = the carrier of S by PARTFUN1:def_2;
then A28: rng (L1 + R1) c= dom R2 ;
then dom (R2 * (L1 + R1)) = dom (L1 + R1) by RELAT_1:27;
then A29: dom ((L2 * R1) + (R2 * (L1 + R1))) = REAL /\ REAL by A26, A27, VFUNCT_1:def_1;
L2 . (R1 /. (x - x0)) = L2 /. (R1 /. (x - x0)) ;
then A30: L2 . (R1 /. (x - x0)) = (L2 * R1) /. (x - x0) by A24, A25, PARTFUN2:5;
A31: R2 /. ((L1 . (x - x0)) + (R1 /. (x - x0))) = R2 /. ((L1 /. (x - x0)) + (R1 /. (x - x0)))
.= R2 /. ((L1 + R1) /. (x - x0)) by A27, VFUNCT_1:def_1
.= (R2 * (L1 + R1)) /. (x - x0) by A27, A28, PARTFUN2:5 ;
A32: (L2 * L1) . (x - x0) = L2 . (L1 . (x - x0)) by A10, FUNCT_1:12;
thus ((f * g) /. x) - ((f * g) /. x0) = (f /. (g /. x)) - ((f * g) /. x0) by A16, A18, PARTFUN2:3
.= (f /. (g /. x)) - (f /. (g /. x0)) by A16, A22, PARTFUN2:3
.= ((diff (f,(g /. x0))) . ((g /. x) - (g /. x0))) + (R2 /. ((g /. x) - (g /. x0))) by A7, A21
.= ((L2 . (L1 . (x - x0))) + (L2 . (R1 /. (x - x0)))) + ((R2 * (L1 + R1)) /. (x - x0)) by A19, A31, VECTSP_1:def_20
.= (L2 . (L1 . (x - x0))) + (((L2 * R1) /. (x - x0)) + ((R2 * (L1 + R1)) /. (x - x0))) by A30, RLVECT_1:def_3
.= (L0 . (x - x0)) + (R0 /. (x - x0)) by A32, A29, VFUNCT_1:def_1 ; ::_thesis: verum
end;
hence A33: f * g is_differentiable_in x0 by A16, NDIFF_3:def_3; ::_thesis: diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0))
(L2 * L1) . 1 = (diff (f,(g /. x0))) . (diff (g,x0)) by A8, A10, FUNCT_1:12;
hence diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) by A33, A16, A17, NDIFF_3:def_4; ::_thesis: verum
end;
theorem Th7: :: NDIFF_5:7
for S being RealNormSpace
for xseq being FinSequence of S
for yseq being FinSequence of REAL st len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) holds
||.(Sum xseq).|| <= Sum yseq
proof
let S be RealNormSpace; ::_thesis: for xseq being FinSequence of S
for yseq being FinSequence of REAL st len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) holds
||.(Sum xseq).|| <= Sum yseq
let xseq be FinSequence of S; ::_thesis: for yseq being FinSequence of REAL st len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) holds
||.(Sum xseq).|| <= Sum yseq
let yseq be FinSequence of REAL ; ::_thesis: ( len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) implies ||.(Sum xseq).|| <= Sum yseq )
assume that
A1: len xseq = len yseq and
A2: for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ; ::_thesis: ||.(Sum xseq).|| <= Sum yseq
defpred S1[ Nat] means for xseq being FinSequence of S
for yseq being FinSequence of REAL st $1 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) holds
||.(Sum xseq).|| <= Sum yseq;
A3: S1[ 0 ]
proof
let xseq be FinSequence of S; ::_thesis: for yseq being FinSequence of REAL st 0 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) holds
||.(Sum xseq).|| <= Sum yseq
let yseq be FinSequence of REAL ; ::_thesis: ( 0 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) implies ||.(Sum xseq).|| <= Sum yseq )
assume A4: ( 0 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) ) ; ::_thesis: ||.(Sum xseq).|| <= Sum yseq
consider Sx being Function of NAT, the carrier of S such that
A5: ( Sum xseq = Sx . (len xseq) & Sx . 0 = 0. S & ( for j being Element of NAT
for v being Element of S st j < len xseq & v = xseq . (j + 1) holds
Sx . (j + 1) = (Sx . j) + v ) ) by RLVECT_1:def_12;
yseq = {} by A4;
hence ||.(Sum xseq).|| <= Sum yseq by A4, A5, RVSUM_1:72; ::_thesis: verum
end;
A6: now__::_thesis:_for_i_being_Element_of_NAT_st_S1[i]_holds_
S1[i_+_1]
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A7: S1[i] ; ::_thesis: S1[i + 1]
now__::_thesis:_for_xseq_being_FinSequence_of_S
for_yseq_being_FinSequence_of_REAL_st_i_+_1_=_len_xseq_&_len_xseq_=_len_yseq_&_(_for_i_being_Element_of_NAT_st_i_in_dom_xseq_holds_
yseq_._i_=_||.(xseq_/._i).||_)_holds_
||.(Sum_xseq).||_<=_Sum_yseq
let xseq be FinSequence of S; ::_thesis: for yseq being FinSequence of REAL st i + 1 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) holds
||.(Sum xseq).|| <= Sum yseq
let yseq be FinSequence of REAL ; ::_thesis: ( i + 1 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) implies ||.(Sum xseq).|| <= Sum yseq )
set xseq0 = xseq | i;
set yseq0 = yseq | i;
assume A8: ( i + 1 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq . i = ||.(xseq /. i).|| ) ) ; ::_thesis: ||.(Sum xseq).|| <= Sum yseq
A9: for k being Element of NAT st k in dom (xseq | i) holds
(yseq | i) . k = ||.((xseq | i) /. k).||
proof
let k be Element of NAT ; ::_thesis: ( k in dom (xseq | i) implies (yseq | i) . k = ||.((xseq | i) /. k).|| )
assume A10: k in dom (xseq | i) ; ::_thesis: (yseq | i) . k = ||.((xseq | i) /. k).||
then A11: ( k in Seg i & k in dom xseq ) by RELAT_1:57;
then A12: yseq . k = ||.(xseq /. k).|| by A8;
xseq /. k = xseq . k by A11, PARTFUN1:def_6;
then xseq /. k = (xseq | i) . k by A11, FUNCT_1:49;
then xseq /. k = (xseq | i) /. k by A10, PARTFUN1:def_6;
hence (yseq | i) . k = ||.((xseq | i) /. k).|| by A11, A12, FUNCT_1:49; ::_thesis: verum
end;
A13: dom xseq = Seg (i + 1) by A8, FINSEQ_1:def_3;
then A14: yseq . (i + 1) = ||.(xseq /. (i + 1)).|| by A8, FINSEQ_1:4;
A15: 1 <= i + 1 by NAT_1:11;
yseq = (yseq | i) ^ <*(yseq /. (i + 1))*> by A8, FINSEQ_5:21;
then yseq = (yseq | i) ^ <*(yseq . (i + 1))*> by A8, A15, FINSEQ_4:15;
then A16: Sum yseq = (Sum (yseq | i)) + (yseq . (i + 1)) by RVSUM_1:74;
A17: len xseq in dom xseq by A13, A8, FINSEQ_1:4;
then reconsider v = xseq . (len xseq) as Element of S by PARTFUN1:4;
A18: v = xseq /. (i + 1) by A8, A17, PARTFUN1:def_6;
A19: i = len (xseq | i) by A8, FINSEQ_1:59, NAT_1:11;
then xseq | i = xseq | (dom (xseq | i)) by FINSEQ_1:def_3;
then A20: Sum xseq = (Sum (xseq | i)) + v by A8, A19, RLVECT_1:38;
A21: ||.((Sum (xseq | i)) + v).|| <= ||.(Sum (xseq | i)).|| + ||.v.|| by NORMSP_1:def_1;
len (xseq | i) = len (yseq | i) by A8, A19, FINSEQ_1:59, NAT_1:11;
then ||.(Sum (xseq | i)).|| <= Sum (yseq | i) by A7, A9, A19;
then ||.(Sum (xseq | i)).|| + ||.v.|| <= (Sum (yseq | i)) + (yseq . (i + 1)) by A14, A18, XREAL_1:6;
hence ||.(Sum xseq).|| <= Sum yseq by A16, A20, A21, XXREAL_0:2; ::_thesis: verum
end;
hence S1[i + 1] ; ::_thesis: verum
end;
for i being Element of NAT holds S1[i] from NAT_1:sch_1(A3, A6);
hence ||.(Sum xseq).|| <= Sum yseq by A1, A2; ::_thesis: verum
end;
theorem Th8: :: NDIFF_5:8
for S being RealNormSpace
for x being Point of S
for N1, N2 being Neighbourhood of x holds N1 /\ N2 is Neighbourhood of x
proof
let S be RealNormSpace; ::_thesis: for x being Point of S
for N1, N2 being Neighbourhood of x holds N1 /\ N2 is Neighbourhood of x
let x be Point of S; ::_thesis: for N1, N2 being Neighbourhood of x holds N1 /\ N2 is Neighbourhood of x
let N1, N2 be Neighbourhood of x; ::_thesis: N1 /\ N2 is Neighbourhood of x
consider N being Neighbourhood of x such that
A1: ( N c= N1 & N c= N2 ) by NDIFF_1:1;
A2: N c= N1 /\ N2 by A1, XBOOLE_1:19;
consider g being Real such that
A3: 0 < g and
A4: { y where y is Point of S : ||.(y - x).|| < g } c= N by NFCONT_1:def_1;
{ y where y is Point of S : ||.(y - x).|| < g } c= N1 /\ N2 by A2, A4, XBOOLE_1:1;
hence N1 /\ N2 is Neighbourhood of x by A3, NFCONT_1:def_1; ::_thesis: verum
end;
theorem Th9: :: NDIFF_5:9
for X being non-empty FinSequence
for x being set st x in product X holds
x is FinSequence
proof
let X be non-empty FinSequence; ::_thesis: for x being set st x in product X holds
x is FinSequence
let x be set ; ::_thesis: ( x in product X implies x is FinSequence )
assume x in product X ; ::_thesis: x is FinSequence
then consider g being Function such that
A1: ( x = g & dom g = dom X & ( for i being set st i in dom X holds
g . i in X . i ) ) by CARD_3:def_5;
dom g = Seg (len X) by A1, FINSEQ_1:def_3;
hence x is FinSequence by A1, FINSEQ_1:def_2; ::_thesis: verum
end;
registration
let G be RealNormSpace-Sequence;
cluster product G -> constituted-FinSeqs ;
coherence
product G is constituted-FinSeqs
proof
let a be Element of (product G); :: according to MONOID_0:def_2 ::_thesis: a is set
product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
hence a is set by Th9; ::_thesis: verum
end;
end;
Lm1: now__::_thesis:_for_G_being_RealLinearSpace-Sequence_holds_dom_(carr_G)_=_dom_G
let G be RealLinearSpace-Sequence; ::_thesis: dom (carr G) = dom G
len (carr G) = len G by PRVECT_2:def_4;
hence dom (carr G) = Seg (len G) by FINSEQ_1:def_3
.= dom G by FINSEQ_1:def_3 ;
::_thesis: verum
end;
definition
let G be RealLinearSpace-Sequence;
let z be Element of product (carr G);
let j be Element of dom G;
:: original: .
redefine funcz . j -> Element of (G . j);
correctness
coherence
z . j is Element of (G . j);
proof
reconsider zz = z as FinSequence by Th9;
dom (carr G) = dom G by Lm1;
then zz . j in (carr G) . j by CARD_3:9;
hence z . j is Element of (G . j) by PRVECT_2:def_4; ::_thesis: verum
end;
end;
theorem Th10: :: NDIFF_5:10
for G being RealNormSpace-Sequence holds the carrier of (product G) = product (carr G)
proof
let G be RealNormSpace-Sequence; ::_thesis: the carrier of (product G) = product (carr G)
product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
hence the carrier of (product G) = product (carr G) ; ::_thesis: verum
end;
theorem Th11: :: NDIFF_5:11
for G being RealNormSpace-Sequence
for i being Element of dom G
for r being set
for x being Function st r in the carrier of (G . i) & x in product (carr G) holds
x +* (i,r) in the carrier of (product G)
proof
let G be RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for r being set
for x being Function st r in the carrier of (G . i) & x in product (carr G) holds
x +* (i,r) in the carrier of (product G)
let i be Element of dom G; ::_thesis: for r being set
for x being Function st r in the carrier of (G . i) & x in product (carr G) holds
x +* (i,r) in the carrier of (product G)
let r be set ; ::_thesis: for x being Function st r in the carrier of (G . i) & x in product (carr G) holds
x +* (i,r) in the carrier of (product G)
let x be Function; ::_thesis: ( r in the carrier of (G . i) & x in product (carr G) implies x +* (i,r) in the carrier of (product G) )
assume A1: ( r in the carrier of (G . i) & x in product (carr G) ) ; ::_thesis: x +* (i,r) in the carrier of (product G)
then consider g being Function such that
A2: ( x = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
set h = x +* (i,r);
set s = i .--> r;
i .--> r = {i} --> r by FUNCOP_1:def_9;
then A3: dom (i .--> r) = {i} by FUNCOP_1:13;
A4: dom (x +* (i,r)) = dom (carr G) by A2, FUNCT_7:30;
for j being set st j in dom (carr G) holds
(x +* (i,r)) . j in (carr G) . j
proof
let j be set ; ::_thesis: ( j in dom (carr G) implies (x +* (i,r)) . j in (carr G) . j )
assume A5: j in dom (carr G) ; ::_thesis: (x +* (i,r)) . j in (carr G) . j
percases ( not j in dom (i .--> r) or j in dom (i .--> r) ) ;
suppose not j in dom (i .--> r) ; ::_thesis: (x +* (i,r)) . j in (carr G) . j
then j <> i by A3, TARSKI:def_1;
then (x +* (i,r)) . j = x . j by FUNCT_7:32;
hence (x +* (i,r)) . j in (carr G) . j by A2, A5; ::_thesis: verum
end;
suppose j in dom (i .--> r) ; ::_thesis: (x +* (i,r)) . j in (carr G) . j
then A6: j = i by TARSKI:def_1;
then (x +* (i,r)) . j = r by A5, A2, FUNCT_7:31;
hence (x +* (i,r)) . j in (carr G) . j by A1, A6, PRVECT_2:def_4; ::_thesis: verum
end;
end;
end;
then x +* (i,r) in product (carr G) by A4, CARD_3:def_5;
hence x +* (i,r) in the carrier of (product G) by Th10; ::_thesis: verum
end;
definition
let G be RealNormSpace-Sequence;
attrG is non-trivial means :Def1: :: NDIFF_5:def 1
for j being Element of dom G holds not G . j is trivial ;
end;
:: deftheorem Def1 defines non-trivial NDIFF_5:def_1_:_
for G being RealNormSpace-Sequence holds
( G is non-trivial iff for j being Element of dom G holds not G . j is trivial );
registration
cluster non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding non-trivial for set ;
correctness
existence
ex b1 being RealNormSpace-Sequence st b1 is non-trivial ;
proof
take G = <* the non trivial RealNormSpace*>; ::_thesis: G is non-trivial
let j be Element of dom G; :: according to NDIFF_5:def_1 ::_thesis: not G . j is trivial
dom G = Seg 1 by FINSEQ_1:38;
then j = 1 by FINSEQ_1:2, TARSKI:def_1;
hence not G . j is trivial by FINSEQ_1:40; ::_thesis: verum
end;
end;
registration
let G be non-trivial RealNormSpace-Sequence;
let i be Element of dom G;
clusterG . i -> non trivial for RealNormSpace;
correctness
coherence
for b1 being RealNormSpace st b1 = G . i holds
not b1 is trivial ;
by Def1;
end;
registration
let G be non-trivial RealNormSpace-Sequence;
cluster product G -> non trivial ;
correctness
coherence
not product G is trivial ;
proof
A1: the carrier of (product G) = product (carr G) by Th10;
ex x, y being set st
( x in product (carr G) & y in product (carr G) & not x = y )
proof
assume A2: for x, y being set st x in product (carr G) & y in product (carr G) holds
x = y ; ::_thesis: contradiction
consider z being set such that
A3: z in product (carr G) by XBOOLE_0:def_1;
consider g being Function such that
A4: ( z = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by A3, CARD_3:def_5;
set i = the Element of dom G;
now__::_thesis:_for_r,_s_being_set_st_r_in_the_carrier_of_(G_._the_Element_of_dom_G)_&_s_in_the_carrier_of_(G_._the_Element_of_dom_G)_holds_
r_=_s
let r, s be set ; ::_thesis: ( r in the carrier of (G . the Element of dom G) & s in the carrier of (G . the Element of dom G) implies r = s )
assume A5: ( r in the carrier of (G . the Element of dom G) & s in the carrier of (G . the Element of dom G) ) ; ::_thesis: r = s
( g +* ( the Element of dom G,r) in the carrier of (product G) & g +* ( the Element of dom G,s) in the carrier of (product G) ) by Th11, A3, A4, A5;
then ( g +* ( the Element of dom G,r) in product (carr G) & g +* ( the Element of dom G,s) in product (carr G) ) by Th10;
then A6: g +* ( the Element of dom G,r) = g +* ( the Element of dom G,s) by A2;
the Element of dom G in dom G ;
then A7: the Element of dom G in dom g by A4, Lm1;
then (g +* ( the Element of dom G,r)) . the Element of dom G = r by FUNCT_7:31;
hence r = s by A6, A7, FUNCT_7:31; ::_thesis: verum
end;
hence contradiction by ZFMISC_1:def_10; ::_thesis: verum
end;
hence not product G is trivial by A1, ZFMISC_1:def_10; ::_thesis: verum
end;
end;
theorem Th12: :: NDIFF_5:12
for G being RealNormSpace-Sequence
for p, q being Point of (product G)
for r0, p0, q0 being Element of product (carr G) st p = p0 & q = q0 holds
( p + q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i) )
proof
let G be RealNormSpace-Sequence; ::_thesis: for p, q being Point of (product G)
for r0, p0, q0 being Element of product (carr G) st p = p0 & q = q0 holds
( p + q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i) )
let p, q be Point of (product G); ::_thesis: for r0, p0, q0 being Element of product (carr G) st p = p0 & q = q0 holds
( p + q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i) )
let r0, p0, q0 be Element of product (carr G); ::_thesis: ( p = p0 & q = q0 implies ( p + q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i) ) )
assume A1: ( p = p0 & q = q0 ) ; ::_thesis: ( p + q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i) )
len (carr G) = len G by PRVECT_2:def_4;
then A2: dom (carr G) = Seg (len G) by FINSEQ_1:def_3
.= dom G by FINSEQ_1:def_3 ;
A3: product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
hereby ::_thesis: ( ( for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i) ) implies p + q = r0 )
assume A4: p + q = r0 ; ::_thesis: for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i)
hereby ::_thesis: verum
let i be Element of dom G; ::_thesis: r0 . i = (p0 . i) + (q0 . i)
reconsider i0 = i as Element of dom (carr G) by A2;
(addop G) . i0 = the addF of (G . i0) by PRVECT_2:def_5;
hence r0 . i = (p0 . i) + (q0 . i) by A1, A4, A3, PRVECT_1:def_8; ::_thesis: verum
end;
end;
assume A5: for i being Element of dom G holds r0 . i = (p0 . i) + (q0 . i) ; ::_thesis: p + q = r0
reconsider pq = p + q as Element of product (carr G) by Th10;
A6: ex g being Function st
( pq = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
A7: ex g being Function st
( r0 = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
now__::_thesis:_for_i0_being_set_st_i0_in_dom_pq_holds_
pq_._i0_=_r0_._i0
let i0 be set ; ::_thesis: ( i0 in dom pq implies pq . i0 = r0 . i0 )
assume A8: i0 in dom pq ; ::_thesis: pq . i0 = r0 . i0
then reconsider i1 = i0 as Element of dom G by A2, A6;
reconsider i = i0 as Element of dom (carr G) by A8, A6;
(addop G) . i = the addF of (G . i) by PRVECT_2:def_5;
then pq . i0 = (p0 . i1) + (q0 . i1) by A1, A3, PRVECT_1:def_8;
hence pq . i0 = r0 . i0 by A5; ::_thesis: verum
end;
hence p + q = r0 by A6, A7, FUNCT_1:2; ::_thesis: verum
end;
theorem Th13: :: NDIFF_5:13
for G being RealNormSpace-Sequence
for p being Point of (product G)
for r being Real
for r0, p0 being Element of product (carr G) st p = p0 holds
( r * p = r0 iff for i being Element of dom G holds r0 . i = r * (p0 . i) )
proof
let G be RealNormSpace-Sequence; ::_thesis: for p being Point of (product G)
for r being Real
for r0, p0 being Element of product (carr G) st p = p0 holds
( r * p = r0 iff for i being Element of dom G holds r0 . i = r * (p0 . i) )
let p be Point of (product G); ::_thesis: for r being Real
for r0, p0 being Element of product (carr G) st p = p0 holds
( r * p = r0 iff for i being Element of dom G holds r0 . i = r * (p0 . i) )
let r be Real; ::_thesis: for r0, p0 being Element of product (carr G) st p = p0 holds
( r * p = r0 iff for i being Element of dom G holds r0 . i = r * (p0 . i) )
let r0, p0 be Element of product (carr G); ::_thesis: ( p = p0 implies ( r * p = r0 iff for i being Element of dom G holds r0 . i = r * (p0 . i) ) )
assume A1: p = p0 ; ::_thesis: ( r * p = r0 iff for i being Element of dom G holds r0 . i = r * (p0 . i) )
hereby ::_thesis: ( ( for i being Element of dom G holds r0 . i = r * (p0 . i) ) implies r * p = r0 )
assume A2: r * p = r0 ; ::_thesis: for i being Element of dom G holds r0 . i = r * (p0 . i)
hereby ::_thesis: verum
let i be Element of dom G; ::_thesis: r0 . i = r * (p0 . i)
reconsider i0 = i as Element of dom (carr G) by Lm1;
A3: (multop G) . i0 = the Mult of (G . i0) by PRVECT_2:def_8;
product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
hence r0 . i = r * (p0 . i) by A1, A2, A3, PRVECT_2:def_2; ::_thesis: verum
end;
end;
assume A4: for i being Element of dom G holds r0 . i = r * (p0 . i) ; ::_thesis: r * p = r0
reconsider rp = r * p as Element of product (carr G) by Th10;
A5: ex g being Function st
( rp = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
A6: ex g being Function st
( r0 = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
now__::_thesis:_for_i0_being_set_st_i0_in_dom_rp_holds_
rp_._i0_=_r0_._i0
let i0 be set ; ::_thesis: ( i0 in dom rp implies rp . i0 = r0 . i0 )
assume A7: i0 in dom rp ; ::_thesis: rp . i0 = r0 . i0
then reconsider i1 = i0 as Element of dom G by Lm1, A5;
reconsider i = i0 as Element of dom (carr G) by A7, A5;
A8: product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
(multop G) . i = the Mult of (G . i) by PRVECT_2:def_8;
then rp . i0 = r * (p0 . i1) by A1, A8, PRVECT_2:def_2;
hence rp . i0 = r0 . i0 by A4; ::_thesis: verum
end;
hence r * p = r0 by A5, A6, FUNCT_1:2; ::_thesis: verum
end;
theorem Th14: :: NDIFF_5:14
for G being RealNormSpace-Sequence
for p0 being Element of product (carr G) holds
( 0. (product G) = p0 iff for i being Element of dom G holds p0 . i = 0. (G . i) )
proof
let G be RealNormSpace-Sequence; ::_thesis: for p0 being Element of product (carr G) holds
( 0. (product G) = p0 iff for i being Element of dom G holds p0 . i = 0. (G . i) )
let p0 be Element of product (carr G); ::_thesis: ( 0. (product G) = p0 iff for i being Element of dom G holds p0 . i = 0. (G . i) )
A1: dom (carr G) = dom G by Lm1;
A2: product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
hence ( 0. (product G) = p0 implies for i being Element of dom G holds p0 . i = 0. (G . i) ) by A1, PRVECT_2:def_7; ::_thesis: ( ( for i being Element of dom G holds p0 . i = 0. (G . i) ) implies 0. (product G) = p0 )
assume A3: for i being Element of dom G holds p0 . i = 0. (G . i) ; ::_thesis: 0. (product G) = p0
now__::_thesis:_for_i0_being_Element_of_dom_(carr_G)_holds_p0_._i0_=_the_ZeroF_of_(G_._i0)
let i0 be Element of dom (carr G); ::_thesis: p0 . i0 = the ZeroF of (G . i0)
reconsider i = i0 as Element of dom G by Lm1;
p0 . i0 = 0. (G . i) by A3;
hence p0 . i0 = the ZeroF of (G . i0) ; ::_thesis: verum
end;
hence 0. (product G) = p0 by A2, PRVECT_2:def_7; ::_thesis: verum
end;
theorem Th15: :: NDIFF_5:15
for G being RealNormSpace-Sequence
for p, q being Point of (product G)
for r0, p0, q0 being Element of product (carr G) st p = p0 & q = q0 holds
( p - q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) )
proof
let G be RealNormSpace-Sequence; ::_thesis: for p, q being Point of (product G)
for r0, p0, q0 being Element of product (carr G) st p = p0 & q = q0 holds
( p - q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) )
let p, q be Point of (product G); ::_thesis: for r0, p0, q0 being Element of product (carr G) st p = p0 & q = q0 holds
( p - q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) )
let r0, p0, q0 be Element of product (carr G); ::_thesis: ( p = p0 & q = q0 implies ( p - q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) ) )
assume A1: ( p = p0 & q = q0 ) ; ::_thesis: ( p - q = r0 iff for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) )
reconsider qq0 = (- 1) * q as Element of product (carr G) by Th10;
A2: p - q = p + ((- 1) * q) by RLVECT_1:16;
hereby ::_thesis: ( ( for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) ) implies p - q = r0 )
assume A3: p - q = r0 ; ::_thesis: for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i)
thus for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) ::_thesis: verum
proof
let i be Element of dom G; ::_thesis: r0 . i = (p0 . i) - (q0 . i)
A4: r0 . i = (p0 . i) + (qq0 . i) by Th12, A3, A1, A2;
- 1 is Real by XREAL_0:def_1;
then qq0 . i = (- 1) * (q0 . i) by A1, Th13;
hence r0 . i = (p0 . i) - (q0 . i) by A4, RLVECT_1:16; ::_thesis: verum
end;
end;
assume A5: for i being Element of dom G holds r0 . i = (p0 . i) - (q0 . i) ; ::_thesis: p - q = r0
now__::_thesis:_for_i_being_Element_of_dom_G_holds_r0_._i_=_(p0_._i)_+_(qq0_._i)
let i be Element of dom G; ::_thesis: r0 . i = (p0 . i) + (qq0 . i)
- 1 is Real by XREAL_0:def_1;
then A6: qq0 . i = (- 1) * (q0 . i) by A1, Th13;
r0 . i = (p0 . i) - (q0 . i) by A5;
hence r0 . i = (p0 . i) + (qq0 . i) by A6, RLVECT_1:16; ::_thesis: verum
end;
hence p - q = r0 by A2, Th12, A1; ::_thesis: verum
end;
begin
definition
let S be RealLinearSpace;
let p, q be Point of S;
func].p,q.[ -> Subset of S equals :: NDIFF_5:def 2
{ (p + (t * (q - p))) where t is Real : ( 0 < t & t < 1 ) } ;
correctness
coherence
{ (p + (t * (q - p))) where t is Real : ( 0 < t & t < 1 ) } is Subset of S;
proof
now__::_thesis:_for_x_being_set_st_x_in__{__(p_+_(t_*_(q_-_p)))_where_t_is_Real_:_(_0_<_t_&_t_<_1_)__}__holds_
x_in_the_carrier_of_S
let x be set ; ::_thesis: ( x in { (p + (t * (q - p))) where t is Real : ( 0 < t & t < 1 ) } implies x in the carrier of S )
assume x in { (p + (t * (q - p))) where t is Real : ( 0 < t & t < 1 ) } ; ::_thesis: x in the carrier of S
then ex t being Real st
( x = p + (t * (q - p)) & 0 < t & t < 1 ) ;
hence x in the carrier of S ; ::_thesis: verum
end;
hence { (p + (t * (q - p))) where t is Real : ( 0 < t & t < 1 ) } is Subset of S by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines ]. NDIFF_5:def_2_:_
for S being RealLinearSpace
for p, q being Point of S holds ].p,q.[ = { (p + (t * (q - p))) where t is Real : ( 0 < t & t < 1 ) } ;
notation
let S be RealLinearSpace;
let p, q be Point of S;
synonym [.p,q.] for LSeg (p,q);
end;
Lm2: now__::_thesis:_for_S_being_RealLinearSpace
for_p,_q_being_Point_of_S
for_z1_being_Real_holds_p_+_(z1_*_(q_-_p))_=_((1_-_z1)_*_p)_+_(z1_*_q)
let S be RealLinearSpace; ::_thesis: for p, q being Point of S
for z1 being Real holds p + (z1 * (q - p)) = ((1 - z1) * p) + (z1 * q)
let p, q be Point of S; ::_thesis: for z1 being Real holds p + (z1 * (q - p)) = ((1 - z1) * p) + (z1 * q)
let z1 be Real; ::_thesis: p + (z1 * (q - p)) = ((1 - z1) * p) + (z1 * q)
thus p + (z1 * (q - p)) = p + ((z1 * q) + (z1 * (- p))) by RLVECT_1:def_5
.= p + ((z1 * q) + (- (z1 * p))) by RLVECT_1:25
.= (p + (- (z1 * p))) + (z1 * q) by RLVECT_1:def_3
.= ((1 * p) - (z1 * p)) + (z1 * q) by RLVECT_1:def_8
.= ((1 - z1) * p) + (z1 * q) by RLVECT_1:35 ; ::_thesis: verum
end;
theorem Th16: :: NDIFF_5:16
for S being RealLinearSpace
for p, q being Point of S holds ].p,q.[ c= [.p,q.]
proof
let S be RealLinearSpace; ::_thesis: for p, q being Point of S holds ].p,q.[ c= [.p,q.]
let p, q be Point of S; ::_thesis: ].p,q.[ c= [.p,q.]
now__::_thesis:_for_z_being_set_st_z_in_].p,q.[_holds_
z_in_[.p,q.]
let z be set ; ::_thesis: ( z in ].p,q.[ implies z in [.p,q.] )
assume z in ].p,q.[ ; ::_thesis: z in [.p,q.]
then consider z1 being Real such that
A1: ( z = p + (z1 * (q - p)) & 0 < z1 & z1 < 1 ) ;
z = ((1 - z1) * p) + (z1 * q) by A1, Lm2;
then z in { (((1 - r) * p) + (r * q)) where r is Real : ( 0 <= r & r <= 1 ) } by A1;
hence z in [.p,q.] by RLTOPSP1:def_2; ::_thesis: verum
end;
hence ].p,q.[ c= [.p,q.] by TARSKI:def_3; ::_thesis: verum
end;
Lm3: for x being Real st ( for e being Real st 0 < e holds
x <= e ) holds
x <= 0
proof
let x be Real; ::_thesis: ( ( for e being Real st 0 < e holds
x <= e ) implies x <= 0 )
assume A1: for e being Real st 0 < e holds
x <= e ; ::_thesis: x <= 0
assume A2: not x <= 0 ; ::_thesis: contradiction
then x <= x / 2 by A1;
then x - (x / 2) <= (x / 2) - (x / 2) by XREAL_1:9;
hence contradiction by A2; ::_thesis: verum
end;
theorem Th17: :: NDIFF_5:17
for T being non trivial RealNormSpace
for R being PartFunc of REAL,T st R is total holds
( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) )
proof
let T be non trivial RealNormSpace; ::_thesis: for R being PartFunc of REAL,T st R is total holds
( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) )
let R be PartFunc of REAL,T; ::_thesis: ( R is total implies ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) ) )
assume A1: R is total ; ::_thesis: ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) )
A2: now__::_thesis:_(_R_is_RestFunc-like_&_ex_r_being_Real_st_
(_r_>_0_&_(_for_d_being_Real_holds_
(_not_d_>_0_or_ex_z_being_Real_st_
(_z_<>_0_&_abs_z_<_d_&_not_||.(R_/._z).||_/_(abs_z)_<_r_)_)_)_)_implies_for_r_being_Real_st_r_>_0_holds_
ex_d_being_Real_st_
(_d_>_0_&_(_for_z_being_Real_st_z_<>_0_&_abs_z_<_d_holds_
||.(R_/._z).||_/_(abs_z)_<_r_)_)_)
assume A3: R is RestFunc-like ; ::_thesis: ( ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & abs z < d & not ||.(R /. z).|| / (abs z) < r ) ) ) ) implies for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) )
assume ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & abs z < d & not ||.(R /. z).|| / (abs z) < r ) ) ) ) ; ::_thesis: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) )
then consider r being Real such that
A4: r > 0 and
A5: for d being Real st d > 0 holds
ex z being Real st
( z <> 0 & abs z < d & not ||.(R /. z).|| / (abs z) < r ) ;
defpred S1[ Element of NAT , Element of REAL ] means ( $2 <> 0 & abs $2 < 1 / ($1 + 1) & not ||.(R /. $2).|| / (abs $2) < r );
A6: now__::_thesis:_for_n_being_Element_of_NAT_ex_z_being_Element_of_REAL_st_S1[n,z]
let n be Element of NAT ; ::_thesis: ex z being Element of REAL st S1[n,z]
1 / (n + 1) is Real by XREAL_0:def_1;
hence ex z being Element of REAL st S1[n,z] by A5; ::_thesis: verum
end;
consider s being Real_Sequence such that
A7: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6);
A8: now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((s_._m)_-_0)_<_p
let p be real number ; ::_thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < p )
assume A9: 0 < p ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < p
consider n being Element of NAT such that
A10: p " < n by SEQ_4:3;
(p ") + 0 < n + 1 by A10, XREAL_1:8;
then A11: 1 / (n + 1) < 1 / (p ") by A9, XREAL_1:76;
take n = n; ::_thesis: for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < p
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((s . m) - 0) < p )
assume n <= m ; ::_thesis: abs ((s . m) - 0) < p
then n + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (n + 1) by XREAL_1:118;
then abs ((s . m) - 0) < 1 / (n + 1) by A7, XXREAL_0:2;
hence abs ((s . m) - 0) < p by A11, XXREAL_0:2; ::_thesis: verum
end;
then A12: s is convergent by SEQ_2:def_6;
then A13: lim s = 0 by A8, SEQ_2:def_7;
s is non-zero by A7, SEQ_1:5;
then reconsider s = s as non-zero 0 -convergent Real_Sequence by A12, A13, FDIFF_1:def_1;
( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0. T ) by A3, NDIFF_3:def_1;
then consider n being Element of NAT such that
A14: for m being Element of NAT st n <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r by A4, NORMSP_1:def_7;
A15: ||.((((s ") (#) (R /* s)) . n) - (0. T)).|| < r by A14;
A16: ||.(((s . n) ") * (R /. (s . n))).|| = (abs ((s . n) ")) * ||.(R /. (s . n)).|| by NORMSP_1:def_1
.= ||.(R /. (s . n)).|| / (abs (s . n)) by COMPLEX1:66 ;
dom R = REAL by A1, PARTFUN1:def_2;
then A17: rng s c= dom R ;
||.((((s ") (#) (R /* s)) . n) - (0. T)).|| = ||.(((s ") (#) (R /* s)) . n).|| by RLVECT_1:13
.= ||.(((s ") . n) * ((R /* s) . n)).|| by NDIFF_1:def_2
.= ||.(((s . n) ") * ((R /* s) . n)).|| by VALUED_1:10
.= ||.(((s . n) ") * (R /. (s . n))).|| by A17, FUNCT_2:109 ;
hence for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) by A7, A15, A16; ::_thesis: verum
end;
now__::_thesis:_(_(_for_r_being_Real_st_r_>_0_holds_
ex_d_being_Real_st_
(_d_>_0_&_(_for_z_being_Real_st_z_<>_0_&_abs_z_<_d_holds_
||.(R_/._z).||_/_(abs_z)_<_r_)_)_)_implies_R_is_RestFunc-like_)
assume A18: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) ; ::_thesis: R is RestFunc-like
now__::_thesis:_for_s_being_non-zero_0_-convergent_Real_Sequence_holds_
(_(s_")_(#)_(R_/*_s)_is_convergent_&_lim_((s_")_(#)_(R_/*_s))_=_0._T_)
let s be non-zero 0 -convergent Real_Sequence; ::_thesis: ( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0. T )
A19: ( s is convergent & lim s = 0 ) ;
A20: now__::_thesis:_for_r_being_Real_st_r_>_0_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
||.((((s_")_(#)_(R_/*_s))_._m)_-_(0._T)).||_<_r
let r be Real; ::_thesis: ( r > 0 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r )
assume r > 0 ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r
then consider d being Real such that
A21: d > 0 and
A22: for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r by A18;
consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < d by A19, A21, SEQ_2:def_7;
take n = n; ::_thesis: for m being Element of NAT st n <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r
thus for m being Element of NAT st n <= m holds
||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r ::_thesis: verum
proof
dom R = REAL by A1, PARTFUN1:def_2;
then A24: rng s c= dom R ;
let m be Element of NAT ; ::_thesis: ( n <= m implies ||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r )
assume n <= m ; ::_thesis: ||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r
then A25: abs ((s . m) - 0) < d by A23;
A26: s . m <> 0 by SEQ_1:5;
||.(R /. (s . m)).|| / (abs (s . m)) = (abs ((s . m) ")) * ||.(R /. (s . m)).|| by COMPLEX1:66
.= ||.(((s . m) ") * (R /. (s . m))).|| by NORMSP_1:def_1
.= ||.(((s . m) ") * ((R /* s) . m)).|| by A24, FUNCT_2:109
.= ||.(((s ") . m) * ((R /* s) . m)).|| by VALUED_1:10
.= ||.(((s ") (#) (R /* s)) . m).|| by NDIFF_1:def_2
.= ||.((((s ") (#) (R /* s)) . m) - (0. T)).|| by RLVECT_1:13 ;
hence ||.((((s ") (#) (R /* s)) . m) - (0. T)).|| < r by A22, A25, A26; ::_thesis: verum
end;
end;
hence (s ") (#) (R /* s) is convergent by NORMSP_1:def_6; ::_thesis: lim ((s ") (#) (R /* s)) = 0. T
hence lim ((s ") (#) (R /* s)) = 0. T by A20, NORMSP_1:def_7; ::_thesis: verum
end;
hence R is RestFunc-like by A1, NDIFF_3:def_1; ::_thesis: verum
end;
hence ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
||.(R /. z).|| / (abs z) < r ) ) ) by A2; ::_thesis: verum
end;
theorem Th18: :: NDIFF_5:18
for R being Function of REAL,REAL holds
( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r ) ) )
proof
let R be Function of REAL,REAL; ::_thesis: ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r ) ) )
A1: now__::_thesis:_(_R_is_RestFunc-like_&_ex_r_being_Real_st_
(_r_>_0_&_(_for_d_being_Real_holds_
(_not_d_>_0_or_ex_z_being_Real_st_
(_z_<>_0_&_abs_z_<_d_&_not_(abs_(R_._z))_/_(abs_z)_<_r_)_)_)_)_implies_for_r_being_Real_st_r_>_0_holds_
ex_d_being_Real_st_
(_d_>_0_&_(_for_z_being_Real_st_z_<>_0_&_abs_z_<_d_holds_
(abs_(R_._z))_/_(abs_z)_<_r_)_)_)
assume A2: R is RestFunc-like ; ::_thesis: ( ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & abs z < d & not (abs (R . z)) / (abs z) < r ) ) ) ) implies for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r ) ) )
assume ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & abs z < d & not (abs (R . z)) / (abs z) < r ) ) ) ) ; ::_thesis: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r ) )
then consider r being Real such that
A3: r > 0 and
A4: for d being Real st d > 0 holds
ex z being Real st
( z <> 0 & abs z < d & not (abs (R . z)) / (abs z) < r ) ;
defpred S1[ Element of NAT , Element of REAL ] means ( $2 <> 0 & abs $2 < 1 / ($1 + 1) & not (abs (R . $2)) / (abs $2) < r );
A5: now__::_thesis:_for_n_being_Element_of_NAT_ex_z_being_Element_of_REAL_st_S1[n,z]
let n be Element of NAT ; ::_thesis: ex z being Element of REAL st S1[n,z]
1 / (n + 1) is Real by XREAL_0:def_1;
hence ex z being Element of REAL st S1[n,z] by A4; ::_thesis: verum
end;
consider s being Real_Sequence such that
A6: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A5);
A7: now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((s_._m)_-_0)_<_p
let p be real number ; ::_thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < p )
assume A8: 0 < p ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < p
consider n being Element of NAT such that
A9: p " < n by SEQ_4:3;
(p ") + 0 < n + 1 by A9, XREAL_1:8;
then A10: 1 / (n + 1) < 1 / (p ") by A8, XREAL_1:76;
take n = n; ::_thesis: for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < p
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((s . m) - 0) < p )
assume n <= m ; ::_thesis: abs ((s . m) - 0) < p
then n + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (n + 1) by XREAL_1:118;
then abs ((s . m) - 0) < 1 / (n + 1) by A6, XXREAL_0:2;
hence abs ((s . m) - 0) < p by A10, XXREAL_0:2; ::_thesis: verum
end;
then A11: s is convergent by SEQ_2:def_6;
then A12: lim s = 0 by A7, SEQ_2:def_7;
s is non-zero by A6, SEQ_1:5;
then reconsider s = s as non-zero 0 -convergent Real_Sequence by A11, A12, FDIFF_1:def_1;
( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0 ) by A2, FDIFF_1:def_2;
then consider n being Element of NAT such that
A13: for m being Element of NAT st n <= m holds
abs ((((s ") (#) (R /* s)) . m) - 0) < r by A3, SEQ_2:def_7;
A14: abs ((((s ") (#) (R /* s)) . n) - 0) < r by A13;
A15: abs (((s . n) ") * (R . (s . n))) = (abs ((s . n) ")) * (abs (R . (s . n))) by COMPLEX1:65
.= (abs (R . (s . n))) / (abs (s . n)) by COMPLEX1:66 ;
abs ((((s ") (#) (R /* s)) . n) - 0) = abs (((s ") . n) * ((R /* s) . n)) by SEQ_1:8
.= abs (((s . n) ") * ((R /* s) . n)) by VALUED_1:10
.= abs (((s . n) ") * (R . (s . n))) by FUNCT_2:115 ;
hence for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r ) ) by A6, A14, A15; ::_thesis: verum
end;
now__::_thesis:_(_(_for_r_being_Real_st_r_>_0_holds_
ex_d_being_Real_st_
(_d_>_0_&_(_for_z_being_Real_st_z_<>_0_&_abs_z_<_d_holds_
(abs_(R_._z))_/_(abs_z)_<_r_)_)_)_implies_R_is_RestFunc-like_)
assume A16: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r ) ) ; ::_thesis: R is RestFunc-like
now__::_thesis:_for_s_being_non-zero_0_-convergent_Real_Sequence_holds_
(_(s_")_(#)_(R_/*_s)_is_convergent_&_lim_((s_")_(#)_(R_/*_s))_=_0_)
let s be non-zero 0 -convergent Real_Sequence; ::_thesis: ( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0 )
A17: ( s is convergent & lim s = 0 ) ;
A18: now__::_thesis:_for_r_being_real_number_st_r_>_0_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((((s_")_(#)_(R_/*_s))_._m)_-_0)_<_r
let r be real number ; ::_thesis: ( r > 0 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((s ") (#) (R /* s)) . m) - 0) < r )
assume A19: r > 0 ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((s ") (#) (R /* s)) . m) - 0) < r
r is Real by XREAL_0:def_1;
then consider d being Real such that
A20: d > 0 and
A21: for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r by A19, A16;
consider n being Element of NAT such that
A22: for m being Element of NAT st n <= m holds
abs ((s . m) - 0) < d by A17, A20, SEQ_2:def_7;
take n = n; ::_thesis: for m being Element of NAT st n <= m holds
abs ((((s ") (#) (R /* s)) . m) - 0) < r
hereby ::_thesis: verum
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((((s ") (#) (R /* s)) . m) - 0) < r )
assume n <= m ; ::_thesis: abs ((((s ") (#) (R /* s)) . m) - 0) < r
then A23: abs ((s . m) - 0) < d by A22;
A24: s . m <> 0 by SEQ_1:5;
(abs (R . (s . m))) / (abs (s . m)) = (abs ((s . m) ")) * (abs (R . (s . m))) by COMPLEX1:66
.= abs (((s . m) ") * (R . (s . m))) by COMPLEX1:65
.= abs (((s . m) ") * ((R /* s) . m)) by FUNCT_2:115
.= abs (((s ") . m) * ((R /* s) . m)) by VALUED_1:10
.= abs ((((s ") (#) (R /* s)) . m) - 0) by SEQ_1:8 ;
hence abs ((((s ") (#) (R /* s)) . m) - 0) < r by A21, A23, A24; ::_thesis: verum
end;
end;
hence (s ") (#) (R /* s) is convergent by SEQ_2:def_6; ::_thesis: lim ((s ") (#) (R /* s)) = 0
hence lim ((s ") (#) (R /* s)) = 0 by A18, SEQ_2:def_7; ::_thesis: verum
end;
hence R is RestFunc-like by FDIFF_1:def_2; ::_thesis: verum
end;
hence ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & abs z < d holds
(abs (R . z)) / (abs z) < r ) ) ) by A1; ::_thesis: verum
end;
Lm4: for T being non trivial RealNormSpace
for f being PartFunc of REAL,T
for g being PartFunc of REAL,REAL st dom f = [.0,1.] & dom g = [.0,1.] & f | [.0,1.] is continuous & g | [.0,1.] is continuous & f is_differentiable_on ].0,1.[ & g is_differentiable_on ].0,1.[ & ( for x being real number st x in ].0,1.[ holds
||.(diff (f,x)).|| <= diff (g,x) ) holds
||.((f /. 1) - (f /. 0)).|| <= (g /. 1) - (g /. 0)
proof
let T be non trivial RealNormSpace; ::_thesis: for f being PartFunc of REAL,T
for g being PartFunc of REAL,REAL st dom f = [.0,1.] & dom g = [.0,1.] & f | [.0,1.] is continuous & g | [.0,1.] is continuous & f is_differentiable_on ].0,1.[ & g is_differentiable_on ].0,1.[ & ( for x being real number st x in ].0,1.[ holds
||.(diff (f,x)).|| <= diff (g,x) ) holds
||.((f /. 1) - (f /. 0)).|| <= (g /. 1) - (g /. 0)
let f be PartFunc of REAL,T; ::_thesis: for g being PartFunc of REAL,REAL st dom f = [.0,1.] & dom g = [.0,1.] & f | [.0,1.] is continuous & g | [.0,1.] is continuous & f is_differentiable_on ].0,1.[ & g is_differentiable_on ].0,1.[ & ( for x being real number st x in ].0,1.[ holds
||.(diff (f,x)).|| <= diff (g,x) ) holds
||.((f /. 1) - (f /. 0)).|| <= (g /. 1) - (g /. 0)
let g be PartFunc of REAL,REAL; ::_thesis: ( dom f = [.0,1.] & dom g = [.0,1.] & f | [.0,1.] is continuous & g | [.0,1.] is continuous & f is_differentiable_on ].0,1.[ & g is_differentiable_on ].0,1.[ & ( for x being real number st x in ].0,1.[ holds
||.(diff (f,x)).|| <= diff (g,x) ) implies ||.((f /. 1) - (f /. 0)).|| <= (g /. 1) - (g /. 0) )
assume A1: ( dom f = [.0,1.] & dom g = [.0,1.] & f | [.0,1.] is continuous & g | [.0,1.] is continuous & f is_differentiable_on ].0,1.[ & g is_differentiable_on ].0,1.[ & ( for x being real number st x in ].0,1.[ holds
||.(diff (f,x)).|| <= diff (g,x) ) ) ; ::_thesis: ||.((f /. 1) - (f /. 0)).|| <= (g /. 1) - (g /. 0)
now__::_thesis:_for_e_being_Real_st_0_<_e_holds_
||.((f_/._1)_-_(f_/._0)).||_-_((g_/._1)_-_(g_/._0))_<=_e
let e be Real; ::_thesis: ( 0 < e implies ||.((f /. 1) - (f /. 0)).|| - ((g /. 1) - (g /. 0)) <= e )
assume A2: 0 < e ; ::_thesis: ||.((f /. 1) - (f /. 0)).|| - ((g /. 1) - (g /. 0)) <= e
set e1 = e / 2;
set B = { x where x is Real : ( x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) } ;
now__::_thesis:_for_z_being_set_st_z_in__{__x_where_x_is_Real_:_(_x_in_[.0,1.]_&_((||.((f_/._x)_-_(f_/._0)).||_-_((g_._x)_-_(g_._0)))_-_((e_/_2)_*_x))_-_(e_/_2)_<=_0_)__}__holds_
z_in_REAL
let z be set ; ::_thesis: ( z in { x where x is Real : ( x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) } implies z in REAL )
assume z in { x where x is Real : ( x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) } ; ::_thesis: z in REAL
then ex x being Real st
( z = x & x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) ;
hence z in REAL ; ::_thesis: verum
end;
then reconsider B = { x where x is Real : ( x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) } as Subset of REAL by TARSKI:def_3;
now__::_thesis:_for_r_being_real_number_st_r_in_B_holds_
abs_r_<_2
let r be real number ; ::_thesis: ( r in B implies abs r < 2 )
assume r in B ; ::_thesis: abs r < 2
then ex x being Real st
( r = x & x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) ;
then A3: ex t being Real st
( r = t & 0 <= t & t <= 1 ) ;
then abs r = r by ABSVALUE:def_1;
hence abs r < 2 by A3, XXREAL_0:2; ::_thesis: verum
end;
then A4: B is real-bounded by SEQ_4:4;
set s = upper_bound B;
A5: ex d being real number st
( 0 < d & d in B )
proof
0 in [.0,1.] ;
then consider d1 being real number such that
A6: ( 0 < d1 & ( for x1 being real number st x1 in [.0,1.] & abs (x1 - 0) < d1 holds
||.((f /. x1) - (f /. 0)).|| < e / 2 ) ) by A2, A1, NFCONT_3:17;
set d2 = d1 / 2;
A7: d1 / 2 < d1 by A6, XREAL_1:216;
take d = min ((d1 / 2),1); ::_thesis: ( 0 < d & d in B )
thus A8: 0 < d by A6, XXREAL_0:21; ::_thesis: d in B
A9: d <= 1 by XXREAL_0:17;
then A10: d in [.0,1.] by A8;
A11: d <= d1 / 2 by XXREAL_0:17;
abs (d - 0) = d by A8, ABSVALUE:def_1;
then abs (d - 0) < d1 by A11, A7, XXREAL_0:2;
then A12: ||.((f /. d) - (f /. 0)).|| < e / 2 by A6, A10;
A13: [.0,d.] c= dom g by A1, A9, XXREAL_1:34;
A14: g | [.0,d.] is continuous by A1, A9, FCONT_1:16, XXREAL_1:34;
A15: ].0,d.[ c= ].0,1.[ by A9, XXREAL_1:46;
then consider x0 being Real such that
A16: ( x0 in ].0,d.[ & diff (g,x0) = ((g . d) - (g . 0)) / (d - 0) ) by A1, A8, A13, A14, FDIFF_1:26, ROLLE:3;
||.(diff (f,x0)).|| <= diff (g,x0) by A1, A16, A15;
then 0 <= (g . d) - (g . 0) by A8, A16;
then 0 + ||.((f /. d) - (f /. 0)).|| <= ((g . d) - (g . 0)) + (e / 2) by A12, XREAL_1:7;
then 0 + ||.((f /. d) - (f /. 0)).|| <= (((g . d) - (g . 0)) + (e / 2)) + ((e / 2) * d) by A8, A2, XREAL_1:7;
then ||.((f /. d) - (f /. 0)).|| - ((((g . d) - (g . 0)) + (e / 2)) + ((e / 2) * d)) <= 0 by XREAL_1:47;
then ((||.((f /. d) - (f /. 0)).|| - ((g . d) - (g . 0))) - ((e / 2) * d)) - (e / 2) <= 0 ;
hence d in B by A10; ::_thesis: verum
end;
then A17: 0 < upper_bound B by A4, SEQ_4:def_1;
now__::_thesis:_for_r_being_real_number_st_r_in_B_holds_
r_<=_1
let r be real number ; ::_thesis: ( r in B implies r <= 1 )
assume r in B ; ::_thesis: r <= 1
then ex x being Real st
( r = x & x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) ;
then ex t being Real st
( r = t & 0 <= t & t <= 1 ) ;
hence r <= 1 ; ::_thesis: verum
end;
then A18: upper_bound B <= 1 by A5, SEQ_4:45;
defpred S1[ Element of NAT , Element of REAL ] means ( $2 in B & abs ((upper_bound B) - $2) <= 1 / ($1 + 1) );
A19: now__::_thesis:_for_x_being_Element_of_NAT_ex_r_being_Element_of_REAL_st_S1[x,r]
let x be Element of NAT ; ::_thesis: ex r being Element of REAL st S1[x,r]
reconsider t = 1 / (1 + x) as real number ;
consider r being real number such that
A20: ( r in B & (upper_bound B) - t < r ) by A4, A5, SEQ_4:def_1;
reconsider r = r as Element of REAL by XREAL_0:def_1;
take r = r; ::_thesis: S1[x,r]
((upper_bound B) - t) + t < r + t by A20, XREAL_1:8;
then A21: (upper_bound B) - r < (t + r) - r by XREAL_1:14;
r <= upper_bound B by A4, A20, SEQ_4:def_1;
then 0 <= (upper_bound B) - r by XREAL_1:48;
hence S1[x,r] by A20, A21, SEQ_2:1; ::_thesis: verum
end;
consider sq being Function of NAT,REAL such that
A22: for x being Element of NAT holds S1[x,sq . x] from FUNCT_2:sch_3(A19);
reconsider sq = sq as Real_Sequence ;
A23: now__::_thesis:_for_p1_being_real_number_st_0_<_p1_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((sq_._m)_-_(upper_bound_B))_<_p1
let p1 be real number ; ::_thesis: ( 0 < p1 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((sq . m) - (upper_bound B)) < p1 )
assume A24: 0 < p1 ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((sq . m) - (upper_bound B)) < p1
set p = p1 / 2;
consider n being Element of NAT such that
A25: 1 / (p1 / 2) < n by SEQ_4:3;
(1 / (p1 / 2)) + 0 < n + 1 by A25, XREAL_1:8;
then A26: 1 / (n + 1) <= 1 / (1 / (p1 / 2)) by A24, XREAL_1:118;
take n = n; ::_thesis: for m being Element of NAT st n <= m holds
abs ((sq . m) - (upper_bound B)) < p1
thus for m being Element of NAT st n <= m holds
abs ((sq . m) - (upper_bound B)) < p1 ::_thesis: verum
proof
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((sq . m) - (upper_bound B)) < p1 )
assume n <= m ; ::_thesis: abs ((sq . m) - (upper_bound B)) < p1
then ( 0 < n + 1 & n + 1 <= m + 1 ) by XREAL_1:6;
then 1 / (m + 1) <= 1 / (n + 1) by XREAL_1:118;
then A27: 1 / (m + 1) <= p1 / 2 by A26, XXREAL_0:2;
( sq . m in B & abs ((upper_bound B) - (sq . m)) <= 1 / (m + 1) ) by A22;
then abs ((sq . m) - (upper_bound B)) <= 1 / (1 + m) by COMPLEX1:60;
then A28: abs ((sq . m) - (upper_bound B)) <= p1 / 2 by A27, XXREAL_0:2;
p1 / 2 < p1 by A24, XREAL_1:216;
hence abs ((sq . m) - (upper_bound B)) < p1 by A28, XXREAL_0:2; ::_thesis: verum
end;
end;
then A29: sq is convergent by SEQ_2:def_6;
then A30: lim sq = upper_bound B by A23, SEQ_2:def_7;
deffunc H1( Real) -> Element of REAL = ((||.((f /. $1) - (f /. 0)).|| - ((g . $1) - (g . 0))) - ((e / 2) * $1)) - (e / 2);
A31: for x being Element of REAL holds H1(x) in REAL ;
consider Lg0 being Function of REAL,REAL such that
A32: for x being Element of REAL holds Lg0 . x = H1(x) from FUNCT_2:sch_8(A31);
set Lg = Lg0 | [.0,1.];
A33: dom Lg0 = REAL by FUNCT_2:def_1;
then A34: dom (Lg0 | [.0,1.]) = [.0,1.] by RELAT_1:62;
now__::_thesis:_for_y_being_set_st_y_in_rng_sq_holds_
y_in_[.0,1.]
let y be set ; ::_thesis: ( y in rng sq implies y in [.0,1.] )
assume y in rng sq ; ::_thesis: y in [.0,1.]
then ex x being set st
( x in NAT & sq . x = y ) by FUNCT_2:11;
then y in B by A22;
then ex x being Real st
( y = x & x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) ;
hence y in [.0,1.] ; ::_thesis: verum
end;
then A35: rng sq c= dom (Lg0 | [.0,1.]) by A34, TARSKI:def_3;
A36: upper_bound B in [.0,1.] by A18, A17;
now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_
ex_t_being_set_st_
(_0_<_t_&_(_for_x1_being_real_number_st_x1_in_dom_(Lg0_|_[.0,1.])_&_abs_(x1_-_(upper_bound_B))_<_t_holds_
abs_(((Lg0_|_[.0,1.])_._x1)_-_((Lg0_|_[.0,1.])_._(upper_bound_B)))_<_r_)_)
let r be real number ; ::_thesis: ( 0 < r implies ex t being set st
( 0 < t & ( for x1 being real number st x1 in dom (Lg0 | [.0,1.]) & abs (x1 - (upper_bound B)) < t holds
abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r ) ) )
set r3 = r / 3;
assume A37: 0 < r ; ::_thesis: ex t being set st
( 0 < t & ( for x1 being real number st x1 in dom (Lg0 | [.0,1.]) & abs (x1 - (upper_bound B)) < t holds
abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r ) )
then consider t1 being real number such that
A38: ( 0 < t1 & ( for x1 being real number st x1 in [.0,1.] & abs (x1 - (upper_bound B)) < t1 holds
||.((f /. x1) - (f /. (upper_bound B))).|| < r / 3 ) ) by A1, A36, NFCONT_3:17;
consider t2 being real number such that
A39: ( 0 < t2 & ( for x1 being real number st x1 in [.0,1.] & abs (x1 - (upper_bound B)) < t2 holds
abs ((g . x1) - (g . (upper_bound B))) < r / 3 ) ) by A37, A36, A1, FCONT_1:14;
set t30 = (r / 3) / (e / 2);
set t3 = ((r / 3) / (e / 2)) / 2;
( 0 < ((r / 3) / (e / 2)) / 2 & ((r / 3) / (e / 2)) / 2 < (r / 3) / (e / 2) ) by A2, A37, XREAL_1:216;
then (e / 2) * (((r / 3) / (e / 2)) / 2) < ((r / 3) / (e / 2)) * (e / 2) by A2, XREAL_1:97;
then A40: (e / 2) * (((r / 3) / (e / 2)) / 2) < r / 3 by A2, XCMPLX_1:87;
take t = min ((min (t1,t2)),(((r / 3) / (e / 2)) / 2)); ::_thesis: ( 0 < t & ( for x1 being real number st x1 in dom (Lg0 | [.0,1.]) & abs (x1 - (upper_bound B)) < t holds
abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r ) )
A41: ( min (t1,t2) <= t1 & min (t1,t2) <= t2 & 0 < min (t1,t2) ) by A38, A39, XXREAL_0:17, XXREAL_0:21;
hence 0 < t by A2, A37, XXREAL_0:21; ::_thesis: for x1 being real number st x1 in dom (Lg0 | [.0,1.]) & abs (x1 - (upper_bound B)) < t holds
abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r
A42: t <= ((r / 3) / (e / 2)) / 2 by XXREAL_0:17;
A43: t <= min (t1,t2) by XXREAL_0:17;
then A44: t <= t1 by A41, XXREAL_0:2;
A45: t <= t2 by A41, A43, XXREAL_0:2;
thus for x1 being real number st x1 in dom (Lg0 | [.0,1.]) & abs (x1 - (upper_bound B)) < t holds
abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r ::_thesis: verum
proof
let x1 be real number ; ::_thesis: ( x1 in dom (Lg0 | [.0,1.]) & abs (x1 - (upper_bound B)) < t implies abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r )
A46: x1 is Element of REAL by XREAL_0:def_1;
assume that
A47: x1 in dom (Lg0 | [.0,1.]) and
A48: abs (x1 - (upper_bound B)) < t ; ::_thesis: abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r
x1 in [.0,1.] by A33, A47, RELAT_1:62;
then A49: (Lg0 | [.0,1.]) . x1 = Lg0 . x1 by FUNCT_1:49
.= ((||.((f /. x1) - (f /. 0)).|| - ((g . x1) - (g . 0))) - ((e / 2) * x1)) - (e / 2) by A32, A46 ;
(Lg0 | [.0,1.]) . (upper_bound B) = Lg0 . (upper_bound B) by A36, FUNCT_1:49;
then (Lg0 | [.0,1.]) . (upper_bound B) = ((||.((f /. (upper_bound B)) - (f /. 0)).|| - ((g . (upper_bound B)) - (g . 0))) - ((e / 2) * (upper_bound B))) - (e / 2) by A32;
then ((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B)) = ((||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||) - ((g . x1) - (g . (upper_bound B)))) - ((e / 2) * (x1 - (upper_bound B))) by A49;
then A50: abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) <= (abs ((||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||) - ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) by COMPLEX1:57;
abs ((||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||) - ((g . x1) - (g . (upper_bound B)))) <= (abs (||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||)) + (abs ((g . x1) - (g . (upper_bound B)))) by COMPLEX1:57;
then (abs ((||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||) - ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) <= ((abs (||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||)) + (abs ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) by XREAL_1:6;
then A51: abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) <= ((abs (||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||)) + (abs ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) by A50, XXREAL_0:2;
A52: abs (||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||) <= ||.(((f /. x1) - (f /. 0)) - ((f /. (upper_bound B)) - (f /. 0))).|| by NORMSP_1:9;
((f /. x1) - (f /. 0)) - ((f /. (upper_bound B)) - (f /. 0)) = (f /. x1) - ((f /. 0) - (- ((f /. (upper_bound B)) - (f /. 0)))) by RLVECT_1:29
.= (f /. x1) - ((f /. 0) + ((f /. (upper_bound B)) - (f /. 0))) by RLVECT_1:17
.= (f /. x1) - ((f /. (upper_bound B)) - ((f /. 0) - (f /. 0))) by RLVECT_1:29
.= (f /. x1) - ((f /. (upper_bound B)) - (0. T)) by RLVECT_1:5
.= (f /. x1) - (f /. (upper_bound B)) by RLVECT_1:13 ;
then (abs (||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||)) + (abs ((g . x1) - (g . (upper_bound B)))) <= ||.((f /. x1) - (f /. (upper_bound B))).|| + (abs ((g . x1) - (g . (upper_bound B)))) by A52, XREAL_1:6;
then ((abs (||.((f /. x1) - (f /. 0)).|| - ||.((f /. (upper_bound B)) - (f /. 0)).||)) + (abs ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) <= (||.((f /. x1) - (f /. (upper_bound B))).|| + (abs ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) by XREAL_1:6;
then A53: abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) <= (||.((f /. x1) - (f /. (upper_bound B))).|| + (abs ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) by A51, XXREAL_0:2;
abs (x1 - (upper_bound B)) < t2 by A48, A45, XXREAL_0:2;
then abs ((g . x1) - (g . (upper_bound B))) < r / 3 by A47, A34, A39;
then A54: ||.((f /. x1) - (f /. (upper_bound B))).|| + (abs ((g . x1) - (g . (upper_bound B)))) < ||.((f /. x1) - (f /. (upper_bound B))).|| + (r / 3) by XREAL_1:8;
abs (x1 - (upper_bound B)) < ((r / 3) / (e / 2)) / 2 by A48, A42, XXREAL_0:2;
then (abs (x1 - (upper_bound B))) * (e / 2) <= (((r / 3) / (e / 2)) / 2) * (e / 2) by A2, XREAL_1:64;
then (abs (x1 - (upper_bound B))) * (abs (e / 2)) <= (((r / 3) / (e / 2)) / 2) * (e / 2) by A2, ABSVALUE:def_1;
then abs ((e / 2) * (x1 - (upper_bound B))) <= (((r / 3) / (e / 2)) / 2) * (e / 2) by COMPLEX1:65;
then A55: abs ((e / 2) * (x1 - (upper_bound B))) < r / 3 by A40, XXREAL_0:2;
abs (x1 - (upper_bound B)) < t1 by A48, A44, XXREAL_0:2;
then ||.((f /. x1) - (f /. (upper_bound B))).|| < r / 3 by A47, A34, A38;
then ||.((f /. x1) - (f /. (upper_bound B))).|| + (r / 3) < (r / 3) + (r / 3) by XREAL_1:8;
then ||.((f /. x1) - (f /. (upper_bound B))).|| + (abs ((g . x1) - (g . (upper_bound B)))) < (r / 3) + (r / 3) by A54, XXREAL_0:2;
then (||.((f /. x1) - (f /. (upper_bound B))).|| + (abs ((g . x1) - (g . (upper_bound B))))) + (abs ((e / 2) * (x1 - (upper_bound B)))) < ((r / 3) + (r / 3)) + (r / 3) by A55, XREAL_1:8;
hence abs (((Lg0 | [.0,1.]) . x1) - ((Lg0 | [.0,1.]) . (upper_bound B))) < r by A53, XXREAL_0:2; ::_thesis: verum
end;
end;
then Lg0 | [.0,1.] is_continuous_in upper_bound B by FCONT_1:3;
then A56: ( (Lg0 | [.0,1.]) /* sq is convergent & (Lg0 | [.0,1.]) . (upper_bound B) = lim ((Lg0 | [.0,1.]) /* sq) ) by A29, A30, A35, FCONT_1:def_1;
A57: for n being Element of NAT holds 0 <= (- ((Lg0 | [.0,1.]) /* sq)) . n
proof
let n be Element of NAT ; ::_thesis: 0 <= (- ((Lg0 | [.0,1.]) /* sq)) . n
(- ((Lg0 | [.0,1.]) /* sq)) . n = - (((Lg0 | [.0,1.]) /* sq) . n) by SEQ_1:10;
then A58: (- ((Lg0 | [.0,1.]) /* sq)) . n = - ((Lg0 | [.0,1.]) . (sq . n)) by A35, FUNCT_2:108;
S1[n,sq . n] by A22;
then A59: ex x being Real st
( sq . n = x & x in [.0,1.] & ((||.((f /. x) - (f /. 0)).|| - ((g . x) - (g . 0))) - ((e / 2) * x)) - (e / 2) <= 0 ) ;
then Lg0 . (sq . n) <= 0 by A32;
then (Lg0 | [.0,1.]) . (sq . n) <= 0 by A59, FUNCT_1:49;
hence 0 <= (- ((Lg0 | [.0,1.]) /* sq)) . n by A58; ::_thesis: verum
end;
- ((Lg0 | [.0,1.]) /* sq) is convergent by A56, SEQ_2:9;
then 0 <= lim (- ((Lg0 | [.0,1.]) /* sq)) by A57, SEQ_2:17;
then 0 <= - (lim ((Lg0 | [.0,1.]) /* sq)) by A56, SEQ_2:10;
then (Lg0 | [.0,1.]) . (upper_bound B) <= 0 by A56;
then Lg0 . (upper_bound B) <= 0 by A36, FUNCT_1:49;
then A60: ((||.((f /. (upper_bound B)) - (f /. 0)).|| - ((g . (upper_bound B)) - (g . 0))) - ((e / 2) * (upper_bound B))) - (e / 2) <= 0 by A32;
A61: upper_bound B = 1
proof
assume upper_bound B <> 1 ; ::_thesis: contradiction
then upper_bound B < 1 by A18, XXREAL_0:1;
then A62: upper_bound B in ].0,1.[ by A17;
then f is_differentiable_in upper_bound B by A1, NDIFF_3:10;
then consider N1 being Neighbourhood of upper_bound B such that
A63: ( N1 c= dom f & ex L1 being LinearFunc of T ex R1 being RestFunc of T st
( diff (f,(upper_bound B)) = L1 . 1 & ( for x being Real st x in N1 holds
(f /. x) - (f /. (upper_bound B)) = (L1 . (x - (upper_bound B))) + (R1 /. (x - (upper_bound B))) ) ) ) by NDIFF_3:def_4;
consider L1 being LinearFunc of T, R1 being RestFunc of T such that
A64: ( diff (f,(upper_bound B)) = L1 . 1 & ( for x being Real st x in N1 holds
(f /. x) - (f /. (upper_bound B)) = (L1 . (x - (upper_bound B))) + (R1 /. (x - (upper_bound B))) ) ) by A63;
g is_differentiable_in upper_bound B by A1, A62, FDIFF_1:9;
then consider N2 being Neighbourhood of upper_bound B such that
A65: ( N2 c= dom g & ex L2 being LinearFunc ex R2 being RestFunc st
( diff (g,(upper_bound B)) = L2 . 1 & ( for x being Real st x in N2 holds
(g . x) - (g . (upper_bound B)) = (L2 . (x - (upper_bound B))) + (R2 . (x - (upper_bound B))) ) ) ) by FDIFF_1:def_5;
consider L2 being LinearFunc, R2 being RestFunc such that
A66: ( diff (g,(upper_bound B)) = L2 . 1 & ( for x being Real st x in N2 holds
(g . x) - (g . (upper_bound B)) = (L2 . (x - (upper_bound B))) + (R2 . (x - (upper_bound B))) ) ) by A65;
consider NN3 being Neighbourhood of upper_bound B such that
A67: ( NN3 c= N1 & NN3 c= N2 ) by RCOMP_1:17;
consider g0 being real number such that
A68: ( 0 < g0 & ].((upper_bound B) - g0),((upper_bound B) + g0).[ c= ].0,1.[ ) by A62, RCOMP_1:19;
reconsider NN4 = ].((upper_bound B) - g0),((upper_bound B) + g0).[ as Neighbourhood of upper_bound B by A68, RCOMP_1:def_6;
consider N3 being Neighbourhood of upper_bound B such that
A69: ( N3 c= NN3 & N3 c= NN4 ) by RCOMP_1:17;
A70: ( N3 c= N1 & N3 c= N2 & N3 c= ].0,1.[ ) by A69, A67, A68, XBOOLE_1:1;
consider d1 being real number such that
A71: ( 0 < d1 & N3 = ].((upper_bound B) - d1),((upper_bound B) + d1).[ ) by RCOMP_1:def_6;
set e2 = (e / 2) / 2;
( R1 is total & R1 is RestFunc-like ) by NDIFF_3:def_1;
then consider d2 being Real such that
A72: ( 0 < d2 & ( for t being Real st t <> 0 & abs t < d2 holds
||.(R1 /. t).|| / (abs t) < (e / 2) / 2 ) ) by A2, Th17;
( R2 is total & R2 is RestFunc-like ) by FDIFF_1:def_2;
then consider d3 being Real such that
A73: ( 0 < d3 & ( for t being Real st t <> 0 & abs t < d3 holds
(abs (R2 . t)) / (abs t) < (e / 2) / 2 ) ) by A2, Th18;
A74: ( min (d1,d2) <= d1 & min (d1,d2) <= d2 & 0 < min (d1,d2) ) by A71, A72, XXREAL_0:17, XXREAL_0:21;
set d40 = min ((min (d1,d2)),d3);
A75: ( min ((min (d1,d2)),d3) <= min (d1,d2) & min ((min (d1,d2)),d3) <= d3 & 0 < min ((min (d1,d2)),d3) ) by A73, A74, XXREAL_0:17, XXREAL_0:21;
set d4 = (min ((min (d1,d2)),d3)) / 2;
A76: ( min ((min (d1,d2)),d3) <= d1 & min ((min (d1,d2)),d3) <= d2 ) by A74, A75, XXREAL_0:2;
(min ((min (d1,d2)),d3)) / 2 < min ((min (d1,d2)),d3) by A75, XREAL_1:216;
then A77: ( 0 < (min ((min (d1,d2)),d3)) / 2 & (min ((min (d1,d2)),d3)) / 2 < d1 & (min ((min (d1,d2)),d3)) / 2 < d2 & (min ((min (d1,d2)),d3)) / 2 < d3 ) by A75, A76, XXREAL_0:2;
then ( (upper_bound B) - d1 < (upper_bound B) + ((min ((min (d1,d2)),d3)) / 2) & (upper_bound B) + ((min ((min (d1,d2)),d3)) / 2) < (upper_bound B) + d1 ) by XREAL_1:8;
then A78: (upper_bound B) + ((min ((min (d1,d2)),d3)) / 2) in N3 by A71;
then A79: (f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B)) = (L1 . (((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2)) - (upper_bound B))) + (R1 /. (((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2)) - (upper_bound B))) by A64, A70;
A80: (g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . (upper_bound B)) = (L2 . (((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2)) - (upper_bound B))) + (R2 . (((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2)) - (upper_bound B))) by A70, A78, A66;
consider df1 being Point of T such that
A81: for p being Real holds L1 . p = p * df1 by NDIFF_3:def_2;
L1 . 1 = 1 * df1 by A81;
then L1 . 1 = df1 by RLVECT_1:def_8;
then A82: L1 . ((min ((min (d1,d2)),d3)) / 2) = ((min ((min (d1,d2)),d3)) / 2) * (diff (f,(upper_bound B))) by A64, A81;
consider df2 being Real such that
A83: for p being Real holds L2 . p = df2 * p by FDIFF_1:def_3;
L2 . 1 = df2 * 1 by A83;
then A84: L2 . ((min ((min (d1,d2)),d3)) / 2) = ((min ((min (d1,d2)),d3)) / 2) * (diff (g,(upper_bound B))) by A66, A83;
A85: ||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| <= ||.(L1 . ((min ((min (d1,d2)),d3)) / 2)).|| + ||.(R1 /. ((min ((min (d1,d2)),d3)) / 2)).|| by A79, NORMSP_1:def_1;
A86: ||.(L1 . ((min ((min (d1,d2)),d3)) / 2)).|| = (abs ((min ((min (d1,d2)),d3)) / 2)) * ||.(diff (f,(upper_bound B))).|| by A82, NORMSP_1:def_1
.= ||.(diff (f,(upper_bound B))).|| * ((min ((min (d1,d2)),d3)) / 2) by A75, ABSVALUE:def_1 ;
A87: 0 < abs ((min ((min (d1,d2)),d3)) / 2) by A75, ABSVALUE:def_1;
abs ((min ((min (d1,d2)),d3)) / 2) < d2 by A77, ABSVALUE:def_1;
then ||.(R1 /. ((min ((min (d1,d2)),d3)) / 2)).|| / (abs ((min ((min (d1,d2)),d3)) / 2)) < (e / 2) / 2 by A72, A75;
then ||.(R1 /. ((min ((min (d1,d2)),d3)) / 2)).|| <= ((e / 2) / 2) * (abs ((min ((min (d1,d2)),d3)) / 2)) by A87, XREAL_1:81;
then ||.(R1 /. ((min ((min (d1,d2)),d3)) / 2)).|| <= ((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2) by A75, ABSVALUE:def_1;
then ||.(L1 . ((min ((min (d1,d2)),d3)) / 2)).|| + ||.(R1 /. ((min ((min (d1,d2)),d3)) / 2)).|| <= (||.(diff (f,(upper_bound B))).|| * ((min ((min (d1,d2)),d3)) / 2)) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) by A86, XREAL_1:6;
then A88: ||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| <= (||.(diff (f,(upper_bound B))).|| * ((min ((min (d1,d2)),d3)) / 2)) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) by A85, XXREAL_0:2;
||.(diff (f,(upper_bound B))).|| * ((min ((min (d1,d2)),d3)) / 2) <= (diff (g,(upper_bound B))) * ((min ((min (d1,d2)),d3)) / 2) by A62, A1, A77, XREAL_1:64;
then (||.(diff (f,(upper_bound B))).|| * ((min ((min (d1,d2)),d3)) / 2)) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) <= ((diff (g,(upper_bound B))) * ((min ((min (d1,d2)),d3)) / 2)) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) by XREAL_1:6;
then A89: ||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| <= ((diff (g,(upper_bound B))) * ((min ((min (d1,d2)),d3)) / 2)) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) by A88, XXREAL_0:2;
abs ((min ((min (d1,d2)),d3)) / 2) < d3 by A77, ABSVALUE:def_1;
then (abs (R2 . ((min ((min (d1,d2)),d3)) / 2))) / (abs ((min ((min (d1,d2)),d3)) / 2)) < (e / 2) / 2 by A73, A75;
then abs (R2 . ((min ((min (d1,d2)),d3)) / 2)) <= ((e / 2) / 2) * (abs ((min ((min (d1,d2)),d3)) / 2)) by A87, XREAL_1:81;
then abs (R2 . ((min ((min (d1,d2)),d3)) / 2)) <= ((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2) by A75, ABSVALUE:def_1;
then - (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) <= R2 . ((min ((min (d1,d2)),d3)) / 2) by ABSVALUE:5;
then (((min ((min (d1,d2)),d3)) / 2) * (diff (g,(upper_bound B)))) - (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) <= (g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . (upper_bound B)) by A80, A84, XREAL_1:6;
then ((min ((min (d1,d2)),d3)) / 2) * (diff (g,(upper_bound B))) <= ((g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . (upper_bound B))) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) by XREAL_1:20;
then ((diff (g,(upper_bound B))) * ((min ((min (d1,d2)),d3)) / 2)) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) <= (((g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . (upper_bound B))) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2))) + (((e / 2) / 2) * ((min ((min (d1,d2)),d3)) / 2)) by XREAL_1:6;
then ||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| <= ((g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . (upper_bound B))) + ((e / 2) * ((min ((min (d1,d2)),d3)) / 2)) by A89, XXREAL_0:2;
then ||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| - (((g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . (upper_bound B))) + ((e / 2) * ((min ((min (d1,d2)),d3)) / 2))) <= 0 by XREAL_1:47;
then A90: (((||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| - (g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2)))) + (g . (upper_bound B))) - ((e / 2) * ((min ((min (d1,d2)),d3)) / 2))) + (((||.((f /. (upper_bound B)) - (f /. 0)).|| - ((g . (upper_bound B)) - (g . 0))) - ((e / 2) * (upper_bound B))) - (e / 2)) <= 0 + 0 by A60;
||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. 0)).|| <= ||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| + ||.((f /. (upper_bound B)) - (f /. 0)).|| by NORMSP_1:10;
then ||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. 0)).|| - ((((g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . 0)) + ((e / 2) * (((min ((min (d1,d2)),d3)) / 2) + (upper_bound B)))) + (e / 2)) <= (||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. (upper_bound B))).|| + ||.((f /. (upper_bound B)) - (f /. 0)).||) - ((((g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . 0)) + ((e / 2) * (((min ((min (d1,d2)),d3)) / 2) + (upper_bound B)))) + (e / 2)) by XREAL_1:9;
then A91: ((||.((f /. ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (f /. 0)).|| - ((g . ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2))) - (g . 0))) - ((e / 2) * ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2)))) - (e / 2) <= 0 by A90;
abs ((0 + 1) - (2 * ((upper_bound B) + ((min ((min (d1,d2)),d3)) / 2)))) < 1 - 0 by A78, A70, RCOMP_1:3;
then (upper_bound B) + ((min ((min (d1,d2)),d3)) / 2) in [.0,1.] by RCOMP_1:2;
then A92: (upper_bound B) + ((min ((min (d1,d2)),d3)) / 2) in B by A91;
(upper_bound B) + 0 < (upper_bound B) + ((min ((min (d1,d2)),d3)) / 2) by A75, XREAL_1:8;
hence contradiction by A92, A4, SEQ_4:def_1; ::_thesis: verum
end;
( 0 in dom g & 1 in dom g ) by A1;
then ( g /. 1 = g . 1 & g /. 0 = g . 0 ) by PARTFUN1:def_6;
then ((||.((f /. 1) - (f /. 0)).|| - ((g /. 1) - (g /. 0))) - e) + e <= 0 + e by A61, A60, XREAL_1:6;
hence ||.((f /. 1) - (f /. 0)).|| - ((g /. 1) - (g /. 0)) <= e ; ::_thesis: verum
end;
then ||.((f /. 1) - (f /. 0)).|| - ((g /. 1) - (g /. 0)) <= 0 by Lm3;
then (||.((f /. 1) - (f /. 0)).|| - ((g /. 1) - (g /. 0))) + ((g /. 1) - (g /. 0)) <= 0 + ((g /. 1) - (g /. 0)) by XREAL_1:6;
hence ||.((f /. 1) - (f /. 0)).|| <= (g /. 1) - (g /. 0) ; ::_thesis: verum
end;
theorem Th19: :: NDIFF_5:19
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for p, q being Point of S
for M being Real st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.(diff (f,x)).|| <= M ) holds
||.((f /. q) - (f /. p)).|| <= M * ||.(q - p).||
proof
let S, T be non trivial RealNormSpace; ::_thesis: for f being PartFunc of S,T
for p, q being Point of S
for M being Real st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.(diff (f,x)).|| <= M ) holds
||.((f /. q) - (f /. p)).|| <= M * ||.(q - p).||
let f be PartFunc of S,T; ::_thesis: for p, q being Point of S
for M being Real st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.(diff (f,x)).|| <= M ) holds
||.((f /. q) - (f /. p)).|| <= M * ||.(q - p).||
let p, q be Point of S; ::_thesis: for M being Real st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.(diff (f,x)).|| <= M ) holds
||.((f /. q) - (f /. p)).|| <= M * ||.(q - p).||
let M be Real; ::_thesis: ( [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.(diff (f,x)).|| <= M ) implies ||.((f /. q) - (f /. p)).|| <= M * ||.(q - p).|| )
assume A1: ( [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.(diff (f,x)).|| <= M ) ) ; ::_thesis: ||.((f /. q) - (f /. p)).|| <= M * ||.(q - p).||
deffunc H1( Element of REAL ) -> Element of the carrier of S = ($1 * (q - p)) + p;
consider pt0 being Function of REAL, the carrier of S such that
A2: for t being Element of REAL holds pt0 . t = H1(t) from FUNCT_2:sch_4();
set pt = pt0 | [.0,1.];
A3: dom pt0 = REAL by FUNCT_2:def_1;
then A4: dom (pt0 | [.0,1.]) = [.0,1.] by RELAT_1:62;
now__::_thesis:_for_t_being_real_number_st_t_in_[.0,1.]_holds_
pt0_/._t_=_(t_*_(q_-_p))_+_p
let t be real number ; ::_thesis: ( t in [.0,1.] implies pt0 /. t = (t * (q - p)) + p )
assume t in [.0,1.] ; ::_thesis: pt0 /. t = (t * (q - p)) + p
A5: t is Element of REAL by XREAL_0:def_1;
then pt0 /. t = pt0 . t by A3, PARTFUN1:def_6;
hence pt0 /. t = (t * (q - p)) + p by A2, A5; ::_thesis: verum
end;
then A6: pt0 | [.0,1.] is continuous by NFCONT_3:33;
A7: ].0,1.[ c= [.0,1.] by XXREAL_1:25;
A8: now__::_thesis:_for_t_being_Real_st_t_in_].0,1.[_holds_
(pt0_|_[.0,1.])_/._t_=_(t_*_(q_-_p))_+_p
let t be Real; ::_thesis: ( t in ].0,1.[ implies (pt0 | [.0,1.]) /. t = (t * (q - p)) + p )
assume t in ].0,1.[ ; ::_thesis: (pt0 | [.0,1.]) /. t = (t * (q - p)) + p
hence (pt0 | [.0,1.]) /. t = pt0 /. t by A4, A7, PARTFUN2:15
.= (t * (q - p)) + p by A2 ;
::_thesis: verum
end;
then A9: ( pt0 | [.0,1.] is_differentiable_on ].0,1.[ & ( for t being Real st t in ].0,1.[ holds
((pt0 | [.0,1.]) `| ].0,1.[) . t = q - p ) ) by A4, A7, NDIFF_3:21;
reconsider phi = f * (pt0 | [.0,1.]) as PartFunc of REAL,T ;
A10: rng (pt0 | [.0,1.]) c= [.p,q.]
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (pt0 | [.0,1.]) or y in [.p,q.] )
assume y in rng (pt0 | [.0,1.]) ; ::_thesis: y in [.p,q.]
then consider x being set such that
A11: ( x in dom (pt0 | [.0,1.]) & y = (pt0 | [.0,1.]) . x ) by FUNCT_1:def_3;
A12: y = pt0 . x by A11, FUNCT_1:47;
reconsider x = x as Element of REAL by A11;
consider r being Real such that
A13: ( x = r & 0 <= r & r <= 1 ) by A11, A4;
y = p + (x * (q - p)) by A2, A12
.= ((1 - x) * p) + (x * q) by Lm2 ;
then y in { (((1 - r1) * p) + (r1 * q)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } by A13;
hence y in [.p,q.] by RLTOPSP1:def_2; ::_thesis: verum
end;
then rng (pt0 | [.0,1.]) c= dom f by A1, XBOOLE_1:1;
then A14: dom phi = [.0,1.] by A4, RELAT_1:27;
A15: for t being real number st t in [.0,1.] holds
phi /. t = f /. (p + (t * (q - p)))
proof
let t be real number ; ::_thesis: ( t in [.0,1.] implies phi /. t = f /. (p + (t * (q - p))) )
assume A16: t in [.0,1.] ; ::_thesis: phi /. t = f /. (p + (t * (q - p)))
then A17: phi /. t = phi . t by A14, PARTFUN1:def_6
.= f . ((pt0 | [.0,1.]) . t) by A16, A14, FUNCT_1:12 ;
(pt0 | [.0,1.]) . t in rng (pt0 | [.0,1.]) by A16, A4, FUNCT_1:def_3;
then A18: (pt0 | [.0,1.]) . t in [.p,q.] by A10;
(pt0 | [.0,1.]) . t = pt0 . t by A16, A4, FUNCT_1:47
.= p + (t * (q - p)) by A2, A16 ;
hence phi /. t = f /. (p + (t * (q - p))) by A17, A18, A1, PARTFUN1:def_6; ::_thesis: verum
end;
now__::_thesis:_for_x0_being_real_number_st_x0_in_dom_phi_holds_
phi_is_continuous_in_x0
let x0 be real number ; ::_thesis: ( x0 in dom phi implies phi is_continuous_in x0 )
assume A19: x0 in dom phi ; ::_thesis: phi is_continuous_in x0
then A20: pt0 | [.0,1.] is_continuous_in x0 by A4, A6, A14, NFCONT_3:def_2;
(pt0 | [.0,1.]) . x0 in rng (pt0 | [.0,1.]) by A4, A19, A14, FUNCT_1:def_3;
then (pt0 | [.0,1.]) . x0 in [.p,q.] by A10;
then (pt0 | [.0,1.]) /. x0 in [.p,q.] by A19, A14, A4, PARTFUN1:def_6;
hence phi is_continuous_in x0 by A1, A19, A20, NFCONT_3:15; ::_thesis: verum
end;
then phi is continuous by NFCONT_3:def_2;
then A21: phi | [.0,1.] is continuous ;
A22: now__::_thesis:_for_x_being_Real_st_x_in_].0,1.[_holds_
(_phi_is_differentiable_in_x_&_diff_(phi,x)_=_(diff_(f,(p_+_(x_*_(q_-_p)))))_._(q_-_p)_)
let x be Real; ::_thesis: ( x in ].0,1.[ implies ( phi is_differentiable_in x & diff (phi,x) = (diff (f,(p + (x * (q - p))))) . (q - p) ) )
assume A23: x in ].0,1.[ ; ::_thesis: ( phi is_differentiable_in x & diff (phi,x) = (diff (f,(p + (x * (q - p))))) . (q - p) )
then A24: pt0 | [.0,1.] is_differentiable_in x by A9, NDIFF_3:10;
((pt0 | [.0,1.]) `| ].0,1.[) . x = q - p by A23, A8, A4, A7, NDIFF_3:21;
then A25: diff ((pt0 | [.0,1.]),x) = q - p by A9, A23, NDIFF_3:def_6;
A26: (pt0 | [.0,1.]) . x = (pt0 | [.0,1.]) /. x by A23, A7, A4, PARTFUN1:def_6;
A27: ex r being Real st
( x = r & 0 < r & r < 1 ) by A23;
A28: (pt0 | [.0,1.]) . x = pt0 . x by A23, A7, A4, FUNCT_1:47;
A29: pt0 . x = p + (x * (q - p)) by A2;
then (pt0 | [.0,1.]) . x in ].p,q.[ by A27, A28;
then A30: f is_differentiable_in (pt0 | [.0,1.]) /. x by A26, A1;
hence phi is_differentiable_in x by A24, Th6; ::_thesis: diff (phi,x) = (diff (f,(p + (x * (q - p))))) . (q - p)
thus diff (phi,x) = (diff (f,(p + (x * (q - p))))) . (q - p) by A25, A26, A28, A29, A30, A24, Th6; ::_thesis: verum
end;
then ( ].0,1.[ c= dom phi & ( for x being Real st x in ].0,1.[ holds
phi is_differentiable_in x ) ) by A14, XXREAL_1:25;
then A31: phi is_differentiable_on ].0,1.[ by NDIFF_3:10;
deffunc H2( Element of REAL ) -> Element of REAL = (M * ||.(q - p).||) * $1;
consider g0 being Function of REAL,REAL such that
A32: for t being Element of REAL holds g0 . t = H2(t) from FUNCT_2:sch_4();
set g = g0 | [.0,1.];
for t being real number st t in [.0,1.] holds
g0 . t = ((M * ||.(q - p).||) * t) + 0 by A32;
then A33: g0 | [.0,1.] is continuous by FCONT_1:41;
dom g0 = REAL by FUNCT_2:def_1;
then A34: dom (g0 | [.0,1.]) = [.0,1.] by RELAT_1:62;
A35: (g0 | [.0,1.]) | [.0,1.] is continuous by A33;
A36: now__::_thesis:_for_t_being_Real_st_t_in_].0,1.[_holds_
(g0_|_[.0,1.])_._t_=_((M_*_||.(q_-_p).||)_*_t)_+_0
let t be Real; ::_thesis: ( t in ].0,1.[ implies (g0 | [.0,1.]) . t = ((M * ||.(q - p).||) * t) + 0 )
assume t in ].0,1.[ ; ::_thesis: (g0 | [.0,1.]) . t = ((M * ||.(q - p).||) * t) + 0
hence (g0 | [.0,1.]) . t = g0 . t by A34, A7, FUNCT_1:47
.= ((M * ||.(q - p).||) * t) + 0 by A32 ;
::_thesis: verum
end;
then A37: ( g0 | [.0,1.] is_differentiable_on ].0,1.[ & ( for t being Real st t in ].0,1.[ holds
((g0 | [.0,1.]) `| ].0,1.[) . t = M * ||.(q - p).|| ) ) by A34, A7, FDIFF_1:23;
for t being real number st t in ].0,1.[ holds
||.(diff (phi,t)).|| <= diff ((g0 | [.0,1.]),t)
proof
let t be real number ; ::_thesis: ( t in ].0,1.[ implies ||.(diff (phi,t)).|| <= diff ((g0 | [.0,1.]),t) )
assume A38: t in ].0,1.[ ; ::_thesis: ||.(diff (phi,t)).|| <= diff ((g0 | [.0,1.]),t)
then A39: ||.(diff (phi,t)).|| = ||.((diff (f,(p + (t * (q - p))))) . (q - p)).|| by A22;
reconsider L = diff (f,(p + (t * (q - p)))) as Lipschitzian LinearOperator of S,T by LOPBAN_1:def_9;
A40: ||.(L . (q - p)).|| <= ||.(diff (f,(p + (t * (q - p))))).|| * ||.(q - p).|| by LOPBAN_1:32;
ex r being Real st
( t = r & 0 < r & r < 1 ) by A38;
then p + (t * (q - p)) in ].p,q.[ ;
then A41: ||.(diff (f,(p + (t * (q - p))))).|| * ||.(q - p).|| <= M * ||.(q - p).|| by A1, XREAL_1:64;
diff ((g0 | [.0,1.]),t) = ((g0 | [.0,1.]) `| ].0,1.[) . t by A38, A37, FDIFF_1:def_7;
then diff ((g0 | [.0,1.]),t) = M * ||.(q - p).|| by A38, A36, A34, A7, FDIFF_1:23;
hence ||.(diff (phi,t)).|| <= diff ((g0 | [.0,1.]),t) by A41, A40, A39, XXREAL_0:2; ::_thesis: verum
end;
then A42: ||.((phi /. 1) - (phi /. 0)).|| <= ((g0 | [.0,1.]) /. 1) - ((g0 | [.0,1.]) /. 0) by Lm4, A14, A21, A31, A34, A35, A37;
A43: ( 1 in [.0,1.] & 0 in [.0,1.] ) ;
then A44: (g0 | [.0,1.]) /. 1 = (g0 | [.0,1.]) . 1 by A34, PARTFUN1:def_6
.= g0 . 1 by A34, A43, FUNCT_1:47
.= (M * ||.(q - p).||) * 1 by A32 ;
A45: (g0 | [.0,1.]) /. 0 = (g0 | [.0,1.]) . 0 by A34, A43, PARTFUN1:def_6
.= g0 . 0 by A34, A43, FUNCT_1:47
.= (M * ||.(q - p).||) * 0 by A32 ;
A46: phi /. 1 = f /. (p + (1 * (q - p))) by A15, A43
.= f /. (p + (q - p)) by RLVECT_1:def_8
.= f /. (q - (p - p)) by RLVECT_1:29
.= f /. (q - (0. S)) by RLVECT_1:15
.= f /. q by RLVECT_1:13 ;
phi /. 0 = f /. (p + (0 * (q - p))) by A15, A43
.= f /. (p + (0. S)) by RLVECT_1:10
.= f /. p by RLVECT_1:4 ;
hence ||.((f /. q) - (f /. p)).|| <= M * ||.(q - p).|| by A42, A44, A45, A46; ::_thesis: verum
end;
theorem Th20: :: NDIFF_5:20
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for p, q being Point of S
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.((diff (f,x)) - L).|| <= M ) holds
||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).||
proof
let S, T be non trivial RealNormSpace; ::_thesis: for f being PartFunc of S,T
for p, q being Point of S
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.((diff (f,x)) - L).|| <= M ) holds
||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).||
let f be PartFunc of S,T; ::_thesis: for p, q being Point of S
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.((diff (f,x)) - L).|| <= M ) holds
||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).||
let p, q be Point of S; ::_thesis: for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.((diff (f,x)) - L).|| <= M ) holds
||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).||
let M be Real; ::_thesis: for L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) st [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.((diff (f,x)) - L).|| <= M ) holds
||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).||
let L be Point of (R_NormSpace_of_BoundedLinearOperators (S,T)); ::_thesis: ( [.p,q.] c= dom f & ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.((diff (f,x)) - L).|| <= M ) implies ||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).|| )
assume that
A1: [.p,q.] c= dom f and
A2: ( ( for x being Point of S st x in [.p,q.] holds
f is_continuous_in x ) & ( for x being Point of S st x in ].p,q.[ holds
f is_differentiable_in x ) & ( for x being Point of S st x in ].p,q.[ holds
||.((diff (f,x)) - L).|| <= M ) ) ; ::_thesis: ||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).||
reconsider LP = L as Lipschitzian LinearOperator of S,T by LOPBAN_1:def_9;
deffunc H1( Point of S) -> Element of the carrier of T = L . ($1 - p);
consider L0 being Function of the carrier of S, the carrier of T such that
A3: for t being Element of the carrier of S holds L0 . t = H1(t) from FUNCT_2:sch_4();
A4: dom L0 = the carrier of S by FUNCT_2:def_1;
now__::_thesis:_for_x1,_x2_being_Point_of_S_st_x1_in_dom_L0_&_x2_in_dom_L0_holds_
||.((L0_/._x1)_-_(L0_/._x2)).||_<=_(||.L.||_+_1)_*_||.(x1_-_x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in dom L0 & x2 in dom L0 implies ||.((L0 /. x1) - (L0 /. x2)).|| <= (||.L.|| + 1) * ||.(x1 - x2).|| )
assume ( x1 in dom L0 & x2 in dom L0 ) ; ::_thesis: ||.((L0 /. x1) - (L0 /. x2)).|| <= (||.L.|| + 1) * ||.(x1 - x2).||
A5: - 1 is Real by XREAL_0:def_1;
( L0 /. x1 = L . (x1 - p) & L0 /. x2 = L . (x2 - p) ) by A3;
then ||.((L0 /. x1) - (L0 /. x2)).|| = ||.((LP . (x1 - p)) + ((- 1) * (LP . (x2 - p)))).|| by RLVECT_1:16
.= ||.((LP . (x1 - p)) + (LP . ((- 1) * (x2 - p)))).|| by A5, LOPBAN_1:def_5
.= ||.(LP . ((x1 - p) + ((- 1) * (x2 - p)))).|| by VECTSP_1:def_20
.= ||.(LP . ((x1 - p) - (x2 - p))).|| by RLVECT_1:16
.= ||.(LP . (x1 - ((x2 - p) + p))).|| by RLVECT_1:27
.= ||.(LP . (x1 - (x2 - (p - p)))).|| by RLVECT_1:29
.= ||.(LP . (x1 - (x2 - (0. S)))).|| by RLVECT_1:15
.= ||.(LP . (x1 - x2)).|| by RLVECT_1:13 ;
then A6: ||.((L0 /. x1) - (L0 /. x2)).|| <= ||.L.|| * ||.(x1 - x2).|| by LOPBAN_1:32;
0 + ||.L.|| < 1 + ||.L.|| by XREAL_1:8;
then ||.L.|| * ||.(x1 - x2).|| <= (||.L.|| + 1) * ||.(x1 - x2).|| by XREAL_1:64;
hence ||.((L0 /. x1) - (L0 /. x2)).|| <= (||.L.|| + 1) * ||.(x1 - x2).|| by A6, XXREAL_0:2; ::_thesis: verum
end;
then L0 is_Lipschitzian_on dom L0 by NFCONT_1:def_9;
then A7: L0 is_continuous_on dom L0 by NFCONT_1:45;
reconsider R = the carrier of S --> (0. T) as PartFunc of S,T ;
A8: dom R = the carrier of S by FUNCOP_1:13;
now__::_thesis:_for_h_being_non-zero_0._S_-convergent_sequence_of_S_holds_
(_(||.h.||_")_(#)_(R_/*_h)_is_convergent_&_lim_((||.h.||_")_(#)_(R_/*_h))_=_0._T_)
let h be non-zero 0. S -convergent sequence of S; ::_thesis: ( (||.h.|| ") (#) (R /* h) is convergent & lim ((||.h.|| ") (#) (R /* h)) = 0. T )
A9: now__::_thesis:_for_n_being_Nat_holds_((||.h.||_")_(#)_(R_/*_h))_._n_=_0._T
let n be Nat; ::_thesis: ((||.h.|| ") (#) (R /* h)) . n = 0. T
A10: R /. (h . n) = R . (h . n) by A8, PARTFUN1:def_6
.= 0. T by FUNCOP_1:7 ;
A11: rng h c= dom R by A8;
A12: n in NAT by ORDINAL1:def_12;
hence ((||.h.|| ") (#) (R /* h)) . n = ((||.h.|| ") . n) * ((R /* h) . n) by NDIFF_1:def_2
.= ((||.h.|| ") . n) * (R /. (h . n)) by A12, A11, FUNCT_2:109
.= 0. T by A10, RLVECT_1:10 ;
::_thesis: verum
end;
then A13: (||.h.|| ") (#) (R /* h) is constant by VALUED_0:def_18;
hence (||.h.|| ") (#) (R /* h) is convergent by NDIFF_1:18; ::_thesis: lim ((||.h.|| ") (#) (R /* h)) = 0. T
((||.h.|| ") (#) (R /* h)) . 0 = 0. T by A9;
hence lim ((||.h.|| ") (#) (R /* h)) = 0. T by A13, NDIFF_1:18; ::_thesis: verum
end;
then reconsider R = R as RestFunc of S,T by NDIFF_1:def_5;
A14: now__::_thesis:_for_x0_being_Point_of_S_holds_
(_L0_is_differentiable_in_x0_&_diff_(L0,x0)_=_L_)
let x0 be Point of S; ::_thesis: ( L0 is_differentiable_in x0 & diff (L0,x0) = L )
set N = the Neighbourhood of x0;
A15: for x being Point of S st x in the Neighbourhood of x0 holds
(L0 /. x) - (L0 /. x0) = (L . (x - x0)) + (R /. (x - x0))
proof
let x be Point of S; ::_thesis: ( x in the Neighbourhood of x0 implies (L0 /. x) - (L0 /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
A16: R /. (x - x0) = R . (x - x0) by A8, PARTFUN1:def_6
.= 0. T by FUNCOP_1:7 ;
A17: - 1 is Real by XREAL_0:def_1;
assume x in the Neighbourhood of x0 ; ::_thesis: (L0 /. x) - (L0 /. x0) = (L . (x - x0)) + (R /. (x - x0))
thus (L0 /. x) - (L0 /. x0) = (L . (x - p)) - (L0 . x0) by A3
.= (L . (x - p)) - (L . (x0 - p)) by A3
.= (LP . (x - p)) + ((- 1) * (LP . (x0 - p))) by RLVECT_1:16
.= (LP . (x - p)) + (LP . ((- 1) * (x0 - p))) by A17, LOPBAN_1:def_5
.= LP . ((x - p) + ((- 1) * (x0 - p))) by VECTSP_1:def_20
.= LP . ((x - p) - (x0 - p)) by RLVECT_1:16
.= LP . (x - ((x0 - p) + p)) by RLVECT_1:27
.= LP . (x - (x0 - (p - p))) by RLVECT_1:29
.= LP . (x - (x0 - (0. S))) by RLVECT_1:15
.= LP . (x - x0) by RLVECT_1:13
.= (L . (x - x0)) + (R /. (x - x0)) by A16, RLVECT_1:4 ; ::_thesis: verum
end;
hence L0 is_differentiable_in x0 by A4, NDIFF_1:def_6; ::_thesis: diff (L0,x0) = L
hence diff (L0,x0) = L by A4, A15, NDIFF_1:def_7; ::_thesis: verum
end;
set g = f - L0;
A18: dom (f - L0) = (dom f) /\ (dom L0) by VFUNCT_1:def_2
.= dom f by A4, XBOOLE_1:28 ;
A19: for x being Point of S st x in dom (f - L0) holds
(f - L0) /. x = (f /. x) - (L . (x - p))
proof
let x be Point of S; ::_thesis: ( x in dom (f - L0) implies (f - L0) /. x = (f /. x) - (L . (x - p)) )
assume x in dom (f - L0) ; ::_thesis: (f - L0) /. x = (f /. x) - (L . (x - p))
hence (f - L0) /. x = (f /. x) - (L0 /. x) by VFUNCT_1:def_2
.= (f /. x) - (L . (x - p)) by A3 ;
::_thesis: verum
end;
A20: for x being Point of S st x in [.p,q.] holds
f - L0 is_continuous_in x
proof
let x be Point of S; ::_thesis: ( x in [.p,q.] implies f - L0 is_continuous_in x )
assume x in [.p,q.] ; ::_thesis: f - L0 is_continuous_in x
then A21: f is_continuous_in x by A2;
L0 | (dom L0) is_continuous_in x by A4, A7, NFCONT_1:def_7;
then L0 is_continuous_in x ;
hence f - L0 is_continuous_in x by A21, NFCONT_1:15; ::_thesis: verum
end;
A22: for x being Point of S st x in ].p,q.[ holds
f - L0 is_differentiable_in x
proof
let x be Point of S; ::_thesis: ( x in ].p,q.[ implies f - L0 is_differentiable_in x )
assume x in ].p,q.[ ; ::_thesis: f - L0 is_differentiable_in x
then ( f is_differentiable_in x & L0 is_differentiable_in x ) by A2, A14;
hence f - L0 is_differentiable_in x by NDIFF_1:36; ::_thesis: verum
end;
for x being Point of S st x in ].p,q.[ holds
||.(diff ((f - L0),x)).|| <= M
proof
let x be Point of S; ::_thesis: ( x in ].p,q.[ implies ||.(diff ((f - L0),x)).|| <= M )
assume A23: x in ].p,q.[ ; ::_thesis: ||.(diff ((f - L0),x)).|| <= M
then A24: f is_differentiable_in x by A2;
( L0 is_differentiable_in x & diff (L0,x) = L ) by A14;
then diff ((f - L0),x) = (diff (f,x)) - L by A24, NDIFF_1:36;
hence ||.(diff ((f - L0),x)).|| <= M by A2, A23; ::_thesis: verum
end;
then A25: ||.(((f - L0) /. q) - ((f - L0) /. p)).|| <= M * ||.(q - p).|| by Th19, A1, A18, A20, A22;
p in [.p,q.] by RLTOPSP1:68;
then (f - L0) /. p = (f /. p) - (L . (p - p)) by A1, A18, A19;
then A26: (f - L0) /. p = (f /. p) - (LP . (0. S)) by RLVECT_1:15
.= (f /. p) - (LP . (0 * p)) by RLVECT_1:10
.= (f /. p) - (0 * (LP . p)) by LOPBAN_1:def_5
.= (f /. p) - (0. T) by RLVECT_1:10
.= f /. p by RLVECT_1:13 ;
q in [.p,q.] by RLTOPSP1:68;
then (f - L0) /. q = (f /. q) - (L . (q - p)) by A1, A18, A19;
then (f /. q) - ((L . (q - p)) + (f /. p)) = ((f - L0) /. q) - ((f - L0) /. p) by A26, RLVECT_1:27;
hence ||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= M * ||.(q - p).|| by A25, RLVECT_1:27; ::_thesis: verum
end;
begin
definition
let G be RealNormSpace-Sequence;
let i be Element of dom G;
func proj i -> Function of (product G),(G . i) means :Def3: :: NDIFF_5:def 3
for x being Element of product (carr G) holds it . x = x . i;
existence
ex b1 being Function of (product G),(G . i) st
for x being Element of product (carr G) holds b1 . x = x . i
proof
deffunc H1( Element of product (carr G)) -> Element of (G . i) = $1 . i;
consider f being Function of (product (carr G)),(G . i) such that
A1: for x being Element of product (carr G) holds f . x = H1(x) from FUNCT_2:sch_4();
product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
then reconsider f = f as Function of (product G),(G . i) ;
take f ; ::_thesis: for x being Element of product (carr G) holds f . x = x . i
thus for x being Element of product (carr G) holds f . x = x . i by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (product G),(G . i) st ( for x being Element of product (carr G) holds b1 . x = x . i ) & ( for x being Element of product (carr G) holds b2 . x = x . i ) holds
b1 = b2
proof
let f, g be Function of the carrier of (product G), the carrier of (G . i); ::_thesis: ( ( for x being Element of product (carr G) holds f . x = x . i ) & ( for x being Element of product (carr G) holds g . x = x . i ) implies f = g )
assume that
A2: for x being Element of product (carr G) holds f . x = x . i and
A3: for x being Element of product (carr G) holds g . x = x . i ; ::_thesis: f = g
A4: product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
now__::_thesis:_for_x1_being_Element_of_the_carrier_of_(product_G)_holds_f_._x1_=_g_._x1
let x1 be Element of the carrier of (product G); ::_thesis: f . x1 = g . x1
reconsider x = x1 as Element of product (carr G) by A4;
f . x1 = x . i by A2;
hence f . x1 = g . x1 by A3; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines proj NDIFF_5:def_3_:_
for G being RealNormSpace-Sequence
for i being Element of dom G
for b3 being Function of (product G),(G . i) holds
( b3 = proj i iff for x being Element of product (carr G) holds b3 . x = x . i );
definition
let G be RealNormSpace-Sequence;
let i be Element of dom G;
let x be Element of (product G);
func reproj (i,x) -> Function of (G . i),(product G) means :Def4: :: NDIFF_5:def 4
for r being Element of (G . i) holds it . r = x +* (i,r);
existence
ex b1 being Function of (G . i),(product G) st
for r being Element of (G . i) holds b1 . r = x +* (i,r)
proof
reconsider x1 = x as Element of product (carr G) by Th10;
defpred S1[ Element of (G . i), Element of the carrier of (product G)] means $2 = x1 +* (i,$1);
A1: for r being Element of (G . i) ex y being Element of the carrier of (product G) st S1[r,y]
proof
let r be Element of (G . i); ::_thesis: ex y being Element of the carrier of (product G) st S1[r,y]
x1 +* (i,r) is Element of the carrier of (product G) by Th11;
hence ex y being Element of the carrier of (product G) st S1[r,y] ; ::_thesis: verum
end;
ex f being Function of the carrier of (G . i), the carrier of (product G) st
for r being Element of (G . i) holds S1[r,f . r] from FUNCT_2:sch_3(A1);
hence ex b1 being Function of (G . i),(product G) st
for r being Element of (G . i) holds b1 . r = x +* (i,r) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (G . i),(product G) st ( for r being Element of (G . i) holds b1 . r = x +* (i,r) ) & ( for r being Element of (G . i) holds b2 . r = x +* (i,r) ) holds
b1 = b2
proof
let f, g be Function of the carrier of (G . i), the carrier of (product G); ::_thesis: ( ( for r being Element of (G . i) holds f . r = x +* (i,r) ) & ( for r being Element of (G . i) holds g . r = x +* (i,r) ) implies f = g )
assume that
A2: for r being Element of (G . i) holds f . r = x +* (i,r) and
A3: for r being Element of (G . i) holds g . r = x +* (i,r) ; ::_thesis: f = g
let r be Element of (G . i); :: according to FUNCT_2:def_8 ::_thesis: f . r = g . r
f . r = x +* (i,r) by A2;
hence f . r = g . r by A3; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines reproj NDIFF_5:def_4_:_
for G being RealNormSpace-Sequence
for i being Element of dom G
for x being Element of (product G)
for b4 being Function of (G . i),(product G) holds
( b4 = reproj (i,x) iff for r being Element of (G . i) holds b4 . r = x +* (i,r) );
definition
let G be non-trivial RealNormSpace-Sequence;
let j be set ;
assume A1: j in dom G ;
func modetrans (G,j) -> Element of dom G equals :Def5: :: NDIFF_5:def 5
j;
correctness
coherence
j is Element of dom G;
by A1;
end;
:: deftheorem Def5 defines modetrans NDIFF_5:def_5_:_
for G being non-trivial RealNormSpace-Sequence
for j being set st j in dom G holds
modetrans (G,j) = j;
definition
let G be non-trivial RealNormSpace-Sequence;
let F be non trivial RealNormSpace;
let i be set ;
let f be PartFunc of (product G),F;
let x be Element of (product G);
predf is_partial_differentiable_in x,i means :Def6: :: NDIFF_5:def 6
f * (reproj ((modetrans (G,i)),x)) is_differentiable_in (proj (modetrans (G,i))) . x;
end;
:: deftheorem Def6 defines is_partial_differentiable_in NDIFF_5:def_6_:_
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for i being set
for f being PartFunc of (product G),F
for x being Element of (product G) holds
( f is_partial_differentiable_in x,i iff f * (reproj ((modetrans (G,i)),x)) is_differentiable_in (proj (modetrans (G,i))) . x );
definition
let G be non-trivial RealNormSpace-Sequence;
let F be non trivial RealNormSpace;
let i be set ;
let f be PartFunc of (product G),F;
let x be Point of (product G);
func partdiff (f,x,i) -> Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F)) equals :: NDIFF_5:def 7
diff ((f * (reproj ((modetrans (G,i)),x))),((proj (modetrans (G,i))) . x));
coherence
diff ((f * (reproj ((modetrans (G,i)),x))),((proj (modetrans (G,i))) . x)) is Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F)) ;
end;
:: deftheorem defines partdiff NDIFF_5:def_7_:_
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for i being set
for f being PartFunc of (product G),F
for x being Point of (product G) holds partdiff (f,x,i) = diff ((f * (reproj ((modetrans (G,i)),x))),((proj (modetrans (G,i))) . x));
begin
definition
let G be non-trivial RealNormSpace-Sequence;
let F be non trivial RealNormSpace;
let i be set ;
let f be PartFunc of (product G),F;
let X be set ;
predf is_partial_differentiable_on X,i means :Def8: :: NDIFF_5:def 8
( X c= dom f & ( for x being Point of (product G) st x in X holds
f | X is_partial_differentiable_in x,i ) );
end;
:: deftheorem Def8 defines is_partial_differentiable_on NDIFF_5:def_8_:_
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for i being set
for f being PartFunc of (product G),F
for X being set holds
( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Point of (product G) st x in X holds
f | X is_partial_differentiable_in x,i ) ) );
theorem Th21: :: NDIFF_5:21
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for xi being Element of (G . i) holds ||.((reproj (i,(0. (product G)))) . xi).|| = ||.xi.||
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for xi being Element of (G . i) holds ||.((reproj (i,(0. (product G)))) . xi).|| = ||.xi.||
let i be Element of dom G; ::_thesis: for xi being Element of (G . i) holds ||.((reproj (i,(0. (product G)))) . xi).|| = ||.xi.||
let xi be Element of (G . i); ::_thesis: ||.((reproj (i,(0. (product G)))) . xi).|| = ||.xi.||
set j = len G;
reconsider i0 = i as Element of NAT ;
Seg (len G) = dom G by FINSEQ_1:def_3;
then A1: ( 1 <= i0 & i0 <= len G ) by FINSEQ_1:1;
set z = 0. (product G);
A2: (reproj (i,(0. (product G)))) . xi = (0. (product G)) +* (i,xi) by Def4;
A3: the carrier of (product G) = product (carr G) by Th10;
then reconsider w = (0. (product G)) +* (i0,xi) as Element of product (carr G) by Th11;
A4: ||.((reproj (i,(0. (product G)))) . xi).|| = |.(normsequence (G,w)).| by A2, PRVECT_2:7;
set q = ||.xi.||;
set q1 = <*||.xi.||*>;
set y = 0* (len G);
A5: len (normsequence (G,w)) = len G by PRVECT_2:def_11;
A6: len (0* (len G)) = len G by CARD_1:def_7;
then A7: (((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0) = Replace ((0* (len G)),i0,||.xi.||) by A1, FINSEQ_7:def_1;
then A8: len ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) = len (0* (len G)) by FINSEQ_7:5;
A9: len (0* (len G)) = len (Replace ((0* (len G)),i0,||.xi.||)) by FINSEQ_7:5;
for k being Nat st 1 <= k & k <= len (normsequence (G,w)) holds
(normsequence (G,w)) . k = ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (normsequence (G,w)) implies (normsequence (G,w)) . k = ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) . k )
assume A10: ( 1 <= k & k <= len (normsequence (G,w)) ) ; ::_thesis: (normsequence (G,w)) . k = ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) . k
then reconsider k1 = k as Element of dom G by A5, FINSEQ_3:25;
A11: k1 in dom G ;
0. (product G) in the carrier of (product G) ;
then 0. (product G) in product (carr G) by Th10;
then consider g being Function such that
A12: ( 0. (product G) = g & dom g = dom (carr G) & ( for y being set st y in dom (carr G) holds
g . y in (carr G) . y ) ) by CARD_3:def_5;
A13: k in dom (0. (product G)) by A11, A12, Lm1;
A14: (normsequence (G,w)) . k = the normF of (G . k1) . (w . k1) by PRVECT_2:def_11;
percases ( k = i0 or k <> i0 ) ;
supposeA15: k = i0 ; ::_thesis: (normsequence (G,w)) . k = ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) . k
then A16: (normsequence (G,w)) . k = ||.xi.|| by A14, A13, FUNCT_7:31;
(Replace ((0* (len G)),i0,||.xi.||)) /. k = ||.xi.|| by A15, A10, A5, A6, FINSEQ_7:8;
hence (normsequence (G,w)) . k = ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) . k by A16, A7, A10, A5, A6, A9, FINSEQ_4:15; ::_thesis: verum
end;
supposeA17: k <> i0 ; ::_thesis: (normsequence (G,w)) . k = ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) . k
then w . k1 = (0. (product G)) . k1 by FUNCT_7:32;
then A18: (normsequence (G,w)) . k = ||.(0. (G . k1)).|| by A14, Th14, A3;
(Replace ((0* (len G)),i0,||.xi.||)) /. k = (0* (len G)) /. k by A17, A10, A5, A6, FINSEQ_7:10;
then (Replace ((0* (len G)),i0,||.xi.||)) . k = (0* (len G)) /. k by A10, A5, A6, A9, FINSEQ_4:15;
then (Replace ((0* (len G)),i0,||.xi.||)) . k = (0* (len G)) . k by A10, A5, A6, FINSEQ_4:15;
hence (normsequence (G,w)) . k = ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) . k by A18, A7; ::_thesis: verum
end;
end;
end;
then A19: normsequence (G,w) = (((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0) by A5, A6, A8, FINSEQ_1:14;
sqrt (Sum (sqr ((0* (len G)) | (i0 -' 1)))) = |.(0* (i0 -' 1)).| by A1, PDIFF_7:2;
then sqrt (Sum (sqr ((0* (len G)) | (i0 -' 1)))) = 0 by EUCLID:7;
then A20: Sum (sqr ((0* (len G)) | (i0 -' 1))) = 0 by RVSUM_1:86, SQUARE_1:24;
sqrt (Sum (sqr ((0* (len G)) /^ i0))) = |.(0* ((len G) -' i0)).| by PDIFF_7:3;
then A21: sqrt (Sum (sqr ((0* (len G)) /^ i0))) = 0 by EUCLID:7;
sqr ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0)) = (sqr (((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>)) ^ (sqr ((0* (len G)) /^ i0)) by RVSUM_1:144
.= ((sqr ((0* (len G)) | (i0 -' 1))) ^ (sqr <*||.xi.||*>)) ^ (sqr ((0* (len G)) /^ i0)) by RVSUM_1:144
.= ((sqr ((0* (len G)) | (i0 -' 1))) ^ <*(||.xi.|| ^2)*>) ^ (sqr ((0* (len G)) /^ i0)) by RVSUM_1:55 ;
then Sum (sqr ((((0* (len G)) | (i0 -' 1)) ^ <*||.xi.||*>) ^ ((0* (len G)) /^ i0))) = (Sum ((sqr ((0* (len G)) | (i0 -' 1))) ^ <*(||.xi.|| ^2)*>)) + (Sum (sqr ((0* (len G)) /^ i0))) by RVSUM_1:75
.= ((Sum (sqr ((0* (len G)) | (i0 -' 1)))) + (||.xi.|| ^2)) + (Sum (sqr ((0* (len G)) /^ i0))) by RVSUM_1:74
.= ||.xi.|| ^2 by A20, A21, RVSUM_1:86, SQUARE_1:24 ;
then ||.((reproj (i,(0. (product G)))) . xi).|| = |.||.xi.||.| by A19, A4, COMPLEX1:72;
hence ||.((reproj (i,(0. (product G)))) . xi).|| = ||.xi.|| by COMPLEX1:43; ::_thesis: verum
end;
theorem Th22: :: NDIFF_5:22
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x being Point of (product G)
for r being Point of (G . i) holds
( ((reproj (i,x)) . r) - x = (reproj (i,(0. (product G)))) . (r - ((proj i) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0. (product G)))) . (((proj i) . x) - r) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x being Point of (product G)
for r being Point of (G . i) holds
( ((reproj (i,x)) . r) - x = (reproj (i,(0. (product G)))) . (r - ((proj i) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0. (product G)))) . (((proj i) . x) - r) )
let i be Element of dom G; ::_thesis: for x being Point of (product G)
for r being Point of (G . i) holds
( ((reproj (i,x)) . r) - x = (reproj (i,(0. (product G)))) . (r - ((proj i) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0. (product G)))) . (((proj i) . x) - r) )
let x be Point of (product G); ::_thesis: for r being Point of (G . i) holds
( ((reproj (i,x)) . r) - x = (reproj (i,(0. (product G)))) . (r - ((proj i) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0. (product G)))) . (((proj i) . x) - r) )
let r be Point of (G . i); ::_thesis: ( ((reproj (i,x)) . r) - x = (reproj (i,(0. (product G)))) . (r - ((proj i) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0. (product G)))) . (((proj i) . x) - r) )
set m = len G;
reconsider xf = x as Element of product (carr G) by Th10;
A1: dom (carr G) = dom G by Lm1;
reconsider Zr = 0. (product G) as Element of product (carr G) by Th10;
reconsider ixr = (reproj (i,x)) . r as Element of product (carr G) by Th10;
reconsider p = ((reproj (i,x)) . r) - x as Element of product (carr G) by Th10;
A2: dom p = dom (carr G) by CARD_3:9;
reconsider q = (reproj (i,(0. (product G)))) . (r - ((proj i) . x)) as Element of product (carr G) by Th10;
A3: dom q = dom (carr G) by CARD_3:9;
reconsider s = x - ((reproj (i,x)) . r) as Element of product (carr G) by Th10;
A4: dom s = dom (carr G) by CARD_3:9;
reconsider t = (reproj (i,(0. (product G)))) . (((proj i) . x) - r) as Element of product (carr G) by Th10;
A5: dom t = dom (carr G) by CARD_3:9;
A6: (reproj (i,x)) . r = x +* (i,r) by Def4;
reconsider xfi = xf . i as Point of (G . i) ;
(reproj (i,(0. (product G)))) . (r - ((proj i) . x)) = (0. (product G)) +* (i,(r - ((proj i) . x))) by Def4;
then A7: q = Zr +* (i,(r - xfi)) by Def3;
(reproj (i,(0. (product G)))) . (((proj i) . x) - r) = (0. (product G)) +* (i,(((proj i) . x) - r)) by Def4;
then A8: t = Zr +* (i,(xfi - r)) by Def3;
set ir = i .--> r;
set irx1 = i .--> (r - xfi);
set irx2 = i .--> (xfi - r);
x in the carrier of (product G) ;
then A9: x in product (carr G) by Th10;
consider g1 being Function such that
A10: ( x = g1 & dom g1 = dom (carr G) & ( for i being set st i in dom (carr G) holds
g1 . i in (carr G) . i ) ) by A9, CARD_3:def_5;
for k being set st k in dom p holds
p . k = q . k
proof
let k be set ; ::_thesis: ( k in dom p implies p . k = q . k )
assume A11: k in dom p ; ::_thesis: p . k = q . k
then reconsider k0 = k as Element of dom G by A1, CARD_3:9;
consider g being Function such that
A12: ( Zr = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
A13: k in dom Zr by A12, A11, CARD_3:9;
A14: k in dom x by A10, A11, CARD_3:9;
percases ( not k in {i} or k in {i} ) ;
suppose not k in {i} ; ::_thesis: p . k = q . k
then A15: k <> i by TARSKI:def_1;
then A16: q . k0 = Zr . k0 by A7, FUNCT_7:32;
p . k = (ixr . k0) - (xf . k0) by Th15
.= (xf . k0) - (xf . k0) by A15, A6, FUNCT_7:32 ;
then p . k = 0. (G . k0) by RLVECT_1:15;
hence p . k = q . k by A16, Th14; ::_thesis: verum
end;
suppose k in {i} ; ::_thesis: p . k = q . k
then A17: k = i by TARSKI:def_1;
then A18: q . k0 = r - xfi by A7, A13, FUNCT_7:31;
p . k = (ixr . k0) - (xf . k0) by Th15;
hence p . k = q . k by A18, A6, A17, A14, FUNCT_7:31; ::_thesis: verum
end;
end;
end;
hence ((reproj (i,x)) . r) - x = (reproj (i,(0. (product G)))) . (r - ((proj i) . x)) by A2, A3, FUNCT_1:2; ::_thesis: x - ((reproj (i,x)) . r) = (reproj (i,(0. (product G)))) . (((proj i) . x) - r)
for k being set st k in dom s holds
s . k = t . k
proof
let k be set ; ::_thesis: ( k in dom s implies s . k = t . k )
assume A19: k in dom s ; ::_thesis: s . k = t . k
then reconsider k0 = k as Element of dom G by A1, CARD_3:9;
consider g being Function such that
A20: ( Zr = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
A21: k in dom Zr by A20, A19, CARD_3:9;
A22: k in dom x by A10, A19, CARD_3:9;
percases ( not k in {i} or k in {i} ) ;
suppose not k in {i} ; ::_thesis: s . k = t . k
then A23: k <> i by TARSKI:def_1;
then A24: t . k0 = Zr . k0 by A8, FUNCT_7:32;
s . k = (xf . k0) - (ixr . k0) by Th15
.= (xf . k0) - (xf . k0) by A6, A23, FUNCT_7:32 ;
then s . k = 0. (G . k0) by RLVECT_1:15;
hence s . k = t . k by A24, Th14; ::_thesis: verum
end;
suppose k in {i} ; ::_thesis: s . k = t . k
then A25: k = i by TARSKI:def_1;
then A26: t . k0 = xfi - r by A8, A21, FUNCT_7:31;
s . k = (xf . k0) - (ixr . k0) by Th15;
hence s . k = t . k by A26, A6, A25, A22, FUNCT_7:31; ::_thesis: verum
end;
end;
end;
hence x - ((reproj (i,x)) . r) = (reproj (i,(0. (product G)))) . (((proj i) . x) - r) by A4, A5, FUNCT_1:2; ::_thesis: verum
end;
theorem Th23: :: NDIFF_5:23
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x being Point of (product G)
for Z being Subset of (product G) st Z is open & x in Z holds
ex N being Neighbourhood of (proj i) . x st
for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x being Point of (product G)
for Z being Subset of (product G) st Z is open & x in Z holds
ex N being Neighbourhood of (proj i) . x st
for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z
let i be Element of dom G; ::_thesis: for x being Point of (product G)
for Z being Subset of (product G) st Z is open & x in Z holds
ex N being Neighbourhood of (proj i) . x st
for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z
let x be Point of (product G); ::_thesis: for Z being Subset of (product G) st Z is open & x in Z holds
ex N being Neighbourhood of (proj i) . x st
for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z
let Z be Subset of (product G); ::_thesis: ( Z is open & x in Z implies ex N being Neighbourhood of (proj i) . x st
for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z )
assume ( Z is open & x in Z ) ; ::_thesis: ex N being Neighbourhood of (proj i) . x st
for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z
then consider r being Real such that
A1: ( 0 < r & { y where y is Point of (product G) : ||.(y - x).|| < r } c= Z ) by NDIFF_1:3;
set N = { y where y is Point of (G . i) : ||.(y - ((proj i) . x)).|| < r } ;
reconsider N = { y where y is Point of (G . i) : ||.(y - ((proj i) . x)).|| < r } as Neighbourhood of (proj i) . x by A1, NFCONT_1:3;
take N ; ::_thesis: for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z
thus for z being Point of (G . i) st z in N holds
(reproj (i,x)) . z in Z ::_thesis: verum
proof
let z be Point of (G . i); ::_thesis: ( z in N implies (reproj (i,x)) . z in Z )
assume z in N ; ::_thesis: (reproj (i,x)) . z in Z
then A2: ex y being Point of (G . i) st
( y = z & ||.(y - ((proj i) . x)).|| < r ) ;
||.(((reproj (i,x)) . z) - x).|| = ||.((reproj (i,(0. (product G)))) . (z - ((proj i) . x))).|| by Th22
.= ||.(z - ((proj i) . x)).|| by Th21 ;
then (reproj (i,x)) . z in { y where y is Point of (product G) : ||.(y - x).|| < r } by A2;
hence (reproj (i,x)) . z in Z by A1; ::_thesis: verum
end;
end;
theorem Th24: :: NDIFF_5:24
for G being non-trivial RealNormSpace-Sequence
for T being non trivial RealNormSpace
for i being set
for f being PartFunc of (product G),T
for Z being Subset of (product G) st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for T being non trivial RealNormSpace
for i being set
for f being PartFunc of (product G),T
for Z being Subset of (product G) st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let T be non trivial RealNormSpace; ::_thesis: for i being set
for f being PartFunc of (product G),T
for Z being Subset of (product G) st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let i be set ; ::_thesis: for f being PartFunc of (product G),T
for Z being Subset of (product G) st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let f be PartFunc of (product G),T; ::_thesis: for Z being Subset of (product G) st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let Z be Subset of (product G); ::_thesis: ( Z is open implies ( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) ) )
assume A1: Z is open ; ::_thesis: ( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
set i0 = modetrans (G,i);
set S = G . (modetrans (G,i));
set RNS = R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),T);
thus ( f is_partial_differentiable_on Z,i implies ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) ) ::_thesis: ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) implies f is_partial_differentiable_on Z,i )
proof
assume A2: f is_partial_differentiable_on Z,i ; ::_thesis: ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) )
hence A3: Z c= dom f by Def8; ::_thesis: for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i
let nx0 be Point of (product G); ::_thesis: ( nx0 in Z implies f is_partial_differentiable_in nx0,i )
reconsider x0 = (proj (modetrans (G,i))) . nx0 as Point of (G . (modetrans (G,i))) ;
assume A4: nx0 in Z ; ::_thesis: f is_partial_differentiable_in nx0,i
then f | Z is_partial_differentiable_in nx0,i by A2, Def8;
then (f | Z) * (reproj ((modetrans (G,i)),nx0)) is_differentiable_in x0 by Def6;
then consider N0 being Neighbourhood of x0 such that
A5: N0 c= dom ((f | Z) * (reproj ((modetrans (G,i)),nx0))) and
A6: ex L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),T)) ex R being RestFunc of (G . (modetrans (G,i))),T st
for x being Point of (G . (modetrans (G,i))) st x in N0 holds
(((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by NDIFF_1:def_6;
consider L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),T)), R being RestFunc of (G . (modetrans (G,i))),T such that
A7: for x being Point of (G . (modetrans (G,i))) st x in N0 holds
(((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A6;
consider N1 being Neighbourhood of x0 such that
A8: for x being Point of (G . (modetrans (G,i))) st x in N1 holds
(reproj ((modetrans (G,i)),nx0)) . x in Z by A1, A4, Th23;
A9: now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i)))_st_x_in_N1_holds_
(reproj_((modetrans_(G,i)),nx0))_._x_in_dom_(f_|_Z)
let x be Point of (G . (modetrans (G,i))); ::_thesis: ( x in N1 implies (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z) )
assume x in N1 ; ::_thesis: (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z)
then (reproj ((modetrans (G,i)),nx0)) . x in Z by A8;
then (reproj ((modetrans (G,i)),nx0)) . x in (dom f) /\ Z by A3, XBOOLE_0:def_4;
hence (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z) by RELAT_1:61; ::_thesis: verum
end;
reconsider N = N0 /\ N1 as Neighbourhood of x0 by Th8;
(f | Z) * (reproj ((modetrans (G,i)),nx0)) c= f * (reproj ((modetrans (G,i)),nx0)) by RELAT_1:29, RELAT_1:59;
then A10: dom ((f | Z) * (reproj ((modetrans (G,i)),nx0))) c= dom (f * (reproj ((modetrans (G,i)),nx0))) by RELAT_1:11;
N c= N0 by XBOOLE_1:17;
then N c= dom ((f | Z) * (reproj ((modetrans (G,i)),nx0))) by A5, XBOOLE_1:1;
then A11: N c= dom (f * (reproj ((modetrans (G,i)),nx0))) by A10, XBOOLE_1:1;
now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i)))_st_x_in_N_holds_
((f_*_(reproj_((modetrans_(G,i)),nx0)))_/._x)_-_((f_*_(reproj_((modetrans_(G,i)),nx0)))_/._x0)_=_(L_._(x_-_x0))_+_(R_/._(x_-_x0))
let x be Point of (G . (modetrans (G,i))); ::_thesis: ( x in N implies ((f * (reproj ((modetrans (G,i)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
assume A12: x in N ; ::_thesis: ((f * (reproj ((modetrans (G,i)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0))
then A13: x in N0 by XBOOLE_0:def_4;
A14: dom (reproj ((modetrans (G,i)),nx0)) = the carrier of (G . (modetrans (G,i))) by FUNCT_2:def_1;
x in N1 by A12, XBOOLE_0:def_4;
then A15: (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z) by A9;
then A16: ( (reproj ((modetrans (G,i)),nx0)) . x in dom f & (reproj ((modetrans (G,i)),nx0)) . x in Z ) by RELAT_1:57;
A17: (reproj ((modetrans (G,i)),nx0)) . x0 in dom (f | Z) by A9, NFCONT_1:4;
then A18: ( (reproj ((modetrans (G,i)),nx0)) . x0 in dom f & (reproj ((modetrans (G,i)),nx0)) . x0 in Z ) by RELAT_1:57;
A19: ((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x = (f | Z) /. ((reproj ((modetrans (G,i)),nx0)) /. x) by A15, A14, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i)),nx0)) /. x) by A16, PARTFUN2:17
.= (f * (reproj ((modetrans (G,i)),nx0))) /. x by A14, A16, PARTFUN2:4 ;
((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x0 = (f | Z) /. ((reproj ((modetrans (G,i)),nx0)) /. x0) by A14, A17, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i)),nx0)) /. x0) by A18, PARTFUN2:17
.= (f * (reproj ((modetrans (G,i)),nx0))) /. x0 by A14, A18, PARTFUN2:4 ;
hence ((f * (reproj ((modetrans (G,i)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A7, A13, A19; ::_thesis: verum
end;
then f * (reproj ((modetrans (G,i)),nx0)) is_differentiable_in x0 by A11, NDIFF_1:def_6;
hence f is_partial_differentiable_in nx0,i by Def6; ::_thesis: verum
end;
assume that
A20: Z c= dom f and
A21: for nx being Point of (product G) st nx in Z holds
f is_partial_differentiable_in nx,i ; ::_thesis: f is_partial_differentiable_on Z,i
now__::_thesis:_for_nx0_being_Point_of_(product_G)_st_nx0_in_Z_holds_
f_|_Z_is_partial_differentiable_in_nx0,i
let nx0 be Point of (product G); ::_thesis: ( nx0 in Z implies f | Z is_partial_differentiable_in nx0,i )
assume A22: nx0 in Z ; ::_thesis: f | Z is_partial_differentiable_in nx0,i
then A23: f is_partial_differentiable_in nx0,i by A21;
reconsider x0 = (proj (modetrans (G,i))) . nx0 as Point of (G . (modetrans (G,i))) ;
f * (reproj ((modetrans (G,i)),nx0)) is_differentiable_in x0 by A23, Def6;
then consider N0 being Neighbourhood of x0 such that
N0 c= dom (f * (reproj ((modetrans (G,i)),nx0))) and
A24: ex L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),T)) ex R being RestFunc of (G . (modetrans (G,i))),T st
for x being Point of (G . (modetrans (G,i))) st x in N0 holds
((f * (reproj ((modetrans (G,i)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by NDIFF_1:def_6;
consider N1 being Neighbourhood of x0 such that
A25: for x being Point of (G . (modetrans (G,i))) st x in N1 holds
(reproj ((modetrans (G,i)),nx0)) . x in Z by A1, A22, Th23;
A26: now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i)))_st_x_in_N1_holds_
(reproj_((modetrans_(G,i)),nx0))_._x_in_dom_(f_|_Z)
let x be Point of (G . (modetrans (G,i))); ::_thesis: ( x in N1 implies (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z) )
assume x in N1 ; ::_thesis: (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z)
then (reproj ((modetrans (G,i)),nx0)) . x in Z by A25;
then (reproj ((modetrans (G,i)),nx0)) . x in (dom f) /\ Z by A20, XBOOLE_0:def_4;
hence (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z) by RELAT_1:61; ::_thesis: verum
end;
A27: N1 c= dom ((f | Z) * (reproj ((modetrans (G,i)),nx0)))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in N1 or z in dom ((f | Z) * (reproj ((modetrans (G,i)),nx0))) )
assume A28: z in N1 ; ::_thesis: z in dom ((f | Z) * (reproj ((modetrans (G,i)),nx0)))
then A29: z in the carrier of (G . (modetrans (G,i))) ;
reconsider x = z as Point of (G . (modetrans (G,i))) by A28;
A30: (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z) by A28, A26;
z in dom (reproj ((modetrans (G,i)),nx0)) by A29, FUNCT_2:def_1;
hence z in dom ((f | Z) * (reproj ((modetrans (G,i)),nx0))) by A30, FUNCT_1:11; ::_thesis: verum
end;
reconsider N = N0 /\ N1 as Neighbourhood of x0 by Th8;
N c= N1 by XBOOLE_1:17;
then A31: N c= dom ((f | Z) * (reproj ((modetrans (G,i)),nx0))) by A27, XBOOLE_1:1;
consider L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),T)), R being RestFunc of (G . (modetrans (G,i))),T such that
A32: for x being Point of (G . (modetrans (G,i))) st x in N0 holds
((f * (reproj ((modetrans (G,i)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A24;
now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i)))_st_x_in_N_holds_
(((f_|_Z)_*_(reproj_((modetrans_(G,i)),nx0)))_/._x)_-_(((f_|_Z)_*_(reproj_((modetrans_(G,i)),nx0)))_/._x0)_=_(L_._(x_-_x0))_+_(R_/._(x_-_x0))
let x be Point of (G . (modetrans (G,i))); ::_thesis: ( x in N implies (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
assume A33: x in N ; ::_thesis: (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0))
then A34: x in N0 by XBOOLE_0:def_4;
A35: dom (reproj ((modetrans (G,i)),nx0)) = the carrier of (G . (modetrans (G,i))) by FUNCT_2:def_1;
x in N1 by A33, XBOOLE_0:def_4;
then A36: (reproj ((modetrans (G,i)),nx0)) . x in dom (f | Z) by A26;
then A37: (reproj ((modetrans (G,i)),nx0)) . x in (dom f) /\ Z by RELAT_1:61;
then A38: (reproj ((modetrans (G,i)),nx0)) . x in dom f by XBOOLE_0:def_4;
A39: (reproj ((modetrans (G,i)),nx0)) . x0 in dom (f | Z) by A26, NFCONT_1:4;
then A40: (reproj ((modetrans (G,i)),nx0)) . x0 in (dom f) /\ Z by RELAT_1:61;
then A41: (reproj ((modetrans (G,i)),nx0)) . x0 in dom f by XBOOLE_0:def_4;
A42: ((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x = (f | Z) /. ((reproj ((modetrans (G,i)),nx0)) /. x) by A36, A35, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i)),nx0)) /. x) by A37, PARTFUN2:16
.= (f * (reproj ((modetrans (G,i)),nx0))) /. x by A35, A38, PARTFUN2:4 ;
((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x0 = (f | Z) /. ((reproj ((modetrans (G,i)),nx0)) /. x0) by A35, A39, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i)),nx0)) /. x0) by A40, PARTFUN2:16
.= (f * (reproj ((modetrans (G,i)),nx0))) /. x0 by A35, A41, PARTFUN2:4 ;
hence (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A42, A34, A32; ::_thesis: verum
end;
then (f | Z) * (reproj ((modetrans (G,i)),nx0)) is_differentiable_in x0 by A31, NDIFF_1:def_6;
hence f | Z is_partial_differentiable_in nx0,i by Def6; ::_thesis: verum
end;
hence f is_partial_differentiable_on Z,i by A20, Def8; ::_thesis: verum
end;
theorem Th25: :: NDIFF_5:25
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for X, i being set st i in dom G & f is_partial_differentiable_on X,i holds
X is Subset of (product G)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for X, i being set st i in dom G & f is_partial_differentiable_on X,i holds
X is Subset of (product G)
let F be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),F
for X, i being set st i in dom G & f is_partial_differentiable_on X,i holds
X is Subset of (product G)
let f be PartFunc of (product G),F; ::_thesis: for X, i being set st i in dom G & f is_partial_differentiable_on X,i holds
X is Subset of (product G)
let X, i be set ; ::_thesis: ( i in dom G & f is_partial_differentiable_on X,i implies X is Subset of (product G) )
assume ( i in dom G & f is_partial_differentiable_on X,i ) ; ::_thesis: X is Subset of (product G)
then X c= dom f by Def8;
hence X is Subset of (product G) by XBOOLE_1:1; ::_thesis: verum
end;
definition
let G be non-trivial RealNormSpace-Sequence;
let S be non trivial RealNormSpace;
let i be set ;
assume A1: i in dom G ;
let f be PartFunc of (product G),S;
let X be set ;
assume A2: f is_partial_differentiable_on X,i ;
funcf `partial| (X,i) -> PartFunc of (product G),(R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),S)) means :Def9: :: NDIFF_5:def 9
( dom it = X & ( for x being Point of (product G) st x in X holds
it /. x = partdiff (f,x,i) ) );
existence
ex b1 being PartFunc of (product G),(R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),S)) st
( dom b1 = X & ( for x being Point of (product G) st x in X holds
b1 /. x = partdiff (f,x,i) ) )
proof
deffunc H1( Element of (product G)) -> Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),S)) = partdiff (f,$1,i);
defpred S1[ Element of (product G)] means $1 in X;
consider F being PartFunc of (product G),(R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),S)) such that
A3: ( ( for x being Point of (product G) holds
( x in dom F iff S1[x] ) ) & ( for x being Point of (product G) st x in dom F holds
F . x = H1(x) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = X & ( for x being Point of (product G) st x in X holds
F /. x = partdiff (f,x,i) ) )
now__::_thesis:_for_y_being_set_st_y_in_X_holds_
y_in_dom_F
A4: X is Subset of (product G) by A2, A1, Th25;
let y be set ; ::_thesis: ( y in X implies y in dom F )
assume y in X ; ::_thesis: y in dom F
hence y in dom F by A3, A4; ::_thesis: verum
end;
then A5: X c= dom F by TARSKI:def_3;
for y being set st y in dom F holds
y in X by A3;
then dom F c= X by TARSKI:def_3;
hence dom F = X by A5, XBOOLE_0:def_10; ::_thesis: for x being Point of (product G) st x in X holds
F /. x = partdiff (f,x,i)
hereby ::_thesis: verum
let x be Point of (product G); ::_thesis: ( x in X implies F /. x = partdiff (f,x,i) )
assume x in X ; ::_thesis: F /. x = partdiff (f,x,i)
then A6: x in dom F by A3;
then F . x = partdiff (f,x,i) by A3;
hence F /. x = partdiff (f,x,i) by A6, PARTFUN1:def_6; ::_thesis: verum
end;
end;
uniqueness
for b1, b2 being PartFunc of (product G),(R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),S)) st dom b1 = X & ( for x being Point of (product G) st x in X holds
b1 /. x = partdiff (f,x,i) ) & dom b2 = X & ( for x being Point of (product G) st x in X holds
b2 /. x = partdiff (f,x,i) ) holds
b1 = b2
proof
let F, H be PartFunc of (product G),(R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),S)); ::_thesis: ( dom F = X & ( for x being Point of (product G) st x in X holds
F /. x = partdiff (f,x,i) ) & dom H = X & ( for x being Point of (product G) st x in X holds
H /. x = partdiff (f,x,i) ) implies F = H )
assume that
A7: dom F = X and
A8: for x being Point of (product G) st x in X holds
F /. x = partdiff (f,x,i) and
A9: dom H = X and
A10: for x being Point of (product G) st x in X holds
H /. x = partdiff (f,x,i) ; ::_thesis: F = H
now__::_thesis:_for_x_being_Point_of_(product_G)_st_x_in_dom_F_holds_
F_/._x_=_H_/._x
let x be Point of (product G); ::_thesis: ( x in dom F implies F /. x = H /. x )
assume A11: x in dom F ; ::_thesis: F /. x = H /. x
then F /. x = partdiff (f,x,i) by A7, A8;
hence F /. x = H /. x by A7, A10, A11; ::_thesis: verum
end;
hence F = H by A7, A9, PARTFUN2:1; ::_thesis: verum
end;
end;
:: deftheorem Def9 defines `partial| NDIFF_5:def_9_:_
for G being non-trivial RealNormSpace-Sequence
for S being non trivial RealNormSpace
for i being set st i in dom G holds
for f being PartFunc of (product G),S
for X being set st f is_partial_differentiable_on X,i holds
for b6 being PartFunc of (product G),(R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),S)) holds
( b6 = f `partial| (X,i) iff ( dom b6 = X & ( for x being Point of (product G) st x in X holds
b6 /. x = partdiff (f,x,i) ) ) );
theorem Th26: :: NDIFF_5:26
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G holds
( (f1 + f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) + (f2 * (reproj ((modetrans (G,i)),x))) & (f1 - f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) - (f2 * (reproj ((modetrans (G,i)),x))) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G holds
( (f1 + f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) + (f2 * (reproj ((modetrans (G,i)),x))) & (f1 - f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) - (f2 * (reproj ((modetrans (G,i)),x))) )
let F be non trivial RealNormSpace; ::_thesis: for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G holds
( (f1 + f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) + (f2 * (reproj ((modetrans (G,i)),x))) & (f1 - f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) - (f2 * (reproj ((modetrans (G,i)),x))) )
let f1, f2 be PartFunc of (product G),F; ::_thesis: for x being Point of (product G)
for i being set st i in dom G holds
( (f1 + f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) + (f2 * (reproj ((modetrans (G,i)),x))) & (f1 - f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) - (f2 * (reproj ((modetrans (G,i)),x))) )
let x be Point of (product G); ::_thesis: for i being set st i in dom G holds
( (f1 + f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) + (f2 * (reproj ((modetrans (G,i)),x))) & (f1 - f2) * (reproj ((modetrans (G,i)),x)) = (f1 * (reproj ((modetrans (G,i)),x))) - (f2 * (reproj ((modetrans (G,i)),x))) )
let i0 be set ; ::_thesis: ( i0 in dom G implies ( (f1 + f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x))) & (f1 - f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x))) ) )
assume i0 in dom G ; ::_thesis: ( (f1 + f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x))) & (f1 - f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x))) )
set i = modetrans (G,i0);
A1: dom (reproj ((modetrans (G,i0)),x)) = the carrier of (G . (modetrans (G,i0))) by FUNCT_2:def_1;
A2: dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
for s being Element of (G . (modetrans (G,i0))) holds
( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) )
proof
let s be Element of (G . (modetrans (G,i0))); ::_thesis: ( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) )
( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff (reproj ((modetrans (G,i0)),x)) . s in (dom f1) /\ (dom f2) ) by A2, A1, FUNCT_1:11;
then ( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff ( (reproj ((modetrans (G,i0)),x)) . s in dom f1 & (reproj ((modetrans (G,i0)),x)) . s in dom f2 ) ) by XBOOLE_0:def_4;
then ( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff ( s in dom (f1 * (reproj ((modetrans (G,i0)),x))) & s in dom (f2 * (reproj ((modetrans (G,i0)),x))) ) ) by A1, FUNCT_1:11;
then ( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff s in (dom (f1 * (reproj ((modetrans (G,i0)),x)))) /\ (dom (f2 * (reproj ((modetrans (G,i0)),x)))) ) by XBOOLE_0:def_4;
hence ( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) ) by VFUNCT_1:def_1; ::_thesis: verum
end;
then for s being set holds
( s in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) ) ;
then A3: dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) = dom ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) by TARSKI:1;
A4: for z being Element of (G . (modetrans (G,i0))) st z in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) holds
((f1 + f2) * (reproj ((modetrans (G,i0)),x))) . z = ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) . z
proof
let z be Element of (G . (modetrans (G,i0))); ::_thesis: ( z in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) implies ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) . z = ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) . z )
assume A5: z in dom ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) ; ::_thesis: ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) . z = ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) . z
then A6: (reproj ((modetrans (G,i0)),x)) . z in dom (f1 + f2) by FUNCT_1:11;
z in (dom (f1 * (reproj ((modetrans (G,i0)),x)))) /\ (dom (f2 * (reproj ((modetrans (G,i0)),x)))) by A3, A5, VFUNCT_1:def_1;
then A7: ( z in dom (f1 * (reproj ((modetrans (G,i0)),x))) & z in dom (f2 * (reproj ((modetrans (G,i0)),x))) ) by XBOOLE_0:def_4;
A8: (reproj ((modetrans (G,i0)),x)) . z in (dom f1) /\ (dom f2) by A2, A5, FUNCT_1:11;
then (reproj ((modetrans (G,i0)),x)) . z in dom f1 by XBOOLE_0:def_4;
then A9: f1 /. ((reproj ((modetrans (G,i0)),x)) . z) = f1 . ((reproj ((modetrans (G,i0)),x)) . z) by PARTFUN1:def_6
.= (f1 * (reproj ((modetrans (G,i0)),x))) . z by A7, FUNCT_1:12
.= (f1 * (reproj ((modetrans (G,i0)),x))) /. z by A7, PARTFUN1:def_6 ;
(reproj ((modetrans (G,i0)),x)) . z in dom f2 by A8, XBOOLE_0:def_4;
then A10: f2 /. ((reproj ((modetrans (G,i0)),x)) . z) = f2 . ((reproj ((modetrans (G,i0)),x)) . z) by PARTFUN1:def_6
.= (f2 * (reproj ((modetrans (G,i0)),x))) . z by A7, FUNCT_1:12
.= (f2 * (reproj ((modetrans (G,i0)),x))) /. z by A7, PARTFUN1:def_6 ;
((f1 + f2) * (reproj ((modetrans (G,i0)),x))) . z = (f1 + f2) . ((reproj ((modetrans (G,i0)),x)) . z) by A5, FUNCT_1:12
.= (f1 + f2) /. ((reproj ((modetrans (G,i0)),x)) . z) by A6, PARTFUN1:def_6
.= (f1 /. ((reproj ((modetrans (G,i0)),x)) . z)) + (f2 /. ((reproj ((modetrans (G,i0)),x)) . z)) by A6, VFUNCT_1:def_1
.= ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) /. z by A3, A5, A9, A10, VFUNCT_1:def_1 ;
hence ((f1 + f2) * (reproj ((modetrans (G,i0)),x))) . z = ((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))) . z by A3, A5, PARTFUN1:def_6; ::_thesis: verum
end;
A11: dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
for s being Element of (G . (modetrans (G,i0))) holds
( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) )
proof
let s be Element of (G . (modetrans (G,i0))); ::_thesis: ( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) )
( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff (reproj ((modetrans (G,i0)),x)) . s in (dom f1) /\ (dom f2) ) by A11, A1, FUNCT_1:11;
then ( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff ( (reproj ((modetrans (G,i0)),x)) . s in dom f1 & (reproj ((modetrans (G,i0)),x)) . s in dom f2 ) ) by XBOOLE_0:def_4;
then ( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff ( s in dom (f1 * (reproj ((modetrans (G,i0)),x))) & s in dom (f2 * (reproj ((modetrans (G,i0)),x))) ) ) by A1, FUNCT_1:11;
then ( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff s in (dom (f1 * (reproj ((modetrans (G,i0)),x)))) /\ (dom (f2 * (reproj ((modetrans (G,i0)),x)))) ) by XBOOLE_0:def_4;
hence ( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) ) by VFUNCT_1:def_2; ::_thesis: verum
end;
then for s being set holds
( s in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) iff s in dom ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) ) ;
then A12: dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) = dom ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) by TARSKI:1;
for z being Element of (G . (modetrans (G,i0))) st z in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) holds
((f1 - f2) * (reproj ((modetrans (G,i0)),x))) . z = ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) . z
proof
let z be Element of (G . (modetrans (G,i0))); ::_thesis: ( z in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) implies ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) . z = ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) . z )
assume A13: z in dom ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) ; ::_thesis: ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) . z = ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) . z
then A14: (reproj ((modetrans (G,i0)),x)) . z in dom (f1 - f2) by FUNCT_1:11;
z in (dom (f1 * (reproj ((modetrans (G,i0)),x)))) /\ (dom (f2 * (reproj ((modetrans (G,i0)),x)))) by A12, A13, VFUNCT_1:def_2;
then A15: ( z in dom (f1 * (reproj ((modetrans (G,i0)),x))) & z in dom (f2 * (reproj ((modetrans (G,i0)),x))) ) by XBOOLE_0:def_4;
A16: (reproj ((modetrans (G,i0)),x)) . z in (dom f1) /\ (dom f2) by A11, A13, FUNCT_1:11;
then (reproj ((modetrans (G,i0)),x)) . z in dom f1 by XBOOLE_0:def_4;
then A17: f1 /. ((reproj ((modetrans (G,i0)),x)) . z) = f1 . ((reproj ((modetrans (G,i0)),x)) . z) by PARTFUN1:def_6
.= (f1 * (reproj ((modetrans (G,i0)),x))) . z by A15, FUNCT_1:12
.= (f1 * (reproj ((modetrans (G,i0)),x))) /. z by A15, PARTFUN1:def_6 ;
(reproj ((modetrans (G,i0)),x)) . z in dom f2 by A16, XBOOLE_0:def_4;
then A18: f2 /. ((reproj ((modetrans (G,i0)),x)) . z) = f2 . ((reproj ((modetrans (G,i0)),x)) . z) by PARTFUN1:def_6
.= (f2 * (reproj ((modetrans (G,i0)),x))) . z by A15, FUNCT_1:12
.= (f2 * (reproj ((modetrans (G,i0)),x))) /. z by A15, PARTFUN1:def_6 ;
thus ((f1 - f2) * (reproj ((modetrans (G,i0)),x))) . z = (f1 - f2) . ((reproj ((modetrans (G,i0)),x)) . z) by A13, FUNCT_1:12
.= (f1 - f2) /. ((reproj ((modetrans (G,i0)),x)) . z) by A14, PARTFUN1:def_6
.= (f1 /. ((reproj ((modetrans (G,i0)),x)) . z)) - (f2 /. ((reproj ((modetrans (G,i0)),x)) . z)) by A14, VFUNCT_1:def_2
.= ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) /. z by A12, A13, A17, A18, VFUNCT_1:def_2
.= ((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))) . z by A12, A13, PARTFUN1:def_6 ; ::_thesis: verum
end;
hence ( (f1 + f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x))) & (f1 - f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x))) ) by A3, A12, A4, PARTFUN1:5; ::_thesis: verum
end;
theorem Th27: :: NDIFF_5:27
for r being Real
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G holds
r (#) (f * (reproj ((modetrans (G,i)),x))) = (r (#) f) * (reproj ((modetrans (G,i)),x))
proof
let r be Real; ::_thesis: for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G holds
r (#) (f * (reproj ((modetrans (G,i)),x))) = (r (#) f) * (reproj ((modetrans (G,i)),x))
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G holds
r (#) (f * (reproj ((modetrans (G,i)),x))) = (r (#) f) * (reproj ((modetrans (G,i)),x))
let F be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G holds
r (#) (f * (reproj ((modetrans (G,i)),x))) = (r (#) f) * (reproj ((modetrans (G,i)),x))
let f be PartFunc of (product G),F; ::_thesis: for x being Point of (product G)
for i being set st i in dom G holds
r (#) (f * (reproj ((modetrans (G,i)),x))) = (r (#) f) * (reproj ((modetrans (G,i)),x))
let x be Point of (product G); ::_thesis: for i being set st i in dom G holds
r (#) (f * (reproj ((modetrans (G,i)),x))) = (r (#) f) * (reproj ((modetrans (G,i)),x))
let i0 be set ; ::_thesis: ( i0 in dom G implies r (#) (f * (reproj ((modetrans (G,i0)),x))) = (r (#) f) * (reproj ((modetrans (G,i0)),x)) )
assume i0 in dom G ; ::_thesis: r (#) (f * (reproj ((modetrans (G,i0)),x))) = (r (#) f) * (reproj ((modetrans (G,i0)),x))
set i = modetrans (G,i0);
A1: dom (r (#) f) = dom f by VFUNCT_1:def_4;
A2: dom (r (#) (f * (reproj ((modetrans (G,i0)),x)))) = dom (f * (reproj ((modetrans (G,i0)),x))) by VFUNCT_1:def_4;
A3: dom (reproj ((modetrans (G,i0)),x)) = the carrier of (G . (modetrans (G,i0))) by FUNCT_2:def_1;
for s being Element of (G . (modetrans (G,i0))) holds
( s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) iff s in dom (f * (reproj ((modetrans (G,i0)),x))) )
proof
let s be Element of (G . (modetrans (G,i0))); ::_thesis: ( s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) iff s in dom (f * (reproj ((modetrans (G,i0)),x))) )
( s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) iff (reproj ((modetrans (G,i0)),x)) . s in dom (r (#) f) ) by A3, FUNCT_1:11;
hence ( s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) iff s in dom (f * (reproj ((modetrans (G,i0)),x))) ) by A1, A3, FUNCT_1:11; ::_thesis: verum
end;
then for s being set holds
( s in dom (r (#) (f * (reproj ((modetrans (G,i0)),x)))) iff s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) ) by A2;
then A4: dom (r (#) (f * (reproj ((modetrans (G,i0)),x)))) = dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) by TARSKI:1;
A5: for s being Element of (G . (modetrans (G,i0))) holds
( s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) iff (reproj ((modetrans (G,i0)),x)) . s in dom (r (#) f) )
proof
let s be Element of (G . (modetrans (G,i0))); ::_thesis: ( s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) iff (reproj ((modetrans (G,i0)),x)) . s in dom (r (#) f) )
dom (reproj ((modetrans (G,i0)),x)) = the carrier of (G . (modetrans (G,i0))) by FUNCT_2:def_1;
hence ( s in dom ((r (#) f) * (reproj ((modetrans (G,i0)),x))) iff (reproj ((modetrans (G,i0)),x)) . s in dom (r (#) f) ) by FUNCT_1:11; ::_thesis: verum
end;
for z being Element of (G . (modetrans (G,i0))) st z in dom (r (#) (f * (reproj ((modetrans (G,i0)),x)))) holds
(r (#) (f * (reproj ((modetrans (G,i0)),x)))) . z = ((r (#) f) * (reproj ((modetrans (G,i0)),x))) . z
proof
let z be Element of (G . (modetrans (G,i0))); ::_thesis: ( z in dom (r (#) (f * (reproj ((modetrans (G,i0)),x)))) implies (r (#) (f * (reproj ((modetrans (G,i0)),x)))) . z = ((r (#) f) * (reproj ((modetrans (G,i0)),x))) . z )
assume A6: z in dom (r (#) (f * (reproj ((modetrans (G,i0)),x)))) ; ::_thesis: (r (#) (f * (reproj ((modetrans (G,i0)),x)))) . z = ((r (#) f) * (reproj ((modetrans (G,i0)),x))) . z
then A7: z in dom (f * (reproj ((modetrans (G,i0)),x))) by VFUNCT_1:def_4;
A8: (reproj ((modetrans (G,i0)),x)) . z in dom f by A1, A5, A4, A6;
then A9: f /. ((reproj ((modetrans (G,i0)),x)) . z) = f . ((reproj ((modetrans (G,i0)),x)) . z) by PARTFUN1:def_6
.= (f * (reproj ((modetrans (G,i0)),x))) . z by A7, FUNCT_1:12
.= (f * (reproj ((modetrans (G,i0)),x))) /. z by A7, PARTFUN1:def_6 ;
A10: (r (#) (f * (reproj ((modetrans (G,i0)),x)))) . z = (r (#) (f * (reproj ((modetrans (G,i0)),x)))) /. z by A6, PARTFUN1:def_6
.= r * (f /. ((reproj ((modetrans (G,i0)),x)) . z)) by A6, A9, VFUNCT_1:def_4 ;
((r (#) f) * (reproj ((modetrans (G,i0)),x))) . z = (r (#) f) . ((reproj ((modetrans (G,i0)),x)) . z) by A4, A6, FUNCT_1:12
.= (r (#) f) /. ((reproj ((modetrans (G,i0)),x)) . z) by A1, A8, PARTFUN1:def_6
.= r * (f /. ((reproj ((modetrans (G,i0)),x)) . z)) by A1, A8, VFUNCT_1:def_4 ;
hence (r (#) (f * (reproj ((modetrans (G,i0)),x)))) . z = ((r (#) f) * (reproj ((modetrans (G,i0)),x))) . z by A10; ::_thesis: verum
end;
hence r (#) (f * (reproj ((modetrans (G,i0)),x))) = (r (#) f) * (reproj ((modetrans (G,i0)),x)) by A4, PARTFUN1:5; ::_thesis: verum
end;
theorem :: NDIFF_5:28
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 + f2 is_partial_differentiable_in x,i & partdiff ((f1 + f2),x,i) = (partdiff (f1,x,i)) + (partdiff (f2,x,i)) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 + f2 is_partial_differentiable_in x,i & partdiff ((f1 + f2),x,i) = (partdiff (f1,x,i)) + (partdiff (f2,x,i)) )
let F be non trivial RealNormSpace; ::_thesis: for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 + f2 is_partial_differentiable_in x,i & partdiff ((f1 + f2),x,i) = (partdiff (f1,x,i)) + (partdiff (f2,x,i)) )
let f1, f2 be PartFunc of (product G),F; ::_thesis: for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 + f2 is_partial_differentiable_in x,i & partdiff ((f1 + f2),x,i) = (partdiff (f1,x,i)) + (partdiff (f2,x,i)) )
let x be Point of (product G); ::_thesis: for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 + f2 is_partial_differentiable_in x,i & partdiff ((f1 + f2),x,i) = (partdiff (f1,x,i)) + (partdiff (f2,x,i)) )
let i0 be set ; ::_thesis: ( i0 in dom G & f1 is_partial_differentiable_in x,i0 & f2 is_partial_differentiable_in x,i0 implies ( f1 + f2 is_partial_differentiable_in x,i0 & partdiff ((f1 + f2),x,i0) = (partdiff (f1,x,i0)) + (partdiff (f2,x,i0)) ) )
set i = modetrans (G,i0);
assume A1: i0 in dom G ; ::_thesis: ( not f1 is_partial_differentiable_in x,i0 or not f2 is_partial_differentiable_in x,i0 or ( f1 + f2 is_partial_differentiable_in x,i0 & partdiff ((f1 + f2),x,i0) = (partdiff (f1,x,i0)) + (partdiff (f2,x,i0)) ) )
then A2: (f1 + f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x))) by Th26;
assume A3: ( f1 is_partial_differentiable_in x,i0 & f2 is_partial_differentiable_in x,i0 ) ; ::_thesis: ( f1 + f2 is_partial_differentiable_in x,i0 & partdiff ((f1 + f2),x,i0) = (partdiff (f1,x,i0)) + (partdiff (f2,x,i0)) )
A4: ( f1 * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x & f2 * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x ) by A3, Def6;
then (f1 + f2) * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x by A2, NDIFF_1:35;
hence f1 + f2 is_partial_differentiable_in x,i0 by Def6; ::_thesis: partdiff ((f1 + f2),x,i0) = (partdiff (f1,x,i0)) + (partdiff (f2,x,i0))
thus (partdiff (f1,x,i0)) + (partdiff (f2,x,i0)) = diff (((f1 * (reproj ((modetrans (G,i0)),x))) + (f2 * (reproj ((modetrans (G,i0)),x)))),((proj (modetrans (G,i0))) . x)) by A4, NDIFF_1:35
.= partdiff ((f1 + f2),x,i0) by A1, Th26 ; ::_thesis: verum
end;
theorem :: NDIFF_5:29
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 - f2 is_partial_differentiable_in x,i & partdiff ((f1 - f2),x,i) = (partdiff (f1,x,i)) - (partdiff (f2,x,i)) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 - f2 is_partial_differentiable_in x,i & partdiff ((f1 - f2),x,i) = (partdiff (f1,x,i)) - (partdiff (f2,x,i)) )
let F be non trivial RealNormSpace; ::_thesis: for f1, f2 being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 - f2 is_partial_differentiable_in x,i & partdiff ((f1 - f2),x,i) = (partdiff (f1,x,i)) - (partdiff (f2,x,i)) )
let f1, f2 be PartFunc of (product G),F; ::_thesis: for x being Point of (product G)
for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 - f2 is_partial_differentiable_in x,i & partdiff ((f1 - f2),x,i) = (partdiff (f1,x,i)) - (partdiff (f2,x,i)) )
let x be Point of (product G); ::_thesis: for i being set st i in dom G & f1 is_partial_differentiable_in x,i & f2 is_partial_differentiable_in x,i holds
( f1 - f2 is_partial_differentiable_in x,i & partdiff ((f1 - f2),x,i) = (partdiff (f1,x,i)) - (partdiff (f2,x,i)) )
let i0 be set ; ::_thesis: ( i0 in dom G & f1 is_partial_differentiable_in x,i0 & f2 is_partial_differentiable_in x,i0 implies ( f1 - f2 is_partial_differentiable_in x,i0 & partdiff ((f1 - f2),x,i0) = (partdiff (f1,x,i0)) - (partdiff (f2,x,i0)) ) )
assume A1: i0 in dom G ; ::_thesis: ( not f1 is_partial_differentiable_in x,i0 or not f2 is_partial_differentiable_in x,i0 or ( f1 - f2 is_partial_differentiable_in x,i0 & partdiff ((f1 - f2),x,i0) = (partdiff (f1,x,i0)) - (partdiff (f2,x,i0)) ) )
set i = modetrans (G,i0);
assume ( f1 is_partial_differentiable_in x,i0 & f2 is_partial_differentiable_in x,i0 ) ; ::_thesis: ( f1 - f2 is_partial_differentiable_in x,i0 & partdiff ((f1 - f2),x,i0) = (partdiff (f1,x,i0)) - (partdiff (f2,x,i0)) )
then A2: ( f1 * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x & f2 * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x ) by Def6;
(f1 - f2) * (reproj ((modetrans (G,i0)),x)) = (f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x))) by A1, Th26;
then (f1 - f2) * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x by A2, NDIFF_1:36;
hence f1 - f2 is_partial_differentiable_in x,i0 by Def6; ::_thesis: partdiff ((f1 - f2),x,i0) = (partdiff (f1,x,i0)) - (partdiff (f2,x,i0))
thus (partdiff (f1,x,i0)) - (partdiff (f2,x,i0)) = diff (((f1 * (reproj ((modetrans (G,i0)),x))) - (f2 * (reproj ((modetrans (G,i0)),x)))),((proj (modetrans (G,i0))) . x)) by A2, NDIFF_1:36
.= partdiff ((f1 - f2),x,i0) by A1, Th26 ; ::_thesis: verum
end;
theorem :: NDIFF_5:30
for r being Real
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f is_partial_differentiable_in x,i holds
( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) )
proof
let r be Real; ::_thesis: for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f is_partial_differentiable_in x,i holds
( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) )
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f is_partial_differentiable_in x,i holds
( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) )
let F be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set st i in dom G & f is_partial_differentiable_in x,i holds
( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) )
let f be PartFunc of (product G),F; ::_thesis: for x being Point of (product G)
for i being set st i in dom G & f is_partial_differentiable_in x,i holds
( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) )
let x be Point of (product G); ::_thesis: for i being set st i in dom G & f is_partial_differentiable_in x,i holds
( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) )
let i0 be set ; ::_thesis: ( i0 in dom G & f is_partial_differentiable_in x,i0 implies ( r (#) f is_partial_differentiable_in x,i0 & partdiff ((r (#) f),x,i0) = r * (partdiff (f,x,i0)) ) )
assume A1: i0 in dom G ; ::_thesis: ( not f is_partial_differentiable_in x,i0 or ( r (#) f is_partial_differentiable_in x,i0 & partdiff ((r (#) f),x,i0) = r * (partdiff (f,x,i0)) ) )
set i = modetrans (G,i0);
assume f is_partial_differentiable_in x,i0 ; ::_thesis: ( r (#) f is_partial_differentiable_in x,i0 & partdiff ((r (#) f),x,i0) = r * (partdiff (f,x,i0)) )
then A2: f * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x by Def6;
r (#) (f * (reproj ((modetrans (G,i0)),x))) = (r (#) f) * (reproj ((modetrans (G,i0)),x)) by A1, Th27;
then (r (#) f) * (reproj ((modetrans (G,i0)),x)) is_differentiable_in (proj (modetrans (G,i0))) . x by A2, NDIFF_1:37;
hence r (#) f is_partial_differentiable_in x,i0 by Def6; ::_thesis: partdiff ((r (#) f),x,i0) = r * (partdiff (f,x,i0))
thus partdiff ((r (#) f),x,i0) = diff ((r (#) (f * (reproj ((modetrans (G,i0)),x)))),((proj (modetrans (G,i0))) . x)) by A1, Th27
.= r * (partdiff (f,x,i0)) by A2, NDIFF_1:37 ; ::_thesis: verum
end;
begin
theorem Th31: :: NDIFF_5:31
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x being Point of (product G) holds ||.((proj i) . x).|| <= ||.x.||
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x being Point of (product G) holds ||.((proj i) . x).|| <= ||.x.||
let i be Element of dom G; ::_thesis: for x being Point of (product G) holds ||.((proj i) . x).|| <= ||.x.||
let x be Point of (product G); ::_thesis: ||.((proj i) . x).|| <= ||.x.||
reconsider y = x as Element of product (carr G) by Th10;
(proj i) . x = y . i by Def3;
hence ||.((proj i) . x).|| <= ||.x.|| by PRVECT_2:10; ::_thesis: verum
end;
registration
let G be non-trivial RealNormSpace-Sequence;
cluster -> len G -element for Element of the carrier of (product G);
coherence
for b1 being Point of (product G) holds b1 is len G -element
proof
let x be Point of (product G); ::_thesis: x is len G -element
A1: the carrier of (product G) = product (carr G) by Th10;
A2: ( dom x = dom (carr G) & ( for i being set st i in dom (carr G) holds
x . i in (carr G) . i ) ) by A1, CARD_3:9;
len (carr G) = len G by PRVECT_2:def_4;
then dom x = Seg (len G) by A2, FINSEQ_1:def_3;
then len x = len G by FINSEQ_1:def_3;
hence x is len G -element by CARD_1:def_7; ::_thesis: verum
end;
end;
theorem Th32: :: NDIFF_5:32
for G being non-trivial RealNormSpace-Sequence
for T being non trivial RealNormSpace
for i being set
for Z being Subset of (product G)
for f being PartFunc of (product G),T st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for T being non trivial RealNormSpace
for i being set
for Z being Subset of (product G)
for f being PartFunc of (product G),T st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let T be non trivial RealNormSpace; ::_thesis: for i being set
for Z being Subset of (product G)
for f being PartFunc of (product G),T st Z is open holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let i0 be set ; ::_thesis: for Z being Subset of (product G)
for f being PartFunc of (product G),T st Z is open holds
( f is_partial_differentiable_on Z,i0 iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i0 ) ) )
let Z be Subset of (product G); ::_thesis: for f being PartFunc of (product G),T st Z is open holds
( f is_partial_differentiable_on Z,i0 iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i0 ) ) )
let f be PartFunc of (product G),T; ::_thesis: ( Z is open implies ( f is_partial_differentiable_on Z,i0 iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i0 ) ) ) )
assume A1: Z is open ; ::_thesis: ( f is_partial_differentiable_on Z,i0 iff ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i0 ) ) )
set i = modetrans (G,i0);
set S = G . (modetrans (G,i0));
set RNS = R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i0))),T);
hereby ::_thesis: ( Z c= dom f & ( for x being Point of (product G) st x in Z holds
f is_partial_differentiable_in x,i0 ) implies f is_partial_differentiable_on Z,i0 )
assume A2: f is_partial_differentiable_on Z,i0 ; ::_thesis: ( Z c= dom f & ( for nx0 being Point of (product G) st nx0 in Z holds
f is_partial_differentiable_in nx0,i0 ) )
hence A3: Z c= dom f by Def8; ::_thesis: for nx0 being Point of (product G) st nx0 in Z holds
f is_partial_differentiable_in nx0,i0
let nx0 be Point of (product G); ::_thesis: ( nx0 in Z implies f is_partial_differentiable_in nx0,i0 )
reconsider x0 = (proj (modetrans (G,i0))) . nx0 as Point of (G . (modetrans (G,i0))) ;
assume A4: nx0 in Z ; ::_thesis: f is_partial_differentiable_in nx0,i0
then f | Z is_partial_differentiable_in nx0,i0 by A2, Def8;
then (f | Z) * (reproj ((modetrans (G,i0)),nx0)) is_differentiable_in x0 by Def6;
then consider N0 being Neighbourhood of x0 such that
A5: N0 c= dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) and
A6: ex L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i0))),T)) ex R being RestFunc of (G . (modetrans (G,i0))),T st
for x being Point of (G . (modetrans (G,i0))) st x in N0 holds
(((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by NDIFF_1:def_6;
consider L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i0))),T)), R being RestFunc of (G . (modetrans (G,i0))),T such that
A7: for x being Point of (G . (modetrans (G,i0))) st x in N0 holds
(((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A6;
consider N1 being Neighbourhood of x0 such that
A8: for x being Point of (G . (modetrans (G,i0))) st x in N1 holds
(reproj ((modetrans (G,i0)),nx0)) . x in Z by A1, A4, Th23;
A9: now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i0)))_st_x_in_N1_holds_
(reproj_((modetrans_(G,i0)),nx0))_._x_in_dom_(f_|_Z)
let x be Point of (G . (modetrans (G,i0))); ::_thesis: ( x in N1 implies (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z) )
assume x in N1 ; ::_thesis: (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z)
then (reproj ((modetrans (G,i0)),nx0)) . x in Z by A8;
then (reproj ((modetrans (G,i0)),nx0)) . x in (dom f) /\ Z by A3, XBOOLE_0:def_4;
hence (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z) by RELAT_1:61; ::_thesis: verum
end;
reconsider N = N0 /\ N1 as Neighbourhood of x0 by Th8;
(f | Z) * (reproj ((modetrans (G,i0)),nx0)) c= f * (reproj ((modetrans (G,i0)),nx0)) by RELAT_1:29, RELAT_1:59;
then A10: dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) c= dom (f * (reproj ((modetrans (G,i0)),nx0))) by RELAT_1:11;
N c= N0 by XBOOLE_1:17;
then N c= dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) by A5, XBOOLE_1:1;
then A11: N c= dom (f * (reproj ((modetrans (G,i0)),nx0))) by A10, XBOOLE_1:1;
A12: dom (reproj ((modetrans (G,i0)),nx0)) = the carrier of (G . (modetrans (G,i0))) by FUNCT_2:def_1;
now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i0)))_st_x_in_N_holds_
((f_*_(reproj_((modetrans_(G,i0)),nx0)))_/._x)_-_((f_*_(reproj_((modetrans_(G,i0)),nx0)))_/._x0)_=_(L_._(x_-_x0))_+_(R_/._(x_-_x0))
let x be Point of (G . (modetrans (G,i0))); ::_thesis: ( x in N implies ((f * (reproj ((modetrans (G,i0)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
assume x in N ; ::_thesis: ((f * (reproj ((modetrans (G,i0)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0))
then A13: ( x in N0 & x in N1 ) by XBOOLE_0:def_4;
then A14: (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z) by A9;
then A15: ( (reproj ((modetrans (G,i0)),nx0)) . x in dom f & (reproj ((modetrans (G,i0)),nx0)) . x in Z ) by RELAT_1:57;
A16: (reproj ((modetrans (G,i0)),nx0)) . x0 in dom (f | Z) by A9, NFCONT_1:4;
then A17: ( (reproj ((modetrans (G,i0)),nx0)) . x0 in dom f & (reproj ((modetrans (G,i0)),nx0)) . x0 in Z ) by RELAT_1:57;
A18: ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x = (f | Z) /. ((reproj ((modetrans (G,i0)),nx0)) /. x) by A14, A12, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i0)),nx0)) /. x) by A15, PARTFUN2:17
.= (f * (reproj ((modetrans (G,i0)),nx0))) /. x by A12, A15, PARTFUN2:4 ;
((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x0 = (f | Z) /. ((reproj ((modetrans (G,i0)),nx0)) /. x0) by A12, A16, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i0)),nx0)) /. x0) by A17, PARTFUN2:17
.= (f * (reproj ((modetrans (G,i0)),nx0))) /. x0 by A12, A17, PARTFUN2:4 ;
hence ((f * (reproj ((modetrans (G,i0)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A7, A13, A18; ::_thesis: verum
end;
then f * (reproj ((modetrans (G,i0)),nx0)) is_differentiable_in x0 by A11, NDIFF_1:def_6;
hence f is_partial_differentiable_in nx0,i0 by Def6; ::_thesis: verum
end;
assume that
A19: Z c= dom f and
A20: for nx being Point of (product G) st nx in Z holds
f is_partial_differentiable_in nx,i0 ; ::_thesis: f is_partial_differentiable_on Z,i0
now__::_thesis:_for_nx0_being_Point_of_(product_G)_st_nx0_in_Z_holds_
f_|_Z_is_partial_differentiable_in_nx0,i0
let nx0 be Point of (product G); ::_thesis: ( nx0 in Z implies f | Z is_partial_differentiable_in nx0,i0 )
assume A21: nx0 in Z ; ::_thesis: f | Z is_partial_differentiable_in nx0,i0
then A22: f is_partial_differentiable_in nx0,i0 by A20;
reconsider x0 = (proj (modetrans (G,i0))) . nx0 as Point of (G . (modetrans (G,i0))) ;
f * (reproj ((modetrans (G,i0)),nx0)) is_differentiable_in x0 by A22, Def6;
then consider N0 being Neighbourhood of x0 such that
N0 c= dom (f * (reproj ((modetrans (G,i0)),nx0))) and
A23: ex L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i0))),T)) ex R being RestFunc of (G . (modetrans (G,i0))),T st
for x being Point of (G . (modetrans (G,i0))) st x in N0 holds
((f * (reproj ((modetrans (G,i0)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by NDIFF_1:def_6;
consider N1 being Neighbourhood of x0 such that
A24: for x being Point of (G . (modetrans (G,i0))) st x in N1 holds
(reproj ((modetrans (G,i0)),nx0)) . x in Z by A1, A21, Th23;
A25: now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i0)))_st_x_in_N1_holds_
(reproj_((modetrans_(G,i0)),nx0))_._x_in_dom_(f_|_Z)
let x be Point of (G . (modetrans (G,i0))); ::_thesis: ( x in N1 implies (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z) )
assume x in N1 ; ::_thesis: (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z)
then (reproj ((modetrans (G,i0)),nx0)) . x in Z by A24;
then (reproj ((modetrans (G,i0)),nx0)) . x in (dom f) /\ Z by A19, XBOOLE_0:def_4;
hence (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z) by RELAT_1:61; ::_thesis: verum
end;
A26: N1 c= dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0)))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in N1 or z in dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) )
assume A27: z in N1 ; ::_thesis: z in dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0)))
then z in the carrier of (G . (modetrans (G,i0))) ;
then A28: z in dom (reproj ((modetrans (G,i0)),nx0)) by FUNCT_2:def_1;
reconsider x = z as Point of (G . (modetrans (G,i0))) by A27;
(reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z) by A27, A25;
hence z in dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) by A28, FUNCT_1:11; ::_thesis: verum
end;
reconsider N = N0 /\ N1 as Neighbourhood of x0 by Th8;
N c= N1 by XBOOLE_1:17;
then A29: N c= dom ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) by A26, XBOOLE_1:1;
consider L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i0))),T)), R being RestFunc of (G . (modetrans (G,i0))),T such that
A30: for x being Point of (G . (modetrans (G,i0))) st x in N0 holds
((f * (reproj ((modetrans (G,i0)),nx0))) /. x) - ((f * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A23;
now__::_thesis:_for_x_being_Point_of_(G_._(modetrans_(G,i0)))_st_x_in_N_holds_
(((f_|_Z)_*_(reproj_((modetrans_(G,i0)),nx0)))_/._x)_-_(((f_|_Z)_*_(reproj_((modetrans_(G,i0)),nx0)))_/._x0)_=_(L_._(x_-_x0))_+_(R_/._(x_-_x0))
let x be Point of (G . (modetrans (G,i0))); ::_thesis: ( x in N implies (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
assume A31: x in N ; ::_thesis: (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0))
then A32: x in N0 by XBOOLE_0:def_4;
A33: dom (reproj ((modetrans (G,i0)),nx0)) = the carrier of (G . (modetrans (G,i0))) by FUNCT_2:def_1;
x in N1 by A31, XBOOLE_0:def_4;
then A34: (reproj ((modetrans (G,i0)),nx0)) . x in dom (f | Z) by A25;
then A35: (reproj ((modetrans (G,i0)),nx0)) . x in (dom f) /\ Z by RELAT_1:61;
then A36: (reproj ((modetrans (G,i0)),nx0)) . x in dom f by XBOOLE_0:def_4;
A37: (reproj ((modetrans (G,i0)),nx0)) . x0 in dom (f | Z) by A25, NFCONT_1:4;
then A38: (reproj ((modetrans (G,i0)),nx0)) . x0 in (dom f) /\ Z by RELAT_1:61;
then A39: (reproj ((modetrans (G,i0)),nx0)) . x0 in dom f by XBOOLE_0:def_4;
A40: ((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x = (f | Z) /. ((reproj ((modetrans (G,i0)),nx0)) /. x) by A34, A33, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i0)),nx0)) /. x) by A35, PARTFUN2:16
.= (f * (reproj ((modetrans (G,i0)),nx0))) /. x by A33, A36, PARTFUN2:4 ;
((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x0 = (f | Z) /. ((reproj ((modetrans (G,i0)),nx0)) /. x0) by A33, A37, PARTFUN2:4
.= f /. ((reproj ((modetrans (G,i0)),nx0)) /. x0) by A38, PARTFUN2:16
.= (f * (reproj ((modetrans (G,i0)),nx0))) /. x0 by A33, A39, PARTFUN2:4 ;
hence (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x) - (((f | Z) * (reproj ((modetrans (G,i0)),nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A40, A32, A30; ::_thesis: verum
end;
then (f | Z) * (reproj ((modetrans (G,i0)),nx0)) is_differentiable_in x0 by A29, NDIFF_1:def_6;
hence f | Z is_partial_differentiable_in nx0,i0 by Def6; ::_thesis: verum
end;
hence f is_partial_differentiable_on Z,i0 by A19, Def8; ::_thesis: verum
end;
theorem Th33: :: NDIFF_5:33
for G being non-trivial RealNormSpace-Sequence
for i, j being Element of dom G
for x being Point of (G . i)
for z being Element of product (carr G) st z = (reproj (i,(0. (product G)))) . x holds
( ( i = j implies z . j = x ) & ( i <> j implies z . j = 0. (G . j) ) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i, j being Element of dom G
for x being Point of (G . i)
for z being Element of product (carr G) st z = (reproj (i,(0. (product G)))) . x holds
( ( i = j implies z . j = x ) & ( i <> j implies z . j = 0. (G . j) ) )
let i, j be Element of dom G; ::_thesis: for x being Point of (G . i)
for z being Element of product (carr G) st z = (reproj (i,(0. (product G)))) . x holds
( ( i = j implies z . j = x ) & ( i <> j implies z . j = 0. (G . j) ) )
let x be Point of (G . i); ::_thesis: for z being Element of product (carr G) st z = (reproj (i,(0. (product G)))) . x holds
( ( i = j implies z . j = x ) & ( i <> j implies z . j = 0. (G . j) ) )
let z be Element of product (carr G); ::_thesis: ( z = (reproj (i,(0. (product G)))) . x implies ( ( i = j implies z . j = x ) & ( i <> j implies z . j = 0. (G . j) ) ) )
assume A1: z = (reproj (i,(0. (product G)))) . x ; ::_thesis: ( ( i = j implies z . j = x ) & ( i <> j implies z . j = 0. (G . j) ) )
reconsider Zr = 0. (product G) as Element of product (carr G) by Th10;
reconsider ixr = (reproj (i,(0. (product G)))) . x as Element of product (carr G) by Th10;
A2: (reproj (i,(0. (product G)))) . x = (0. (product G)) +* (i,x) by Def4;
set ix = i .--> x;
consider g being Function such that
A3: ( Zr = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
A4: dom Zr = dom G by A3, Lm1;
now__::_thesis:_(_i_<>_j_implies_z_._j_=_0._(G_._j)_)
assume i <> j ; ::_thesis: z . j = 0. (G . j)
then z . j = Zr . j by A1, A2, FUNCT_7:32;
hence z . j = 0. (G . j) by Th14; ::_thesis: verum
end;
hence ( ( i = j implies z . j = x ) & ( i <> j implies z . j = 0. (G . j) ) ) by A1, A2, A4, FUNCT_7:31; ::_thesis: verum
end;
theorem Th34: :: NDIFF_5:34
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x, y being Point of (G . i) holds (reproj (i,(0. (product G)))) . (x + y) = ((reproj (i,(0. (product G)))) . x) + ((reproj (i,(0. (product G)))) . y)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x, y being Point of (G . i) holds (reproj (i,(0. (product G)))) . (x + y) = ((reproj (i,(0. (product G)))) . x) + ((reproj (i,(0. (product G)))) . y)
let i be Element of dom G; ::_thesis: for x, y being Point of (G . i) holds (reproj (i,(0. (product G)))) . (x + y) = ((reproj (i,(0. (product G)))) . x) + ((reproj (i,(0. (product G)))) . y)
let x, y be Point of (G . i); ::_thesis: (reproj (i,(0. (product G)))) . (x + y) = ((reproj (i,(0. (product G)))) . x) + ((reproj (i,(0. (product G)))) . y)
reconsider v = (reproj (i,(0. (product G)))) . (x + y) as Element of product (carr G) by Th10;
reconsider s = (reproj (i,(0. (product G)))) . x as Element of product (carr G) by Th10;
reconsider t = (reproj (i,(0. (product G)))) . y as Element of product (carr G) by Th10;
for j being Element of dom G holds v . j = (s . j) + (t . j)
proof
let j be Element of dom G; ::_thesis: v . j = (s . j) + (t . j)
percases ( i = j or i <> j ) ;
supposeA1: i = j ; ::_thesis: v . j = (s . j) + (t . j)
then reconsider yy = y as Point of (G . j) ;
v . j = x + y by Th33, A1;
then v . j = (s . j) + yy by Th33, A1;
hence v . j = (s . j) + (t . j) by Th33, A1; ::_thesis: verum
end;
supposeA2: i <> j ; ::_thesis: v . j = (s . j) + (t . j)
then v . j = 0. (G . j) by Th33;
then v . j = (0. (G . j)) + (0. (G . j)) by RLVECT_1:def_4;
then v . j = (s . j) + (0. (G . j)) by Th33, A2;
hence v . j = (s . j) + (t . j) by Th33, A2; ::_thesis: verum
end;
end;
end;
hence (reproj (i,(0. (product G)))) . (x + y) = ((reproj (i,(0. (product G)))) . x) + ((reproj (i,(0. (product G)))) . y) by Th12; ::_thesis: verum
end;
theorem Th35: :: NDIFF_5:35
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x, y being Point of (product G) holds (proj i) . (x + y) = ((proj i) . x) + ((proj i) . y)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x, y being Point of (product G) holds (proj i) . (x + y) = ((proj i) . x) + ((proj i) . y)
let i be Element of dom G; ::_thesis: for x, y being Point of (product G) holds (proj i) . (x + y) = ((proj i) . x) + ((proj i) . y)
let x, y be Point of (product G); ::_thesis: (proj i) . (x + y) = ((proj i) . x) + ((proj i) . y)
reconsider v = x + y as Element of product (carr G) by Th10;
reconsider s = x as Element of product (carr G) by Th10;
reconsider t = y as Element of product (carr G) by Th10;
( (proj i) . (x + y) = v . i & (proj i) . x = s . i & (proj i) . y = t . i ) by Def3;
hence (proj i) . (x + y) = ((proj i) . x) + ((proj i) . y) by Th12; ::_thesis: verum
end;
theorem :: NDIFF_5:36
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x, y being Point of (G . i) holds (reproj (i,(0. (product G)))) . (x - y) = ((reproj (i,(0. (product G)))) . x) - ((reproj (i,(0. (product G)))) . y)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x, y being Point of (G . i) holds (reproj (i,(0. (product G)))) . (x - y) = ((reproj (i,(0. (product G)))) . x) - ((reproj (i,(0. (product G)))) . y)
let i be Element of dom G; ::_thesis: for x, y being Point of (G . i) holds (reproj (i,(0. (product G)))) . (x - y) = ((reproj (i,(0. (product G)))) . x) - ((reproj (i,(0. (product G)))) . y)
let x, y be Point of (G . i); ::_thesis: (reproj (i,(0. (product G)))) . (x - y) = ((reproj (i,(0. (product G)))) . x) - ((reproj (i,(0. (product G)))) . y)
reconsider v = (reproj (i,(0. (product G)))) . (x - y) as Element of product (carr G) by Th10;
reconsider s = (reproj (i,(0. (product G)))) . x as Element of product (carr G) by Th10;
reconsider t = (reproj (i,(0. (product G)))) . y as Element of product (carr G) by Th10;
for j being Element of dom G holds v . j = (s . j) - (t . j)
proof
let j be Element of dom G; ::_thesis: v . j = (s . j) - (t . j)
percases ( i = j or i <> j ) ;
supposeA1: i = j ; ::_thesis: v . j = (s . j) - (t . j)
then reconsider yy = y as Point of (G . j) ;
v . j = x - y by Th33, A1;
then v . j = (s . j) - yy by Th33, A1;
hence v . j = (s . j) - (t . j) by Th33, A1; ::_thesis: verum
end;
supposeA2: i <> j ; ::_thesis: v . j = (s . j) - (t . j)
then v . j = 0. (G . j) by Th33;
then v . j = (0. (G . j)) - (0. (G . j)) by RLVECT_1:13;
then v . j = (s . j) - (0. (G . j)) by Th33, A2;
hence v . j = (s . j) - (t . j) by Th33, A2; ::_thesis: verum
end;
end;
end;
hence (reproj (i,(0. (product G)))) . (x - y) = ((reproj (i,(0. (product G)))) . x) - ((reproj (i,(0. (product G)))) . y) by Th15; ::_thesis: verum
end;
theorem Th37: :: NDIFF_5:37
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x, y being Point of (product G) holds (proj i) . (x - y) = ((proj i) . x) - ((proj i) . y)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x, y being Point of (product G) holds (proj i) . (x - y) = ((proj i) . x) - ((proj i) . y)
let i be Element of dom G; ::_thesis: for x, y being Point of (product G) holds (proj i) . (x - y) = ((proj i) . x) - ((proj i) . y)
let x, y be Point of (product G); ::_thesis: (proj i) . (x - y) = ((proj i) . x) - ((proj i) . y)
reconsider v = x - y as Element of product (carr G) by Th10;
reconsider s = x as Element of product (carr G) by Th10;
reconsider t = y as Element of product (carr G) by Th10;
( (proj i) . (x - y) = v . i & (proj i) . x = s . i & (proj i) . y = t . i ) by Def3;
hence (proj i) . (x - y) = ((proj i) . x) - ((proj i) . y) by Th15; ::_thesis: verum
end;
theorem Th38: :: NDIFF_5:38
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x being Point of (G . i) st x <> 0. (G . i) holds
(reproj (i,(0. (product G)))) . x <> 0. (product G)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x being Point of (G . i) st x <> 0. (G . i) holds
(reproj (i,(0. (product G)))) . x <> 0. (product G)
let i be Element of dom G; ::_thesis: for x being Point of (G . i) st x <> 0. (G . i) holds
(reproj (i,(0. (product G)))) . x <> 0. (product G)
let x be Point of (G . i); ::_thesis: ( x <> 0. (G . i) implies (reproj (i,(0. (product G)))) . x <> 0. (product G) )
assume A1: x <> 0. (G . i) ; ::_thesis: (reproj (i,(0. (product G)))) . x <> 0. (product G)
assume A2: (reproj (i,(0. (product G)))) . x = 0. (product G) ; ::_thesis: contradiction
reconsider v = (reproj (i,(0. (product G)))) . x as Element of product (carr G) by Th10;
x = v . i by Th33;
hence contradiction by A1, Th14, A2; ::_thesis: verum
end;
theorem Th39: :: NDIFF_5:39
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x being Point of (G . i)
for a being Element of REAL holds (reproj (i,(0. (product G)))) . (a * x) = a * ((reproj (i,(0. (product G)))) . x)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x being Point of (G . i)
for a being Element of REAL holds (reproj (i,(0. (product G)))) . (a * x) = a * ((reproj (i,(0. (product G)))) . x)
let i be Element of dom G; ::_thesis: for x being Point of (G . i)
for a being Element of REAL holds (reproj (i,(0. (product G)))) . (a * x) = a * ((reproj (i,(0. (product G)))) . x)
let x be Point of (G . i); ::_thesis: for a being Element of REAL holds (reproj (i,(0. (product G)))) . (a * x) = a * ((reproj (i,(0. (product G)))) . x)
let a be Element of REAL ; ::_thesis: (reproj (i,(0. (product G)))) . (a * x) = a * ((reproj (i,(0. (product G)))) . x)
reconsider v = (reproj (i,(0. (product G)))) . (a * x) as Element of product (carr G) by Th10;
reconsider s = (reproj (i,(0. (product G)))) . x as Element of product (carr G) by Th10;
for j being Element of dom G holds v . j = a * (s . j)
proof
let j be Element of dom G; ::_thesis: v . j = a * (s . j)
percases ( i = j or i <> j ) ;
supposeA1: i = j ; ::_thesis: v . j = a * (s . j)
then v . j = a * x by Th33;
hence v . j = a * (s . j) by Th33, A1; ::_thesis: verum
end;
supposeA2: i <> j ; ::_thesis: v . j = a * (s . j)
then v . j = 0. (G . j) by Th33;
then v . j = a * (0. (G . j)) by RLVECT_1:10;
hence v . j = a * (s . j) by Th33, A2; ::_thesis: verum
end;
end;
end;
hence (reproj (i,(0. (product G)))) . (a * x) = a * ((reproj (i,(0. (product G)))) . x) by Th13; ::_thesis: verum
end;
theorem Th40: :: NDIFF_5:40
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x being Point of (product G)
for a being Element of REAL holds (proj i) . (a * x) = a * ((proj i) . x)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x being Point of (product G)
for a being Element of REAL holds (proj i) . (a * x) = a * ((proj i) . x)
let i be Element of dom G; ::_thesis: for x being Point of (product G)
for a being Element of REAL holds (proj i) . (a * x) = a * ((proj i) . x)
let x be Point of (product G); ::_thesis: for a being Element of REAL holds (proj i) . (a * x) = a * ((proj i) . x)
let a be Element of REAL ; ::_thesis: (proj i) . (a * x) = a * ((proj i) . x)
reconsider v = a * x as Element of product (carr G) by Th10;
reconsider s = x as Element of product (carr G) by Th10;
( (proj i) . (a * x) = v . i & (proj i) . x = s . i ) by Def3;
hence (proj i) . (a * x) = a * ((proj i) . x) by Th13; ::_thesis: verum
end;
theorem Th41: :: NDIFF_5:41
for G being non-trivial RealNormSpace-Sequence
for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for x being Point of (product G)
for i being set st f is_differentiable_in x holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for x being Point of (product G)
for i being set st f is_differentiable_in x holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) )
let S be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),S
for x being Point of (product G)
for i being set st f is_differentiable_in x holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) )
let f be PartFunc of (product G),S; ::_thesis: for x being Point of (product G)
for i being set st f is_differentiable_in x holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) )
let x be Point of (product G); ::_thesis: for i being set st f is_differentiable_in x holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) )
let i0 be set ; ::_thesis: ( f is_differentiable_in x implies ( f is_partial_differentiable_in x,i0 & partdiff (f,x,i0) = (diff (f,x)) * (reproj ((modetrans (G,i0)),(0. (product G)))) ) )
assume A1: f is_differentiable_in x ; ::_thesis: ( f is_partial_differentiable_in x,i0 & partdiff (f,x,i0) = (diff (f,x)) * (reproj ((modetrans (G,i0)),(0. (product G)))) )
set i = modetrans (G,i0);
consider N being Neighbourhood of x such that
A2: ( N c= dom f & ex R being RestFunc of (product G),S st
for y being Point of (product G) st y in N holds
(f /. y) - (f /. x) = ((diff (f,x)) . (y - x)) + (R /. (y - x)) ) by A1, NDIFF_1:def_7;
consider R being RestFunc of (product G),S such that
A3: for y being Point of (product G) st y in N holds
(f /. y) - (f /. x) = ((diff (f,x)) . (y - x)) + (R /. (y - x)) by A2;
consider r0 being Real such that
A4: ( 0 < r0 & { z where z is Point of (product G) : ||.(z - x).|| < r0 } c= N ) by NFCONT_1:def_1;
set u = f * (reproj ((modetrans (G,i0)),x));
reconsider x0 = (proj (modetrans (G,i0))) . x as Point of (G . (modetrans (G,i0))) ;
set Z = 0. (product G);
set Nx0 = { z where z is Point of (G . (modetrans (G,i0))) : ||.(z - x0).|| < r0 } ;
now__::_thesis:_for_s_being_set_st_s_in__{__z_where_z_is_Point_of_(G_._(modetrans_(G,i0)))_:_||.(z_-_x0).||_<_r0__}__holds_
s_in_the_carrier_of_(G_._(modetrans_(G,i0)))
let s be set ; ::_thesis: ( s in { z where z is Point of (G . (modetrans (G,i0))) : ||.(z - x0).|| < r0 } implies s in the carrier of (G . (modetrans (G,i0))) )
assume s in { z where z is Point of (G . (modetrans (G,i0))) : ||.(z - x0).|| < r0 } ; ::_thesis: s in the carrier of (G . (modetrans (G,i0)))
then ex z being Point of (G . (modetrans (G,i0))) st
( s = z & ||.(z - x0).|| < r0 ) ;
hence s in the carrier of (G . (modetrans (G,i0))) ; ::_thesis: verum
end;
then { z where z is Point of (G . (modetrans (G,i0))) : ||.(z - x0).|| < r0 } is Subset of (G . (modetrans (G,i0))) by TARSKI:def_3;
then reconsider Nx0 = { z where z is Point of (G . (modetrans (G,i0))) : ||.(z - x0).|| < r0 } as Neighbourhood of x0 by A4, NFCONT_1:def_1;
A5: for xi being Element of (G . (modetrans (G,i0))) st xi in Nx0 holds
(reproj ((modetrans (G,i0)),x)) . xi in N
proof
let xi be Element of (G . (modetrans (G,i0))); ::_thesis: ( xi in Nx0 implies (reproj ((modetrans (G,i0)),x)) . xi in N )
assume xi in Nx0 ; ::_thesis: (reproj ((modetrans (G,i0)),x)) . xi in N
then A6: ex z being Point of (G . (modetrans (G,i0))) st
( xi = z & ||.(z - x0).|| < r0 ) ;
((reproj ((modetrans (G,i0)),x)) . xi) - x = (reproj ((modetrans (G,i0)),(0. (product G)))) . (xi - x0) by Th22;
then ||.(((reproj ((modetrans (G,i0)),x)) . xi) - x).|| < r0 by Th21, A6;
then (reproj ((modetrans (G,i0)),x)) . xi in { z where z is Point of (product G) : ||.(z - x).|| < r0 } ;
hence (reproj ((modetrans (G,i0)),x)) . xi in N by A4; ::_thesis: verum
end;
A7: R is total by NDIFF_1:def_5;
then A8: dom R = the carrier of (product G) by PARTFUN1:def_2;
reconsider R1 = R * (reproj ((modetrans (G,i0)),(0. (product G)))) as PartFunc of (G . (modetrans (G,i0))),S ;
A9: dom (reproj ((modetrans (G,i0)),(0. (product G)))) = the carrier of (G . (modetrans (G,i0))) by FUNCT_2:def_1;
A10: dom R1 = the carrier of (G . (modetrans (G,i0))) by A7, PARTFUN1:def_2;
for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Point of (G . (modetrans (G,i0))) st z <> 0. (G . (modetrans (G,i0))) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) )
proof
let r be Real; ::_thesis: ( r > 0 implies ex d being Real st
( d > 0 & ( for z being Point of (G . (modetrans (G,i0))) st z <> 0. (G . (modetrans (G,i0))) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) ) )
assume r > 0 ; ::_thesis: ex d being Real st
( d > 0 & ( for z being Point of (G . (modetrans (G,i0))) st z <> 0. (G . (modetrans (G,i0))) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) )
then consider d being Real such that
A11: ( d > 0 & ( for z being Point of (product G) st z <> 0. (product G) & ||.z.|| < d holds
(||.z.|| ") * ||.(R /. z).|| < r ) ) by A7, NDIFF_1:23;
take d ; ::_thesis: ( d > 0 & ( for z being Point of (G . (modetrans (G,i0))) st z <> 0. (G . (modetrans (G,i0))) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) )
now__::_thesis:_for_z_being_Point_of_(G_._(modetrans_(G,i0)))_st_z_<>_0._(G_._(modetrans_(G,i0)))_&_||.z.||_<_d_holds_
(||.z.||_")_*_||.(R1_/._z).||_<_r
let z be Point of (G . (modetrans (G,i0))); ::_thesis: ( z <> 0. (G . (modetrans (G,i0))) & ||.z.|| < d implies (||.z.|| ") * ||.(R1 /. z).|| < r )
assume A12: ( z <> 0. (G . (modetrans (G,i0))) & ||.z.|| < d ) ; ::_thesis: (||.z.|| ") * ||.(R1 /. z).|| < r
A13: ||.((reproj ((modetrans (G,i0)),(0. (product G)))) . z).|| = ||.z.|| by Th21;
R /. ((reproj ((modetrans (G,i0)),(0. (product G)))) . z) = R . ((reproj ((modetrans (G,i0)),(0. (product G)))) . z) by A8, PARTFUN1:def_6;
then R /. ((reproj ((modetrans (G,i0)),(0. (product G)))) . z) = R1 . z by A9, FUNCT_1:13;
then R /. ((reproj ((modetrans (G,i0)),(0. (product G)))) . z) = R1 /. z by A10, PARTFUN1:def_6;
hence (||.z.|| ") * ||.(R1 /. z).|| < r by A11, A13, A12, Th38; ::_thesis: verum
end;
hence ( d > 0 & ( for z being Point of (G . (modetrans (G,i0))) st z <> 0. (G . (modetrans (G,i0))) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) ) by A11; ::_thesis: verum
end;
then reconsider R1 = R1 as RestFunc of (G . (modetrans (G,i0))),S by A7, NDIFF_1:23;
reconsider dfx = diff (f,x) as Lipschitzian LinearOperator of (product G),S by LOPBAN_1:def_9;
reconsider LD1 = dfx * (reproj ((modetrans (G,i0)),(0. (product G)))) as Function of (G . (modetrans (G,i0))),S ;
A14: now__::_thesis:_for_x,_y_being_Element_of_(G_._(modetrans_(G,i0)))_holds_LD1_._(x_+_y)_=_(LD1_._x)_+_(LD1_._y)
let x, y be Element of (G . (modetrans (G,i0))); ::_thesis: LD1 . (x + y) = (LD1 . x) + (LD1 . y)
LD1 . (x + y) = dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . (x + y)) by FUNCT_2:15;
then LD1 . (x + y) = dfx . (((reproj ((modetrans (G,i0)),(0. (product G)))) . x) + ((reproj ((modetrans (G,i0)),(0. (product G)))) . y)) by Th34;
then LD1 . (x + y) = (dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . x)) + (dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . y)) by VECTSP_1:def_20;
then LD1 . (x + y) = (LD1 . x) + (dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . y)) by FUNCT_2:15;
hence LD1 . (x + y) = (LD1 . x) + (LD1 . y) by FUNCT_2:15; ::_thesis: verum
end;
now__::_thesis:_for_x_being_Element_of_(G_._(modetrans_(G,i0)))
for_a_being_Real_holds_LD1_._(a_*_x)_=_a_*_(LD1_._x)
let x be Element of (G . (modetrans (G,i0))); ::_thesis: for a being Real holds LD1 . (a * x) = a * (LD1 . x)
let a be Real; ::_thesis: LD1 . (a * x) = a * (LD1 . x)
LD1 . (a * x) = dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . (a * x)) by FUNCT_2:15;
then LD1 . (a * x) = dfx . (a * ((reproj ((modetrans (G,i0)),(0. (product G)))) . x)) by Th39;
then LD1 . (a * x) = a * (dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . x)) by LOPBAN_1:def_5;
hence LD1 . (a * x) = a * (LD1 . x) by FUNCT_2:15; ::_thesis: verum
end;
then reconsider LD1 = LD1 as LinearOperator of (G . (modetrans (G,i0))),S by A14, VECTSP_1:def_20, LOPBAN_1:def_5;
consider K0 being Real such that
A15: ( 0 <= K0 & ( for x being VECTOR of (product G) holds ||.(dfx . x).|| <= K0 * ||.x.|| ) ) by LOPBAN_1:def_8;
now__::_thesis:_for_r_being_VECTOR_of_(G_._(modetrans_(G,i0)))_holds_||.(LD1_._r).||_<=_K0_*_||.r.||
let r be VECTOR of (G . (modetrans (G,i0))); ::_thesis: ||.(LD1 . r).|| <= K0 * ||.r.||
||.(dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . r)).|| <= K0 * ||.((reproj ((modetrans (G,i0)),(0. (product G)))) . r).|| by A15;
then ||.(dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . r)).|| <= K0 * ||.r.|| by Th21;
hence ||.(LD1 . r).|| <= K0 * ||.r.|| by FUNCT_2:15; ::_thesis: verum
end;
then LD1 is Lipschitzian by A15, LOPBAN_1:def_8;
then reconsider LD1 = LD1 as Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i0))),S)) by LOPBAN_1:def_9;
now__::_thesis:_for_s_being_set_st_s_in_(reproj_((modetrans_(G,i0)),x))_.:_Nx0_holds_
s_in_dom_f
let s be set ; ::_thesis: ( s in (reproj ((modetrans (G,i0)),x)) .: Nx0 implies s in dom f )
assume s in (reproj ((modetrans (G,i0)),x)) .: Nx0 ; ::_thesis: s in dom f
then ex t being Element of (G . (modetrans (G,i0))) st
( t in Nx0 & s = (reproj ((modetrans (G,i0)),x)) . t ) by FUNCT_2:65;
then s in N by A5;
hence s in dom f by A2; ::_thesis: verum
end;
then A16: (reproj ((modetrans (G,i0)),x)) .: Nx0 c= dom f by TARSKI:def_3;
dom (reproj ((modetrans (G,i0)),x)) = the carrier of (G . (modetrans (G,i0))) by FUNCT_2:def_1;
then A17: Nx0 c= dom (f * (reproj ((modetrans (G,i0)),x))) by A16, FUNCT_3:3;
A18: for y being Point of (G . (modetrans (G,i0))) st y in Nx0 holds
((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0))
proof
let y be Point of (G . (modetrans (G,i0))); ::_thesis: ( y in Nx0 implies ((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0)) )
assume A19: y in Nx0 ; ::_thesis: ((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0))
then A20: (reproj ((modetrans (G,i0)),x)) . y in N by A5;
A21: (reproj ((modetrans (G,i0)),x)) . x0 = x +* ((modetrans (G,i0)),x0) by Def4;
A22: the carrier of (product G) = product (carr G) by Th10;
x . (modetrans (G,i0)) = x0 by Def3, A22;
then A23: x = x +* ((modetrans (G,i0)),x0) by FUNCT_7:35;
A24: (reproj ((modetrans (G,i0)),x)) . x0 in N by A5, NFCONT_1:4;
(f * (reproj ((modetrans (G,i0)),x))) /. y = (f * (reproj ((modetrans (G,i0)),x))) . y by A19, A17, PARTFUN1:def_6;
then (f * (reproj ((modetrans (G,i0)),x))) /. y = f . ((reproj ((modetrans (G,i0)),x)) . y) by FUNCT_2:15;
then A25: (f * (reproj ((modetrans (G,i0)),x))) /. y = f /. ((reproj ((modetrans (G,i0)),x)) . y) by A20, A2, PARTFUN1:def_6;
R /. ((reproj ((modetrans (G,i0)),(0. (product G)))) . (y - x0)) = R . ((reproj ((modetrans (G,i0)),(0. (product G)))) . (y - x0)) by A8, PARTFUN1:def_6;
then R /. ((reproj ((modetrans (G,i0)),(0. (product G)))) . (y - x0)) = R1 . (y - x0) by A9, FUNCT_1:13;
then A26: R /. ((reproj ((modetrans (G,i0)),(0. (product G)))) . (y - x0)) = R1 /. (y - x0) by A10, PARTFUN1:def_6;
x0 in Nx0 by NFCONT_1:4;
then (f * (reproj ((modetrans (G,i0)),x))) /. x0 = (f * (reproj ((modetrans (G,i0)),x))) . x0 by A17, PARTFUN1:def_6;
then (f * (reproj ((modetrans (G,i0)),x))) /. x0 = f . ((reproj ((modetrans (G,i0)),x)) . x0) by FUNCT_2:15;
then ((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = (f /. ((reproj ((modetrans (G,i0)),x)) . y)) - (f /. x) by A25, A23, A24, A2, A21, PARTFUN1:def_6;
then ((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = ((diff (f,x)) . (((reproj ((modetrans (G,i0)),x)) . y) - x)) + (R /. (((reproj ((modetrans (G,i0)),x)) . y) - x)) by A3, A19, A5;
then ((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = (dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . (y - x0))) + (R /. (((reproj ((modetrans (G,i0)),x)) . y) - x)) by Th22;
then ((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = (dfx . ((reproj ((modetrans (G,i0)),(0. (product G)))) . (y - x0))) + (R /. ((reproj ((modetrans (G,i0)),(0. (product G)))) . (y - x0))) by Th22;
hence ((f * (reproj ((modetrans (G,i0)),x))) /. y) - ((f * (reproj ((modetrans (G,i0)),x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0)) by A26, FUNCT_2:15; ::_thesis: verum
end;
then A27: f * (reproj ((modetrans (G,i0)),x)) is_differentiable_in x0 by A17, NDIFF_1:def_6;
hence f is_partial_differentiable_in x,i0 by Def6; ::_thesis: partdiff (f,x,i0) = (diff (f,x)) * (reproj ((modetrans (G,i0)),(0. (product G))))
thus partdiff (f,x,i0) = (diff (f,x)) * (reproj ((modetrans (G,i0)),(0. (product G)))) by A27, A17, A18, NDIFF_1:def_7; ::_thesis: verum
end;
Lm5: for G being non-trivial RealNormSpace-Sequence
for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for x being Point of (product G) ex L being Lipschitzian LinearOperator of (product G),S st
for h being Element of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for x being Point of (product G) ex L being Lipschitzian LinearOperator of (product G),S st
for h being Element of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
let S be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),S
for x being Point of (product G) ex L being Lipschitzian LinearOperator of (product G),S st
for h being Element of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
let f be PartFunc of (product G),S; ::_thesis: for x being Point of (product G) ex L being Lipschitzian LinearOperator of (product G),S st
for h being Element of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
let x be Point of (product G); ::_thesis: ex L being Lipschitzian LinearOperator of (product G),S st
for h being Element of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
set m = len G;
defpred S1[ set , set ] means ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . $1) ) & $2 = Sum w );
A1: for v being Element of (product G) ex y being Element of S st S1[v,y]
proof
let v be Element of (product G); ::_thesis: ex y being Element of S st S1[v,y]
defpred S2[ set , set ] means ex i being Element of NAT st
( i = $1 & $2 = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . v) );
A2: for i being Nat st i in Seg (len G) holds
ex r being Element of S st S2[i,r]
proof
let i be Nat; ::_thesis: ( i in Seg (len G) implies ex r being Element of S st S2[i,r] )
assume i in Seg (len G) ; ::_thesis: ex r being Element of S st S2[i,r]
reconsider i0 = i as Element of NAT by ORDINAL1:def_12;
(partdiff (f,x,i0)) . ((proj (modetrans (G,i0))) . v) in the carrier of S ;
hence ex r being Element of S st S2[i,r] ; ::_thesis: verum
end;
consider w being FinSequence of S such that
A3: ( dom w = Seg (len G) & ( for i being Nat st i in Seg (len G) holds
S2[i,w . i] ) ) from FINSEQ_1:sch_5(A2);
A4: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_Seg_(len_G)_holds_
w_._i_=_(partdiff_(f,x,i))_._((proj_(modetrans_(G,i)))_._v)
let i be Element of NAT ; ::_thesis: ( i in Seg (len G) implies w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . v) )
assume i in Seg (len G) ; ::_thesis: w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . v)
then S2[i,w . i] by A3;
hence w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . v) ; ::_thesis: verum
end;
reconsider w0 = Sum w as Element of S ;
ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . v) ) & w0 = Sum w ) by A4, A3;
hence ex y0 being Element of S st S1[v,y0] ; ::_thesis: verum
end;
consider L being Function of (product G),S such that
A5: for h being Element of (product G) holds S1[h,L . h] from FUNCT_2:sch_3(A1);
A6: for s, t being Element of (product G) holds L . (s + t) = (L . s) + (L . t)
proof
let s, t be Element of (product G); ::_thesis: L . (s + t) = (L . s) + (L . t)
consider w being FinSequence of S such that
A7: ( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . s) ) & L . s = Sum w ) by A5;
consider v being FinSequence of S such that
A8: ( dom v = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
v . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . t) ) & L . t = Sum v ) by A5;
consider u being FinSequence of S such that
A9: ( dom u = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
u . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . (s + t)) ) & L . (s + t) = Sum u ) by A5;
A10: len w = len G by A7, FINSEQ_1:def_3;
A11: len v = len G by A8, FINSEQ_1:def_3;
A12: len u = len G by A9, FINSEQ_1:def_3;
now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_w_holds_
u_._i_=_(w_/._i)_+_(v_/._i)
let i be Element of NAT ; ::_thesis: ( i in dom w implies u . i = (w /. i) + (v /. i) )
assume A13: i in dom w ; ::_thesis: u . i = (w /. i) + (v /. i)
then A14: ( 1 <= i & i <= len G ) by A7, FINSEQ_1:1;
then w /. i = w . i by A10, FINSEQ_4:15;
then A15: w /. i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . s) by A7, A13;
v /. i = v . i by A14, A11, FINSEQ_4:15;
then A16: v /. i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . t) by A7, A8, A13;
A17: partdiff (f,x,i) is Lipschitzian LinearOperator of (G . (modetrans (G,i))),S by LOPBAN_1:def_9;
u . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . (s + t)) by A7, A9, A13;
then u . i = (partdiff (f,x,i)) . (((proj (modetrans (G,i))) . s) + ((proj (modetrans (G,i))) . t)) by Th35;
hence u . i = (w /. i) + (v /. i) by A15, A16, A17, VECTSP_1:def_20; ::_thesis: verum
end;
hence L . (s + t) = (L . s) + (L . t) by A9, A7, A8, A10, A11, A12, RLVECT_2:2; ::_thesis: verum
end;
for s being Element of (product G)
for r being Real holds L . (r * s) = r * (L . s)
proof
let s be Element of (product G); ::_thesis: for r being Real holds L . (r * s) = r * (L . s)
let r be Real; ::_thesis: L . (r * s) = r * (L . s)
consider w being FinSequence of S such that
A18: ( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . s) ) & L . s = Sum w ) by A5;
consider u being FinSequence of S such that
A19: ( dom u = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
u . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . (r * s)) ) & L . (r * s) = Sum u ) by A5;
A20: ( len w = len G & len u = len G ) by A18, A19, FINSEQ_1:def_3;
now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_w_holds_
u_._i_=_r_*_(w_/._i)
let i be Element of NAT ; ::_thesis: ( i in dom w implies u . i = r * (w /. i) )
assume A21: i in dom w ; ::_thesis: u . i = r * (w /. i)
then ( 1 <= i & i <= len G ) by A18, FINSEQ_1:1;
then w /. i = w . i by A20, FINSEQ_4:15;
then A22: w /. i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . s) by A18, A21;
A23: partdiff (f,x,i) is Lipschitzian LinearOperator of (G . (modetrans (G,i))),S by LOPBAN_1:def_9;
u . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . (r * s)) by A18, A19, A21;
then u . i = (partdiff (f,x,i)) . (r * ((proj (modetrans (G,i))) . s)) by Th40;
hence u . i = r * (w /. i) by A22, A23, LOPBAN_1:def_5; ::_thesis: verum
end;
hence L . (r * s) = r * (L . s) by A18, A19, A20, RLVECT_2:3; ::_thesis: verum
end;
then reconsider L = L as LinearOperator of (product G),S by A6, VECTSP_1:def_20, LOPBAN_1:def_5;
defpred S2[ set , set ] means ex i being Element of NAT st
( i = $1 & $2 = ||.(partdiff (f,x,i)).|| );
A24: for i being Nat st i in Seg (len G) holds
ex r being Element of REAL st S2[i,r]
proof
let i be Nat; ::_thesis: ( i in Seg (len G) implies ex r being Element of REAL st S2[i,r] )
assume i in Seg (len G) ; ::_thesis: ex r being Element of REAL st S2[i,r]
reconsider i0 = i as Element of NAT by ORDINAL1:def_12;
reconsider r = ||.(partdiff (f,x,i0)).|| as Element of REAL ;
S2[i,r] ;
hence ex r being Element of REAL st S2[i,r] ; ::_thesis: verum
end;
consider Kw being FinSequence of REAL such that
A25: ( dom Kw = Seg (len G) & ( for i being Nat st i in Seg (len G) holds
S2[i,Kw . i] ) ) from FINSEQ_1:sch_5(A24);
A26: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_Seg_(len_G)_holds_
Kw_._i_=_||.(partdiff_(f,x,i)).||
let i be Element of NAT ; ::_thesis: ( i in Seg (len G) implies Kw . i = ||.(partdiff (f,x,i)).|| )
assume i in Seg (len G) ; ::_thesis: Kw . i = ||.(partdiff (f,x,i)).||
then S2[i,Kw . i] by A25;
hence Kw . i = ||.(partdiff (f,x,i)).|| ; ::_thesis: verum
end;
A27: now__::_thesis:_for_i_being_Nat_st_i_in_dom_Kw_holds_
0_<=_Kw_._i
let i be Nat; ::_thesis: ( i in dom Kw implies 0 <= Kw . i )
assume i in dom Kw ; ::_thesis: 0 <= Kw . i
then Kw . i = ||.(partdiff (f,x,i)).|| by A26, A25;
hence 0 <= Kw . i ; ::_thesis: verum
end;
set LK = Sum Kw;
A28: 0 <= Sum Kw by A27, RVSUM_1:84;
for s being Element of (product G) holds ||.(L . s).|| <= (Sum Kw) * ||.s.||
proof
let s be Element of (product G); ::_thesis: ||.(L . s).|| <= (Sum Kw) * ||.s.||
consider w being FinSequence of S such that
A29: ( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . s) ) & L . s = Sum w ) by A5;
defpred S3[ set , set ] means ex i being Element of NAT st
( i = $1 & $2 = ||.(partdiff (f,x,i)).|| * ||.s.|| );
A30: for i being Nat st i in Seg (len G) holds
ex r being Element of REAL st S3[i,r]
proof
let i be Nat; ::_thesis: ( i in Seg (len G) implies ex r being Element of REAL st S3[i,r] )
assume i in Seg (len G) ; ::_thesis: ex r being Element of REAL st S3[i,r]
reconsider i0 = i as Element of NAT by ORDINAL1:def_12;
reconsider r = ||.(partdiff (f,x,i0)).|| * ||.s.|| as Element of REAL ;
S3[i,r] ;
hence ex r being Element of REAL st S3[i,r] ; ::_thesis: verum
end;
consider Dw being FinSequence of REAL such that
A31: ( dom Dw = Seg (len G) & ( for i being Nat st i in Seg (len G) holds
S3[i,Dw . i] ) ) from FINSEQ_1:sch_5(A30);
A32: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_Seg_(len_G)_holds_
Dw_._i_=_||.(partdiff_(f,x,i)).||_*_||.s.||
let i be Element of NAT ; ::_thesis: ( i in Seg (len G) implies Dw . i = ||.(partdiff (f,x,i)).|| * ||.s.|| )
assume i in Seg (len G) ; ::_thesis: Dw . i = ||.(partdiff (f,x,i)).|| * ||.s.||
then S3[i,Dw . i] by A31;
hence Dw . i = ||.(partdiff (f,x,i)).|| * ||.s.|| ; ::_thesis: verum
end;
defpred S4[ set , set ] means ex i being Element of NAT st
( i = $1 & $2 = ||.(w /. i).|| );
A33: for i being Nat st i in Seg (len G) holds
ex r being Element of REAL st S4[i,r]
proof
let i be Nat; ::_thesis: ( i in Seg (len G) implies ex r being Element of REAL st S4[i,r] )
assume i in Seg (len G) ; ::_thesis: ex r being Element of REAL st S4[i,r]
reconsider i0 = i as Element of NAT by ORDINAL1:def_12;
reconsider r = ||.(w /. i0).|| as Element of REAL ;
S4[i,r] ;
hence ex r being Element of REAL st S4[i,r] ; ::_thesis: verum
end;
consider yseq being FinSequence of REAL such that
A34: ( dom yseq = Seg (len G) & ( for i being Nat st i in Seg (len G) holds
S4[i,yseq . i] ) ) from FINSEQ_1:sch_5(A33);
A35: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_Seg_(len_G)_holds_
yseq_._i_=_||.(w_/._i).||
let i be Element of NAT ; ::_thesis: ( i in Seg (len G) implies yseq . i = ||.(w /. i).|| )
assume i in Seg (len G) ; ::_thesis: yseq . i = ||.(w /. i).||
then S4[i,yseq . i] by A34;
hence yseq . i = ||.(w /. i).|| ; ::_thesis: verum
end;
len w = len yseq by A29, A34, FINSEQ_3:29;
then A36: ||.(L . s).|| <= Sum yseq by A29, A35, Th7;
len G = len yseq by A34, FINSEQ_1:def_3;
then A37: yseq is Element of (len G) -tuples_on REAL by FINSEQ_2:92;
len Dw = len G by A31, FINSEQ_1:def_3;
then A38: Dw is Element of (len G) -tuples_on REAL by FINSEQ_2:92;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_(len_G)_holds_
yseq_._i_<=_Dw_._i
let i be Nat; ::_thesis: ( i in Seg (len G) implies yseq . i <= Dw . i )
assume A39: i in Seg (len G) ; ::_thesis: yseq . i <= Dw . i
then A40: yseq . i = ||.(w /. i).|| by A35;
w /. i = w . i by A39, A29, PARTFUN1:def_6;
then A41: ||.(w /. i).|| = ||.((partdiff (f,x,i)) . ((proj (modetrans (G,i))) . s)).|| by A29, A39;
reconsider DF1 = partdiff (f,x,i) as Lipschitzian LinearOperator of (G . (modetrans (G,i))),S by LOPBAN_1:def_9;
A42: ||.(DF1 . ((proj (modetrans (G,i))) . s)).|| <= ||.(partdiff (f,x,i)).|| * ||.((proj (modetrans (G,i))) . s).|| by LOPBAN_1:32;
product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
then reconsider ss = s as Element of product (carr G) ;
reconsider xi = (proj (modetrans (G,i))) . s as Point of (G . (modetrans (G,i))) ;
xi = ss . (modetrans (G,i)) by Def3;
then ||.(partdiff (f,x,i)).|| * ||.((proj (modetrans (G,i))) . s).|| <= ||.(partdiff (f,x,i)).|| * ||.s.|| by PRVECT_2:10, XREAL_1:64;
then ||.(w /. i).|| <= ||.(partdiff (f,x,i)).|| * ||.s.|| by A41, A42, XXREAL_0:2;
hence yseq . i <= Dw . i by A32, A39, A40; ::_thesis: verum
end;
then A43: Sum yseq <= Sum Dw by A37, A38, RVSUM_1:82;
len Kw = len G by A25, FINSEQ_1:def_3;
then reconsider KKw = Kw as Element of (len G) -tuples_on REAL by FINSEQ_2:92;
||.s.|| * KKw in (len G) -tuples_on REAL ;
then ex t being Element of REAL * st
( t = ||.s.|| * KKw & len t = len G ) ;
then A44: dom Dw = dom (||.s.|| * Kw) by A31, FINSEQ_1:def_3;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_Dw_holds_
Dw_._k_=_(||.s.||_*_Kw)_._k
let k be Nat; ::_thesis: ( k in dom Dw implies Dw . k = (||.s.|| * Kw) . k )
assume A45: k in dom Dw ; ::_thesis: Dw . k = (||.s.|| * Kw) . k
then Dw . k = ||.(partdiff (f,x,k)).|| * ||.s.|| by A32, A31;
then Dw . k = ||.s.|| * (Kw . k) by A26, A45, A31;
hence Dw . k = (||.s.|| * Kw) . k by RVSUM_1:45; ::_thesis: verum
end;
then Dw = ||.s.|| * Kw by A44, FINSEQ_1:13;
then Sum Dw = (Sum Kw) * ||.s.|| by RVSUM_1:87;
hence ||.(L . s).|| <= (Sum Kw) * ||.s.|| by A36, A43, XXREAL_0:2; ::_thesis: verum
end;
then reconsider L = L as Lipschitzian LinearOperator of (product G),S by A28, LOPBAN_1:def_8;
take L ; ::_thesis: for h being Element of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
thus for h being Element of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w ) by A5; ::_thesis: verum
end;
theorem Th42: :: NDIFF_5:42
for S being RealNormSpace
for h, g being FinSequence of S st len h = (len g) + 1 & ( for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ) holds
(h /. 1) - (h /. (len h)) = Sum g
proof
let S be RealNormSpace; ::_thesis: for h, g being FinSequence of S st len h = (len g) + 1 & ( for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ) holds
(h /. 1) - (h /. (len h)) = Sum g
let h, g be FinSequence of S; ::_thesis: ( len h = (len g) + 1 & ( for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ) implies (h /. 1) - (h /. (len h)) = Sum g )
assume that
A1: len h = (len g) + 1 and
A2: for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ; ::_thesis: (h /. 1) - (h /. (len h)) = Sum g
consider F being Function of NAT, the carrier of S such that
A3: ( Sum g = F . (len g) & F . 0 = 0. S & ( for j being Element of NAT
for v being Element of S st j < len g & v = g . (j + 1) holds
F . (j + 1) = (F . j) + v ) ) by RLVECT_1:def_12;
percases ( len g = 0 or len g > 0 ) ;
suppose len g = 0 ; ::_thesis: (h /. 1) - (h /. (len h)) = Sum g
hence (h /. 1) - (h /. (len h)) = Sum g by A3, A1, RLVECT_1:15; ::_thesis: verum
end;
supposeA4: len g > 0 ; ::_thesis: (h /. 1) - (h /. (len h)) = Sum g
defpred S1[ Nat] means ( $1 <= len g implies F . $1 = (h /. 1) - (h /. ($1 + 1)) );
A5: S1[1]
proof
assume A6: 1 <= len g ; ::_thesis: F . 1 = (h /. 1) - (h /. (1 + 1))
then 1 in Seg (len g) ;
then A7: 1 in dom g by FINSEQ_1:def_3;
reconsider zz0 = 0 as Element of NAT ;
g /. 1 = g . (zz0 + 1) by A7, PARTFUN1:def_6;
then F . (zz0 + 1) = (F . 0) + (g /. 1) by A3, A6
.= g /. 1 by A3, RLVECT_1:4 ;
hence F . 1 = (h /. 1) - (h /. (1 + 1)) by A7, A2; ::_thesis: verum
end;
A8: for j being Nat st 1 <= j & S1[j] holds
S1[j + 1]
proof
let j be Nat; ::_thesis: ( 1 <= j & S1[j] implies S1[j + 1] )
assume 1 <= j ; ::_thesis: ( not S1[j] or S1[j + 1] )
assume A9: S1[j] ; ::_thesis: S1[j + 1]
assume A10: j + 1 <= len g ; ::_thesis: F . (j + 1) = (h /. 1) - (h /. ((j + 1) + 1))
then A11: j < len g by NAT_1:13;
1 <= j + 1 by XREAL_1:38;
then A12: j + 1 in dom g by A10, FINSEQ_3:25;
then A13: g . (j + 1) = g /. (j + 1) by PARTFUN1:def_6;
j is Element of NAT by ORDINAL1:def_12;
then F . (j + 1) = (F . j) + (g /. (j + 1)) by A13, A11, A3
.= (F . j) + ((h /. (j + 1)) - (h /. ((j + 1) + 1))) by A2, A12
.= (((h /. 1) - (h /. (j + 1))) + (h /. (j + 1))) - (h /. ((j + 1) + 1)) by A9, A10, NAT_1:13, RLVECT_1:28
.= ((h /. 1) - ((h /. (j + 1)) - (h /. (j + 1)))) - (h /. ((j + 1) + 1)) by RLVECT_1:29
.= ((h /. 1) - (0. S)) - (h /. ((j + 1) + 1)) by RLVECT_1:15 ;
hence F . (j + 1) = (h /. 1) - (h /. ((j + 1) + 1)) by RLVECT_1:13; ::_thesis: verum
end;
A14: 1 <= len g by A4, NAT_1:14;
for i being Nat st 1 <= i holds
S1[i] from NAT_1:sch_8(A5, A8);
hence (h /. 1) - (h /. (len h)) = Sum g by A3, A1, A14; ::_thesis: verum
end;
end;
end;
theorem :: NDIFF_5:43
for G being non-trivial RealNormSpace-Sequence
for x, y being Element of product (carr G)
for Z being set holds x +* (y | Z) is Element of product (carr G) by CARD_3:79;
theorem Th44: :: NDIFF_5:44
for G being non-trivial RealNormSpace-Sequence
for x, y being Point of (product G)
for Z, x0 being Element of product (carr G)
for X being set st Z = 0. (product G) & x0 = x & y = Z +* (x0 | X) holds
||.y.|| <= ||.x.||
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for x, y being Point of (product G)
for Z, x0 being Element of product (carr G)
for X being set st Z = 0. (product G) & x0 = x & y = Z +* (x0 | X) holds
||.y.|| <= ||.x.||
let x, y be Point of (product G); ::_thesis: for Z, x0 being Element of product (carr G)
for X being set st Z = 0. (product G) & x0 = x & y = Z +* (x0 | X) holds
||.y.|| <= ||.x.||
let Z, x0 be Element of product (carr G); ::_thesis: for X being set st Z = 0. (product G) & x0 = x & y = Z +* (x0 | X) holds
||.y.|| <= ||.x.||
let X be set ; ::_thesis: ( Z = 0. (product G) & x0 = x & y = Z +* (x0 | X) implies ||.y.|| <= ||.x.|| )
assume A1: ( Z = 0. (product G) & x0 = x & y = Z +* (x0 | X) ) ; ::_thesis: ||.y.|| <= ||.x.||
reconsider y0 = y as Element of product (carr G) by Th10;
A2: ||.y.|| = (productnorm G) . y by PRVECT_2:def_13
.= |.(normsequence (G,y0)).| by PRVECT_2:def_12 ;
A3: ||.x.|| = (productnorm G) . x by PRVECT_2:def_13
.= |.(normsequence (G,x0)).| by A1, PRVECT_2:def_12 ;
reconsider Ny = normsequence (G,y0) as len G -element FinSequence of REAL ;
reconsider Nx = normsequence (G,x0) as len G -element FinSequence of REAL ;
A4: ( len Nx = len G & len Ny = len G ) by CARD_1:def_7;
for k being Element of NAT st k in Seg (len Ny) holds
( 0 <= Ny . k & Ny . k <= Nx . k )
proof
let k be Element of NAT ; ::_thesis: ( k in Seg (len Ny) implies ( 0 <= Ny . k & Ny . k <= Nx . k ) )
assume A5: k in Seg (len Ny) ; ::_thesis: ( 0 <= Ny . k & Ny . k <= Nx . k )
then reconsider k1 = k as Element of dom G by A4, FINSEQ_1:def_3;
x0 is Element of the carrier of (product G) by Th10;
then reconsider xx = x0 as len G -element FinSequence ;
dom xx = Seg (len G) by FINSEQ_1:89;
then A6: k in dom x0 by A5, CARD_1:def_7;
reconsider yk = y0 . k1, xk = x0 . k1 as Element of the carrier of (G . k1) ;
A7: Nx . k = the normF of (G . k1) . (x0 . k1) by PRVECT_2:def_11;
A8: Ny . k = ||.yk.|| by PRVECT_2:def_11;
hence 0 <= Ny . k ; ::_thesis: Ny . k <= Nx . k
A9: Nx . k = ||.xk.|| by PRVECT_2:def_11;
percases ( k1 in X or not k1 in X ) ;
suppose k1 in X ; ::_thesis: Ny . k <= Nx . k
then A10: k1 in dom (x0 | X) by A6, RELAT_1:57;
then y0 . k1 = (x0 | X) . k1 by A1, FUNCT_4:13;
then y0 . k1 = x0 . k1 by A10, FUNCT_1:47;
hence Ny . k <= Nx . k by A7, PRVECT_2:def_11; ::_thesis: verum
end;
suppose not k1 in X ; ::_thesis: Ny . k <= Nx . k
then not k1 in dom (x0 | X) ;
then y0 . k1 = Z . k1 by A1, FUNCT_4:11;
then y0 . k1 = 0. (G . k1) by A1, Th14;
hence Ny . k <= Nx . k by A8, A9; ::_thesis: verum
end;
end;
end;
hence ||.y.|| <= ||.x.|| by A2, A3, A4, PRVECT_2:2; ::_thesis: verum
end;
theorem Th45: :: NDIFF_5:45
for G being non-trivial RealNormSpace-Sequence
for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for x, y being Point of (product G) ex h being FinSequence of (product G) ex g being FinSequence of S ex Z, y0 being Element of product (carr G) st
( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for x, y being Point of (product G) ex h being FinSequence of (product G) ex g being FinSequence of S ex Z, y0 being Element of product (carr G) st
( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
let S be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),S
for x, y being Point of (product G) ex h being FinSequence of (product G) ex g being FinSequence of S ex Z, y0 being Element of product (carr G) st
( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
let f be PartFunc of (product G),S; ::_thesis: for x, y being Point of (product G) ex h being FinSequence of (product G) ex g being FinSequence of S ex Z, y0 being Element of product (carr G) st
( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
let x, y be Point of (product G); ::_thesis: ex h being FinSequence of (product G) ex g being FinSequence of S ex Z, y0 being Element of product (carr G) st
( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
set m = len G;
A1: the carrier of (product G) = product (carr G) by Th10;
reconsider Z0 = 0. (product G) as Element of product (carr G) by Th10;
reconsider y0 = y as Element of product (carr G) by Th10;
reconsider y1 = y as len G -element FinSequence ;
reconsider Z1 = 0. (product G) as len G -element FinSequence ;
len y1 = len G by CARD_1:def_7;
then A2: dom y1 = dom G by FINSEQ_3:29;
len Z1 = len G by CARD_1:def_7;
then A3: dom Z1 = dom G by FINSEQ_3:29;
defpred S1[ Nat, set ] means $2 = Z0 +* (y0 | (Seg (((len G) + 1) -' $1)));
A4: for k being Nat st k in Seg ((len G) + 1) holds
ex x being Element of (product G) st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg ((len G) + 1) implies ex x being Element of (product G) st S1[k,x] )
assume k in Seg ((len G) + 1) ; ::_thesis: ex x being Element of (product G) st S1[k,x]
Z0 +* (y0 | (Seg (((len G) + 1) -' k))) is Element of product (carr G) by CARD_3:79;
hence ex x being Element of (product G) st S1[k,x] by A1; ::_thesis: verum
end;
consider h being FinSequence of (product G) such that
A5: ( dom h = Seg ((len G) + 1) & ( for j being Nat st j in Seg ((len G) + 1) holds
S1[j,h . j] ) ) from FINSEQ_1:sch_5(A4);
A6: now__::_thesis:_for_j_being_Nat_st_j_in_dom_h_holds_
h_/._j_=_Z0_+*_(y0_|_(Seg_(((len_G)_+_1)_-'_j)))
let j be Nat; ::_thesis: ( j in dom h implies h /. j = Z0 +* (y0 | (Seg (((len G) + 1) -' j))) )
assume A7: j in dom h ; ::_thesis: h /. j = Z0 +* (y0 | (Seg (((len G) + 1) -' j)))
then h /. j = h . j by PARTFUN1:def_6;
hence h /. j = Z0 +* (y0 | (Seg (((len G) + 1) -' j))) by A7, A5; ::_thesis: verum
end;
deffunc H1( Nat) -> Element of the carrier of S = f /. (x + (h /. $1));
consider z being FinSequence of S such that
A8: ( len z = (len G) + 1 & ( for j being Nat st j in dom z holds
z . j = H1(j) ) ) from FINSEQ_2:sch_1();
A9: now__::_thesis:_for_j_being_Nat_st_j_in_dom_z_holds_
z_/._j_=_f_/._(x_+_(h_/._j))
let j be Nat; ::_thesis: ( j in dom z implies z /. j = f /. (x + (h /. j)) )
assume A10: j in dom z ; ::_thesis: z /. j = f /. (x + (h /. j))
then z /. j = z . j by PARTFUN1:def_6;
hence z /. j = f /. (x + (h /. j)) by A10, A8; ::_thesis: verum
end;
deffunc H2( Nat) -> Element of the carrier of S = (z /. $1) - (z /. ($1 + 1));
consider g being FinSequence of S such that
A11: ( len g = len G & ( for j being Nat st j in dom g holds
g . j = H2(j) ) ) from FINSEQ_2:sch_1();
A12: now__::_thesis:_for_j_being_Nat_st_j_in_dom_g_holds_
g_/._j_=_(z_/._j)_-_(z_/._(j_+_1))
let j be Nat; ::_thesis: ( j in dom g implies g /. j = (z /. j) - (z /. (j + 1)) )
assume A13: j in dom g ; ::_thesis: g /. j = (z /. j) - (z /. (j + 1))
then g /. j = g . j by PARTFUN1:def_6;
hence g /. j = (z /. j) - (z /. (j + 1)) by A13, A11; ::_thesis: verum
end;
A14: ((len G) + 1) -' 1 = ((len G) + 1) - 1 by NAT_1:11, XREAL_1:233;
reconsider zz0 = 0 as Element of NAT ;
1 <= (len G) + 1 by NAT_1:11;
then A15: 1 in dom h by A5;
then h /. 1 = Z0 +* (y0 | (Seg (((len G) + 1) -' 1))) by A6
.= Z0 +* (y0 | (dom G)) by A14, FINSEQ_1:def_3
.= Z0 +* y0 by A2, RELAT_1:69 ;
then A16: h /. 1 = y by A2, A3, FUNCT_4:19;
A17: ((len G) + 1) -' (len z) = ((len G) + 1) - (len z) by A8, XREAL_1:233;
( 1 <= len z & len z <= (len G) + 1 ) by A8, NAT_1:14;
then A18: len z in dom h by A5;
then A19: h /. (len z) = Z0 +* (y0 | (Seg 0)) by A6, A17, A8
.= Z0 +* {}
.= 0. (product G) ;
A20: dom h = dom z by A5, A8, FINSEQ_1:def_3;
then A21: z /. 1 = f /. (x + y) by A9, A16, A15;
z /. (len z) = f /. (x + (h /. (len z))) by A9, A20, A18;
then A22: z /. (len z) = f /. x by A19, RLVECT_1:def_4;
take h ; ::_thesis: ex g being FinSequence of S ex Z, y0 being Element of product (carr G) st
( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
take g ; ::_thesis: ex Z, y0 being Element of product (carr G) st
( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
take Z0 ; ::_thesis: ex y0 being Element of product (carr G) st
( y0 = y & Z0 = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z0 +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
take y0 ; ::_thesis: ( y0 = y & Z0 = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z0 +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g )
A23: now__::_thesis:_for_i_being_Nat_st_i_in_dom_g_holds_
g_/._i_=_(f_/._(x_+_(h_/._i)))_-_(f_/._(x_+_(h_/._(i_+_1))))
let i be Nat; ::_thesis: ( i in dom g implies g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) )
assume A24: i in dom g ; ::_thesis: g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1))))
then A25: i in Seg (len G) by A11, FINSEQ_1:def_3;
then ( 1 <= i & i <= len G ) by FINSEQ_1:1;
then A26: i + 1 <= (len G) + 1 by XREAL_1:6;
len G <= (len G) + 1 by NAT_1:11;
then Seg (len G) c= Seg ((len G) + 1) by FINSEQ_1:5;
then A27: z /. i = f /. (x + (h /. i)) by A9, A5, A25, A20;
1 <= i + 1 by NAT_1:11;
then i + 1 in Seg ((len G) + 1) by A26;
then i + 1 in dom z by A8, FINSEQ_1:def_3;
then z /. (i + 1) = f /. (x + (h /. (i + 1))) by A9;
hence g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) by A12, A24, A27; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat
for_hi_being_Element_of_(product_G)_st_i_in_dom_h_&_h_/._i_=_hi_holds_
||.hi.||_<=_||.y.||
let i be Nat; ::_thesis: for hi being Element of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.||
let hi be Element of (product G); ::_thesis: ( i in dom h & h /. i = hi implies ||.hi.|| <= ||.y.|| )
assume A28: ( i in dom h & h /. i = hi ) ; ::_thesis: ||.hi.|| <= ||.y.||
then h /. i = Z0 +* (y0 | (Seg (((len G) + 1) -' i))) by A6;
hence ||.hi.|| <= ||.y.|| by Th44, A28; ::_thesis: verum
end;
hence ( y0 = y & Z0 = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z0 +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g ) by A6, A21, A22, A23, A8, A12, Th42, A5, A11, FINSEQ_1:def_3; ::_thesis: verum
end;
theorem Th46: :: NDIFF_5:46
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
(proj i) . y = xi
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
(proj i) . y = xi
let i be Element of dom G; ::_thesis: for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
(proj i) . y = xi
let x, y be Point of (product G); ::_thesis: for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
(proj i) . y = xi
let xi be Point of (G . i); ::_thesis: ( y = (reproj (i,x)) . xi implies (proj i) . y = xi )
assume A1: y = (reproj (i,x)) . xi ; ::_thesis: (proj i) . y = xi
A2: y = x +* (i,xi) by A1, Def4;
x in the carrier of (product G) ;
then x in product (carr G) by Th10;
then consider g being Function such that
A3: ( x = g & dom g = dom (carr G) & ( for y being set st y in dom (carr G) holds
g . y in (carr G) . y ) ) by CARD_3:def_5;
A4: i in dom G ;
A5: i in dom x by Lm1, A4, A3;
y is Element of product (carr G) by Th10;
then (proj i) . y = (x +* (i,xi)) . i by A2, Def3;
hence (proj i) . y = xi by A5, FUNCT_7:31; ::_thesis: verum
end;
theorem Th47: :: NDIFF_5:47
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for y being Point of (product G)
for q being Point of (G . i) st q = (proj i) . y holds
y = (reproj (i,y)) . q
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for y being Point of (product G)
for q being Point of (G . i) st q = (proj i) . y holds
y = (reproj (i,y)) . q
let i be Element of dom G; ::_thesis: for y being Point of (product G)
for q being Point of (G . i) st q = (proj i) . y holds
y = (reproj (i,y)) . q
let y be Point of (product G); ::_thesis: for q being Point of (G . i) st q = (proj i) . y holds
y = (reproj (i,y)) . q
let q be Point of (G . i); ::_thesis: ( q = (proj i) . y implies y = (reproj (i,y)) . q )
assume A1: q = (proj i) . y ; ::_thesis: y = (reproj (i,y)) . q
reconsider z1 = (reproj (i,y)) . q as len G -element FinSequence ;
reconsider z2 = y as len G -element FinSequence ;
A2: dom z1 = Seg (len G) by FINSEQ_1:89
.= dom z2 by FINSEQ_1:89 ;
for k being Nat st k in dom z1 holds
z1 . k = z2 . k
proof
let k be Nat; ::_thesis: ( k in dom z1 implies z1 . k = z2 . k )
assume k in dom z1 ; ::_thesis: z1 . k = z2 . k
product G = NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) by PRVECT_2:6;
then A3: q = y . i by A1, Def3;
percases ( k = i or k <> i ) ;
supposeA4: k = i ; ::_thesis: z1 . k = z2 . k
then (y +* (i,q)) . k = q by A3, FUNCT_7:35;
hence z1 . k = z2 . k by A4, A3, Def4; ::_thesis: verum
end;
suppose k <> i ; ::_thesis: z1 . k = z2 . k
then (y +* (i,q)) . k = y . k by FUNCT_7:32;
hence z1 . k = z2 . k by Def4; ::_thesis: verum
end;
end;
end;
hence y = (reproj (i,y)) . q by A2, FINSEQ_1:13; ::_thesis: verum
end;
theorem Th48: :: NDIFF_5:48
for G being non-trivial RealNormSpace-Sequence
for i being Element of dom G
for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i being Element of dom G
for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
let i be Element of dom G; ::_thesis: for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
let x, y be Point of (product G); ::_thesis: for xi being Point of (G . i) st y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
let xi be Point of (G . i); ::_thesis: ( y = (reproj (i,x)) . xi implies reproj (i,x) = reproj (i,y) )
assume A1: y = (reproj (i,x)) . xi ; ::_thesis: reproj (i,x) = reproj (i,y)
for v being Element of (G . i) holds (reproj (i,x)) . v = (reproj (i,y)) . v
proof
let v be Element of (G . i); ::_thesis: (reproj (i,x)) . v = (reproj (i,y)) . v
A2: ( (reproj (i,x)) . v = x +* (i,v) & (reproj (i,y)) . v = y +* (i,v) ) by Def4;
reconsider xv = (reproj (i,x)) . v, yv = (reproj (i,y)) . v as len G -element FinSequence ;
A3: ( dom xv = Seg (len G) & dom yv = Seg (len G) ) by FINSEQ_1:89;
then A4: dom xv = dom G by FINSEQ_1:def_3;
for k being Nat st k in dom xv holds
xv . k = yv . k
proof
let k be Nat; ::_thesis: ( k in dom xv implies xv . k = yv . k )
assume A5: k in dom xv ; ::_thesis: xv . k = yv . k
( x in the carrier of (product G) & y in the carrier of (product G) ) ;
then A6: ( x in product (carr G) & y in product (carr G) ) by Th10;
then consider g being Function such that
A7: ( x = g & dom g = dom (carr G) & ( for i being set st i in dom (carr G) holds
g . i in (carr G) . i ) ) by CARD_3:def_5;
consider g1 being Function such that
A8: ( y = g1 & dom g1 = dom (carr G) & ( for i being set st i in dom (carr G) holds
g1 . i in (carr G) . i ) ) by A6, CARD_3:def_5;
A9: ( k in dom y & k in dom x ) by A7, A8, Lm1, A5, A4;
percases ( k = i or k <> i ) ;
suppose k = i ; ::_thesis: xv . k = yv . k
then ( (y +* (i,v)) . k = v & (x +* (i,v)) . k = v ) by A9, FUNCT_7:31;
hence yv . k = xv . k by A2; ::_thesis: verum
end;
supposeA10: k <> i ; ::_thesis: xv . k = yv . k
A11: ( yv . k = y . k & xv . k = x . k ) by A2, A10, FUNCT_7:32;
y = x +* (i,xi) by A1, Def4;
hence yv . k = xv . k by A11, A10, FUNCT_7:32; ::_thesis: verum
end;
end;
end;
hence (reproj (i,x)) . v = (reproj (i,y)) . v by A3, FINSEQ_1:13; ::_thesis: verum
end;
hence reproj (i,x) = reproj (i,y) by FUNCT_2:def_8; ::_thesis: verum
end;
theorem Th49: :: NDIFF_5:49
for G being non-trivial RealNormSpace-Sequence
for i, j being Element of dom G
for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi & i <> j holds
(proj j) . x = (proj j) . y
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for i, j being Element of dom G
for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi & i <> j holds
(proj j) . x = (proj j) . y
let i, j be Element of dom G; ::_thesis: for x, y being Point of (product G)
for xi being Point of (G . i) st y = (reproj (i,x)) . xi & i <> j holds
(proj j) . x = (proj j) . y
let x, y be Point of (product G); ::_thesis: for xi being Point of (G . i) st y = (reproj (i,x)) . xi & i <> j holds
(proj j) . x = (proj j) . y
let xi be Point of (G . i); ::_thesis: ( y = (reproj (i,x)) . xi & i <> j implies (proj j) . x = (proj j) . y )
assume A1: ( y = (reproj (i,x)) . xi & i <> j ) ; ::_thesis: (proj j) . x = (proj j) . y
reconsider y1 = y as Element of product (carr G) by Th10;
A2: y = x +* (i,xi) by A1, Def4;
set ix = i .--> xi;
A3: the carrier of (product G) = product (carr G) by Th10;
y1 . j = x . j by A2, A1, FUNCT_7:32;
then (proj j) . y = x . j by Def3;
hence (proj j) . x = (proj j) . y by A3, Def3; ::_thesis: verum
end;
theorem :: NDIFF_5:50
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for i being Element of dom G
for x being Point of (product G)
for xi being Point of (G . i)
for f being PartFunc of (product G),F
for g being PartFunc of (G . i),F st (proj i) . x = xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,x,i)
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for i being Element of dom G
for x being Point of (product G)
for xi being Point of (G . i)
for f being PartFunc of (product G),F
for g being PartFunc of (G . i),F st (proj i) . x = xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,x,i)
let F be non trivial RealNormSpace; ::_thesis: for i being Element of dom G
for x being Point of (product G)
for xi being Point of (G . i)
for f being PartFunc of (product G),F
for g being PartFunc of (G . i),F st (proj i) . x = xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,x,i)
let i be Element of dom G; ::_thesis: for x being Point of (product G)
for xi being Point of (G . i)
for f being PartFunc of (product G),F
for g being PartFunc of (G . i),F st (proj i) . x = xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,x,i)
let x be Point of (product G); ::_thesis: for xi being Point of (G . i)
for f being PartFunc of (product G),F
for g being PartFunc of (G . i),F st (proj i) . x = xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,x,i)
let xi be Point of (G . i); ::_thesis: for f being PartFunc of (product G),F
for g being PartFunc of (G . i),F st (proj i) . x = xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,x,i)
let f be PartFunc of (product G),F; ::_thesis: for g being PartFunc of (G . i),F st (proj i) . x = xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,x,i)
let g be PartFunc of (G . i),F; ::_thesis: ( (proj i) . x = xi & g = f * (reproj (i,x)) implies diff (g,xi) = partdiff (f,x,i) )
i = modetrans (G,i) by Def5;
hence ( (proj i) . x = xi & g = f * (reproj (i,x)) implies diff (g,xi) = partdiff (f,x,i) ) ; ::_thesis: verum
end;
theorem Th51: :: NDIFF_5:51
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F))
for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F))
for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
let F be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),F
for x being Point of (product G)
for i being set
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F))
for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
let f be PartFunc of (product G),F; ::_thesis: for x being Point of (product G)
for i being set
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F))
for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
let x be Point of (product G); ::_thesis: for i being set
for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F))
for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
let i be set ; ::_thesis: for M being Real
for L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F))
for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
let M be Real; ::_thesis: for L being Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F))
for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
let L be Point of (R_NormSpace_of_BoundedLinearOperators ((G . (modetrans (G,i))),F)); ::_thesis: for p, q being Point of (G . (modetrans (G,i))) st i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) holds
||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
let p, q be Point of (G . (modetrans (G,i))); ::_thesis: ( i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) implies ||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).|| )
assume A1: ( i in dom G & ( for h being Point of (G . (modetrans (G,i))) st h in ].p,q.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
(reproj ((modetrans (G,i)),x)) . h in dom f ) & ( for h being Point of (G . (modetrans (G,i))) st h in [.p,q.] holds
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . h,i ) ) ; ::_thesis: ||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).||
set m = len G;
set S = G . (modetrans (G,i));
set g = f * (reproj ((modetrans (G,i)),x));
A2: now__::_thesis:_for_h_being_set_st_h_in_[.p,q.]_holds_
h_in_dom_(f_*_(reproj_((modetrans_(G,i)),x)))
let h be set ; ::_thesis: ( h in [.p,q.] implies h in dom (f * (reproj ((modetrans (G,i)),x))) )
assume A3: h in [.p,q.] ; ::_thesis: h in dom (f * (reproj ((modetrans (G,i)),x)))
then reconsider h1 = h as Point of (G . (modetrans (G,i))) ;
A4: dom (reproj ((modetrans (G,i)),x)) = the carrier of (G . (modetrans (G,i))) by FUNCT_2:def_1;
(reproj ((modetrans (G,i)),x)) . h1 in dom f by A1, A3;
hence h in dom (f * (reproj ((modetrans (G,i)),x))) by A4, FUNCT_1:11; ::_thesis: verum
end;
then A5: [.p,q.] c= dom (f * (reproj ((modetrans (G,i)),x))) by TARSKI:def_3;
A6: now__::_thesis:_for_x0_being_Point_of_(G_._(modetrans_(G,i)))_st_x0_in_[.p,q.]_holds_
f_*_(reproj_((modetrans_(G,i)),x))_is_differentiable_in_x0
let x0 be Point of (G . (modetrans (G,i))); ::_thesis: ( x0 in [.p,q.] implies f * (reproj ((modetrans (G,i)),x)) is_differentiable_in x0 )
assume A7: x0 in [.p,q.] ; ::_thesis: f * (reproj ((modetrans (G,i)),x)) is_differentiable_in x0
set y = (reproj ((modetrans (G,i)),x)) . x0;
A8: (proj (modetrans (G,i))) . ((reproj ((modetrans (G,i)),x)) . x0) = x0 by Th46;
f is_partial_differentiable_in (reproj ((modetrans (G,i)),x)) . x0,i by A1, A7;
then f * (reproj ((modetrans (G,i)),((reproj ((modetrans (G,i)),x)) . x0))) is_differentiable_in x0 by A8, Def6;
hence f * (reproj ((modetrans (G,i)),x)) is_differentiable_in x0 by Th48; ::_thesis: verum
end;
now__::_thesis:_for_z_being_set_st_z_in_].p,q.[_holds_
z_in_[.p,q.]
let z be set ; ::_thesis: ( z in ].p,q.[ implies z in [.p,q.] )
assume z in ].p,q.[ ; ::_thesis: z in [.p,q.]
then consider z1 being Real such that
A9: ( z = p + (z1 * (q - p)) & 0 < z1 & z1 < 1 ) ;
z = ((1 - z1) * p) + (z1 * q) by A9, Lm2;
then z in { (((1 - r1) * p) + (r1 * q)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } by A9;
hence z in [.p,q.] by RLTOPSP1:def_2; ::_thesis: verum
end;
then ].p,q.[ c= [.p,q.] by TARSKI:def_3;
then A10: for x being Point of (G . (modetrans (G,i))) st x in ].p,q.[ holds
f * (reproj ((modetrans (G,i)),x)) is_differentiable_in x by A6;
A11: for x being Point of (G . (modetrans (G,i))) st x in [.p,q.] holds
f * (reproj ((modetrans (G,i)),x)) is_continuous_in x by A6, NDIFF_1:44;
A12: now__::_thesis:_for_h_being_Point_of_(G_._(modetrans_(G,i)))_st_h_in_].p,q.[_holds_
||.((diff_((f_*_(reproj_((modetrans_(G,i)),x))),h))_-_L).||_<=_M
let h be Point of (G . (modetrans (G,i))); ::_thesis: ( h in ].p,q.[ implies ||.((diff ((f * (reproj ((modetrans (G,i)),x))),h)) - L).|| <= M )
set y = (reproj ((modetrans (G,i)),x)) . h;
assume h in ].p,q.[ ; ::_thesis: ||.((diff ((f * (reproj ((modetrans (G,i)),x))),h)) - L).|| <= M
then A13: ||.((partdiff (f,((reproj ((modetrans (G,i)),x)) . h),i)) - L).|| <= M by A1;
(proj (modetrans (G,i))) . ((reproj ((modetrans (G,i)),x)) . h) = h by Th46;
hence ||.((diff ((f * (reproj ((modetrans (G,i)),x))),h)) - L).|| <= M by A13, Th48; ::_thesis: verum
end;
A14: ( p in dom (f * (reproj ((modetrans (G,i)),x))) & q in dom (f * (reproj ((modetrans (G,i)),x))) ) by A2, RLTOPSP1:68;
( f /. ((reproj ((modetrans (G,i)),x)) . p) = f /. ((reproj ((modetrans (G,i)),x)) /. p) & f /. ((reproj ((modetrans (G,i)),x)) . q) = f /. ((reproj ((modetrans (G,i)),x)) /. q) ) ;
then ( f /. ((reproj ((modetrans (G,i)),x)) . p) = (f * (reproj ((modetrans (G,i)),x))) /. p & f /. ((reproj ((modetrans (G,i)),x)) . q) = (f * (reproj ((modetrans (G,i)),x))) /. q ) by A14, PARTFUN2:3;
hence ||.(((f /. ((reproj ((modetrans (G,i)),x)) . q)) - (f /. ((reproj ((modetrans (G,i)),x)) . p))) - (L . (q - p))).|| <= M * ||.(q - p).|| by A12, Th20, A5, A10, A11; ::_thesis: verum
end;
theorem Th52: :: NDIFF_5:52
for G being non-trivial RealNormSpace-Sequence
for x, y, z, w being Point of (product G)
for i being Element of dom G
for d being Real
for p, q, r being Point of (G . i) st ||.(y - x).|| < d & ||.(z - x).|| < d & p = (proj i) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
||.(w - x).|| < d
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for x, y, z, w being Point of (product G)
for i being Element of dom G
for d being Real
for p, q, r being Point of (G . i) st ||.(y - x).|| < d & ||.(z - x).|| < d & p = (proj i) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
||.(w - x).|| < d
let x, y, z, w be Point of (product G); ::_thesis: for i being Element of dom G
for d being Real
for p, q, r being Point of (G . i) st ||.(y - x).|| < d & ||.(z - x).|| < d & p = (proj i) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
||.(w - x).|| < d
let i be Element of dom G; ::_thesis: for d being Real
for p, q, r being Point of (G . i) st ||.(y - x).|| < d & ||.(z - x).|| < d & p = (proj i) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
||.(w - x).|| < d
let d be Real; ::_thesis: for p, q, r being Point of (G . i) st ||.(y - x).|| < d & ||.(z - x).|| < d & p = (proj i) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
||.(w - x).|| < d
let p, q, r be Point of (G . i); ::_thesis: ( ||.(y - x).|| < d & ||.(z - x).|| < d & p = (proj i) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r implies ||.(w - x).|| < d )
assume that
A1: ( ||.(y - x).|| < d & ||.(z - x).|| < d ) and
A2: ( p = (proj i) . y & z = (reproj (i,y)) . q ) and
A3: r in [.p,q.] and
A4: w = (reproj (i,y)) . r ; ::_thesis: ||.(w - x).|| < d
set wx = w - x;
set yx = y - x;
set zx = z - x;
reconsider xi = (proj i) . x as Point of (G . i) ;
r in { (((1 - t) * p) + (t * q)) where t is Real : ( 0 <= t & t <= 1 ) } by A3, RLTOPSP1:def_2;
then consider t being Real such that
A5: ( r = ((1 - t) * p) + (t * q) & 0 <= t & t <= 1 ) ;
A6: ( r = p + (t * (q - p)) & 0 <= t & t <= 1 ) by A5, Lm2;
reconsider wx0 = w - x, yx0 = y - x, zx0 = z - x as Element of product (carr G) by Th10;
reconsider Nwx = normsequence (G,wx0) as len G -element FinSequence of REAL ;
reconsider Nyx = normsequence (G,yx0) as len G -element FinSequence of REAL ;
reconsider Nzx = normsequence (G,zx0) as len G -element FinSequence of REAL ;
set tyz = ((1 - t) * (y - x)) + (t * (z - x));
reconsider tyz0 = ((1 - t) * (y - x)) + (t * (z - x)) as Element of product (carr G) by Th10;
reconsider Ntyz = normsequence (G,tyz0) as len G -element FinSequence of REAL ;
A7: 1 = (1 - t) + t ;
r = p + ((t * q) - (t * p)) by A6, RLVECT_1:34
.= (p + (- (t * p))) + (t * q) by RLVECT_1:def_3
.= ((1 * p) - (t * p)) + (t * q) by RLVECT_1:def_8
.= ((1 - t) * p) + (t * q) by RLVECT_1:35 ;
then A8: r - xi = (((1 - t) * p) + (t * q)) - (1 * xi) by RLVECT_1:def_8
.= (((1 - t) * p) + (t * q)) - (((1 - t) * xi) + (t * xi)) by A7, RLVECT_1:def_6
.= ((((1 - t) * p) + (t * q)) - (t * xi)) - ((1 - t) * xi) by RLVECT_1:27
.= (((1 - t) * p) + ((t * q) - (t * xi))) - ((1 - t) * xi) by RLVECT_1:28
.= ((t * q) - (t * xi)) + (((1 - t) * p) - ((1 - t) * xi)) by RLVECT_1:def_3
.= (t * (q - xi)) + (((1 - t) * p) - ((1 - t) * xi)) by RLVECT_1:34
.= (t * (q - xi)) + ((1 - t) * (p - xi)) by RLVECT_1:34 ;
reconsider Swx = w - x as len G -element FinSequence ;
reconsider Syz = ((1 - t) * (y - x)) + (t * (z - x)) as len G -element FinSequence ;
A9: ( dom Swx = Seg (len G) & dom Syz = Seg (len G) ) by FINSEQ_1:89;
A10: for k being Nat st k in dom Swx holds
Swx . k = Syz . k
proof
let k be Nat; ::_thesis: ( k in dom Swx implies Swx . k = Syz . k )
assume k in dom Swx ; ::_thesis: Swx . k = Syz . k
then reconsider k0 = k as Element of dom G by A9, FINSEQ_1:def_3;
percases ( k = i or k <> i ) ;
supposeA11: k = i ; ::_thesis: Swx . k = Syz . k
then Swx . k = (proj i) . wx0 by Def3;
then A12: Swx . k = ((proj i) . w) - ((proj i) . x) by Th37;
A13: (proj i) . z = q by A2, Th46;
Syz . k = (proj i) . tyz0 by A11, Def3;
then Syz . k = ((proj i) . ((1 - t) * (y - x))) + ((proj i) . (t * (z - x))) by Th35;
then Syz . k = ((1 - t) * ((proj i) . (y - x))) + ((proj i) . (t * (z - x))) by Th40;
then Syz . k = ((1 - t) * ((proj i) . (y - x))) + (t * ((proj i) . (z - x))) by Th40;
then Syz . k = ((1 - t) * (((proj i) . y) - ((proj i) . x))) + (t * ((proj i) . (z - x))) by Th37;
then Syz . k = ((1 - t) * (p - xi)) + (t * (q - xi)) by A2, A13, Th37;
hence Swx . k = Syz . k by A12, A8, A4, Th46; ::_thesis: verum
end;
suppose k <> i ; ::_thesis: Swx . k = Syz . k
then A14: ( (proj k0) . y = (proj k0) . w & (proj k0) . z = (proj k0) . y ) by A2, A4, Th49;
Swx . k = (proj k0) . wx0 by Def3;
then A15: Swx . k = ((proj k0) . w) - ((proj k0) . x) by Th37;
Syz . k = (proj k0) . tyz0 by Def3
.= ((proj k0) . ((1 - t) * (y - x))) + ((proj k0) . (t * (z - x))) by Th35
.= ((1 - t) * ((proj k0) . (y - x))) + ((proj k0) . (t * (z - x))) by Th40
.= ((1 - t) * ((proj k0) . (y - x))) + (t * ((proj k0) . (z - x))) by Th40 ;
then Syz . k = ((1 - t) * (((proj k0) . y) - ((proj k0) . x))) + (t * ((proj k0) . (z - x))) by Th37;
then Syz . k = ((1 - t) * (((proj k0) . y) - ((proj k0) . x))) + (t * (((proj k0) . y) - ((proj k0) . x))) by A14, Th37;
then Syz . k = (((1 - t) * ((proj k0) . y)) - ((1 - t) * ((proj k0) . x))) + (t * (((proj k0) . y) - ((proj k0) . x))) by RLVECT_1:34;
then Syz . k = (((1 - t) * ((proj k0) . y)) - ((1 - t) * ((proj k0) . x))) + ((t * ((proj k0) . y)) - (t * ((proj k0) . x))) by RLVECT_1:34;
then Syz . k = ((((1 - t) * ((proj k0) . y)) - ((1 - t) * ((proj k0) . x))) + (t * ((proj k0) . y))) - (t * ((proj k0) . x)) by RLVECT_1:def_3;
then Syz . k = (((1 - t) * ((proj k0) . y)) - (((1 - t) * ((proj k0) . x)) - (t * ((proj k0) . y)))) - (t * ((proj k0) . x)) by RLVECT_1:29;
then Syz . k = (((1 - t) * ((proj k0) . y)) + ((t * ((proj k0) . y)) + (- ((1 - t) * ((proj k0) . x))))) - (t * ((proj k0) . x)) by RLVECT_1:33;
then Syz . k = ((((1 - t) * ((proj k0) . y)) + (t * ((proj k0) . y))) + (- ((1 - t) * ((proj k0) . x)))) - (t * ((proj k0) . x)) by RLVECT_1:def_3;
then Syz . k = ((((1 - t) + t) * ((proj k0) . y)) + (- ((1 - t) * ((proj k0) . x)))) - (t * ((proj k0) . x)) by RLVECT_1:def_6;
then Syz . k = (((proj k0) . y) + (- ((1 - t) * ((proj k0) . x)))) - (t * ((proj k0) . x)) by RLVECT_1:def_8;
then Syz . k = ((proj k0) . y) + ((- ((1 - t) * ((proj k0) . x))) - (t * ((proj k0) . x))) by RLVECT_1:28;
then Syz . k = ((proj k0) . y) + (- ((t * ((proj k0) . x)) + ((1 - t) * ((proj k0) . x)))) by RLVECT_1:30;
then Syz . k = ((proj k0) . y) + (- ((t + (1 - t)) * ((proj k0) . x))) by RLVECT_1:def_6;
hence Swx . k = Syz . k by A15, A14, RLVECT_1:def_8; ::_thesis: verum
end;
end;
end;
A16: ( len Nwx = len G & len Ntyz = len G ) by CARD_1:def_7;
for k being Element of NAT st k in Seg (len Nwx) holds
( 0 <= Nwx . k & Nwx . k <= Ntyz . k )
proof
let k be Element of NAT ; ::_thesis: ( k in Seg (len Nwx) implies ( 0 <= Nwx . k & Nwx . k <= Ntyz . k ) )
assume A17: k in Seg (len Nwx) ; ::_thesis: ( 0 <= Nwx . k & Nwx . k <= Ntyz . k )
then reconsider k1 = k as Element of dom G by A16, FINSEQ_1:def_3;
reconsider wxk = wx0 . k1 as Element of (G . k1) ;
A18: Nwx . k = ||.wxk.|| by PRVECT_2:def_11;
wx0 . k1 = Syz . k by A10, A17, A16, A9;
hence ( 0 <= Nwx . k & Nwx . k <= Ntyz . k ) by A18, PRVECT_2:def_11; ::_thesis: verum
end;
then A19: |.Nwx.| <= |.Ntyz.| by A16, PRVECT_2:2;
A20: ||.(w - x).|| = (productnorm G) . (w - x) by PRVECT_2:def_13;
||.(((1 - t) * (y - x)) + (t * (z - x))).|| = (productnorm G) . (((1 - t) * (y - x)) + (t * (z - x))) by PRVECT_2:def_13
.= |.(normsequence (G,tyz0)).| by PRVECT_2:def_12 ;
then A21: ||.(w - x).|| <= ||.(((1 - t) * (y - x)) + (t * (z - x))).|| by A19, A20, PRVECT_2:def_12;
A22: ||.(((1 - t) * (y - x)) + (t * (z - x))).|| <= ((abs (1 - t)) * ||.(y - x).||) + ((abs t) * ||.(z - x).||) by NORMSP_1:5;
1 - t >= 0 by A6, XREAL_1:48;
then A23: ( abs (1 - t) = 1 - t & abs t = t ) by A6, ABSVALUE:def_1;
((abs (1 - t)) * ||.(y - x).||) + ((abs t) * ||.(z - x).||) < d
proof
percases ( t = 1 or t = 0 or ( t <> 1 & t <> 0 ) ) ;
suppose ( t = 1 or t = 0 ) ; ::_thesis: ((abs (1 - t)) * ||.(y - x).||) + ((abs t) * ||.(z - x).||) < d
hence ((abs (1 - t)) * ||.(y - x).||) + ((abs t) * ||.(z - x).||) < d by A1, A23; ::_thesis: verum
end;
suppose ( t <> 1 & t <> 0 ) ; ::_thesis: ((abs (1 - t)) * ||.(y - x).||) + ((abs t) * ||.(z - x).||) < d
then ( 0 < t & t < 1 ) by A6, XXREAL_0:1;
then ( 0 < t & 1 - t > 0 ) by XREAL_1:50;
then ( (abs (1 - t)) * ||.(y - x).|| < (1 - t) * d & (abs t) * ||.(z - x).|| < t * d ) by A1, A23, XREAL_1:68;
then ((abs (1 - t)) * ||.(y - x).||) + ((abs t) * ||.(z - x).||) < ((1 - t) * d) + (t * d) by XREAL_1:8;
hence ((abs (1 - t)) * ||.(y - x).||) + ((abs t) * ||.(z - x).||) < d ; ::_thesis: verum
end;
end;
end;
then ||.(((1 - t) * (y - x)) + (t * (z - x))).|| < d by A22, XXREAL_0:2;
hence ||.(w - x).|| < d by A21, XXREAL_0:2; ::_thesis: verum
end;
theorem Th53: :: NDIFF_5:53
for G being non-trivial RealNormSpace-Sequence
for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for X being Subset of (product G)
for x, y, z being Point of (product G)
for i being set
for p, q being Point of (G . (modetrans (G,i)))
for d, r being Real st i in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i)) - (partdiff (f,x,i))).|| <= r ) & z = (reproj ((modetrans (G,i)),y)) . p & q = (proj (modetrans (G,i))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i)) . (p - q))).|| <= ||.(p - q).|| * r
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for X being Subset of (product G)
for x, y, z being Point of (product G)
for i being set
for p, q being Point of (G . (modetrans (G,i)))
for d, r being Real st i in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i)) - (partdiff (f,x,i))).|| <= r ) & z = (reproj ((modetrans (G,i)),y)) . p & q = (proj (modetrans (G,i))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i)) . (p - q))).|| <= ||.(p - q).|| * r
let S be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),S
for X being Subset of (product G)
for x, y, z being Point of (product G)
for i being set
for p, q being Point of (G . (modetrans (G,i)))
for d, r being Real st i in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i)) - (partdiff (f,x,i))).|| <= r ) & z = (reproj ((modetrans (G,i)),y)) . p & q = (proj (modetrans (G,i))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i)) . (p - q))).|| <= ||.(p - q).|| * r
let f be PartFunc of (product G),S; ::_thesis: for X being Subset of (product G)
for x, y, z being Point of (product G)
for i being set
for p, q being Point of (G . (modetrans (G,i)))
for d, r being Real st i in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i)) - (partdiff (f,x,i))).|| <= r ) & z = (reproj ((modetrans (G,i)),y)) . p & q = (proj (modetrans (G,i))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i)) . (p - q))).|| <= ||.(p - q).|| * r
let X be Subset of (product G); ::_thesis: for x, y, z being Point of (product G)
for i being set
for p, q being Point of (G . (modetrans (G,i)))
for d, r being Real st i in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i)) - (partdiff (f,x,i))).|| <= r ) & z = (reproj ((modetrans (G,i)),y)) . p & q = (proj (modetrans (G,i))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i)) . (p - q))).|| <= ||.(p - q).|| * r
let x, y, z be Point of (product G); ::_thesis: for i being set
for p, q being Point of (G . (modetrans (G,i)))
for d, r being Real st i in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i)) - (partdiff (f,x,i))).|| <= r ) & z = (reproj ((modetrans (G,i)),y)) . p & q = (proj (modetrans (G,i))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i)) . (p - q))).|| <= ||.(p - q).|| * r
let i0 be set ; ::_thesis: for p, q being Point of (G . (modetrans (G,i0)))
for d, r being Real st i0 in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i0 ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i0)) - (partdiff (f,x,i0))).|| <= r ) & z = (reproj ((modetrans (G,i0)),y)) . p & q = (proj (modetrans (G,i0))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i0)) . (p - q))).|| <= ||.(p - q).|| * r
let p, q be Point of (G . (modetrans (G,i0))); ::_thesis: for d, r being Real st i0 in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i0 ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i0)) - (partdiff (f,x,i0))).|| <= r ) & z = (reproj ((modetrans (G,i0)),y)) . p & q = (proj (modetrans (G,i0))) . y holds
||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i0)) . (p - q))).|| <= ||.(p - q).|| * r
let d, r be Real; ::_thesis: ( i0 in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i0 ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i0)) - (partdiff (f,x,i0))).|| <= r ) & z = (reproj ((modetrans (G,i0)),y)) . p & q = (proj (modetrans (G,i0))) . y implies ||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i0)) . (p - q))).|| <= ||.(p - q).|| * r )
assume A1: ( i0 in dom G & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= dom f & ( for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i0 ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
z in X ) & ( for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,i0)) - (partdiff (f,x,i0))).|| <= r ) & z = (reproj ((modetrans (G,i0)),y)) . p & q = (proj (modetrans (G,i0))) . y ) ; ::_thesis: ||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i0)) . (p - q))).|| <= ||.(p - q).|| * r
set i = modetrans (G,i0);
A2: y = (reproj ((modetrans (G,i0)),y)) . q by A1, Th47;
A3: now__::_thesis:_for_h_being_Point_of_(G_._(modetrans_(G,i0)))_st_h_in_[.q,p.]_holds_
(reproj_((modetrans_(G,i0)),y))_._h_in_dom_f
let h be Point of (G . (modetrans (G,i0))); ::_thesis: ( h in [.q,p.] implies (reproj ((modetrans (G,i0)),y)) . h in dom f )
assume h in [.q,p.] ; ::_thesis: (reproj ((modetrans (G,i0)),y)) . h in dom f
then ||.(((reproj ((modetrans (G,i0)),y)) . h) - x).|| < d by A1, Th52;
then (reproj ((modetrans (G,i0)),y)) . h in X by A1;
hence (reproj ((modetrans (G,i0)),y)) . h in dom f by A1; ::_thesis: verum
end;
A4: now__::_thesis:_for_h_being_Point_of_(G_._(modetrans_(G,i0)))_st_h_in_[.q,p.]_holds_
f_is_partial_differentiable_in_(reproj_((modetrans_(G,i0)),y))_._h,i0
let h be Point of (G . (modetrans (G,i0))); ::_thesis: ( h in [.q,p.] implies f is_partial_differentiable_in (reproj ((modetrans (G,i0)),y)) . h,i0 )
assume h in [.q,p.] ; ::_thesis: f is_partial_differentiable_in (reproj ((modetrans (G,i0)),y)) . h,i0
then ||.(((reproj ((modetrans (G,i0)),y)) . h) - x).|| < d by A1, Th52;
hence f is_partial_differentiable_in (reproj ((modetrans (G,i0)),y)) . h,i0 by A1; ::_thesis: verum
end;
for h being Point of (G . (modetrans (G,i0))) st h in ].q,p.[ holds
||.((partdiff (f,((reproj ((modetrans (G,i0)),y)) . h),i0)) - (partdiff (f,x,i0))).|| <= r
proof
let h be Point of (G . (modetrans (G,i0))); ::_thesis: ( h in ].q,p.[ implies ||.((partdiff (f,((reproj ((modetrans (G,i0)),y)) . h),i0)) - (partdiff (f,x,i0))).|| <= r )
assume A5: h in ].q,p.[ ; ::_thesis: ||.((partdiff (f,((reproj ((modetrans (G,i0)),y)) . h),i0)) - (partdiff (f,x,i0))).|| <= r
].q,p.[ c= [.q,p.] by Th16;
then ||.(((reproj ((modetrans (G,i0)),y)) . h) - x).|| < d by A1, A5, Th52;
hence ||.((partdiff (f,((reproj ((modetrans (G,i0)),y)) . h),i0)) - (partdiff (f,x,i0))).|| <= r by A1; ::_thesis: verum
end;
hence ||.(((f /. z) - (f /. y)) - ((partdiff (f,x,i0)) . (p - q))).|| <= ||.(p - q).|| * r by A2, A1, Th51, A3, A4; ::_thesis: verum
end;
theorem Th54: :: NDIFF_5:54
for G being non-trivial RealNormSpace-Sequence
for h being FinSequence of (product G)
for y, x being Point of (product G)
for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y))
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for h being FinSequence of (product G)
for y, x being Point of (product G)
for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y))
let h be FinSequence of (product G); ::_thesis: for y, x being Point of (product G)
for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y))
let y, x be Point of (product G); ::_thesis: for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y))
let y0, Z be Element of product (carr G); ::_thesis: for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y))
let j be Element of NAT ; ::_thesis: ( y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) implies x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) )
assume that
A1: y = y0 and
A2: Z = 0. (product G) and
A3: len h = (len G) + 1 and
A4: ( 1 <= j & j <= len G ) and
A5: for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ; ::_thesis: x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y))
len G <= len h by A3, NAT_1:11;
then j <= len h by A4, XXREAL_0:2;
then j in Seg (len h) by A4;
then j in dom h by FINSEQ_1:def_3;
then A6: h /. j = Z +* (y0 | (Seg (((len G) + 1) -' j))) by A5;
( 1 <= j + 1 & j + 1 <= len h ) by A3, A4, NAT_1:12, XREAL_1:6;
then j + 1 in Seg (len h) ;
then j + 1 in dom h by FINSEQ_1:def_3;
then A7: h /. (j + 1) = Z +* (y0 | (Seg (((len G) + 1) -' (j + 1)))) by A5;
j in Seg (len G) by A4;
then ((len G) -' j) + 1 in Seg (len G) by NAT_2:6;
then ((len G) + 1) -' j in Seg (len G) by A4, NAT_D:38;
then ((len G) + 1) -' j in dom G by FINSEQ_1:def_3;
then A8: modetrans (G,(((len G) + 1) -' j)) = ((len G) + 1) -' j by Def5;
set xh = x + (h /. (j + 1));
reconsider x1 = x, y1 = y as Element of product (carr G) by Th10;
reconsider xy = x + y as Element of product (carr G) by Th10;
x + (h /. (j + 1)) is Element of product (carr G) by Th10;
then consider g being Function such that
A9: ( x + (h /. (j + 1)) = g & dom g = dom (carr G) & ( for y being set st y in dom (carr G) holds
g . y in (carr G) . y ) ) by CARD_3:def_5;
A10: dom (x + (h /. (j + 1))) = dom G by A9, Lm1;
(proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y) = xy . (((len G) + 1) -' j) by A8, Def3;
then A11: (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) = (x + (h /. (j + 1))) +* ((modetrans (G,(((len G) + 1) -' j))),(xy . (((len G) + 1) -' j))) by Def4
.= (x + (h /. (j + 1))) +* ((modetrans (G,(((len G) + 1) -' j))) .--> (xy . (((len G) + 1) -' j))) by A10, FUNCT_7:def_3
.= (x + (h /. (j + 1))) +* ({(((len G) + 1) -' j)} --> (xy . (((len G) + 1) -' j))) by A8, FUNCOP_1:def_9 ;
reconsider F1 = x + (h /. j) as len G -element FinSequence ;
reconsider F2 = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) as len G -element FinSequence ;
reconsider h1 = h /. j as Element of product (carr G) by Th10;
reconsider xh1 = x + (h /. j) as Element of product (carr G) by Th10;
reconsider h2 = h /. (j + 1) as Element of product (carr G) by Th10;
A12: ( len F1 = len G & len F2 = len G ) by CARD_1:def_7;
for k being Nat st 1 <= k & k <= len F1 holds
F1 . k = F2 . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len F1 implies F1 . k = F2 . k )
assume A13: ( 1 <= k & k <= len F1 ) ; ::_thesis: F1 . k = F2 . k
then A14: k in Seg (len F1) by FINSEQ_1:1;
then reconsider k1 = k as Element of dom G by A12, FINSEQ_1:def_3;
(proj k1) . xh1 = ((proj k1) . x) + ((proj k1) . (h /. j)) by Th35;
then A15: F1 . k = ((proj k1) . x) + ((proj k1) . (h /. j)) by Def3;
y0 is Element of the carrier of (product G) by Th10;
then A16: dom y0 = Seg (len G) by FINSEQ_1:89;
A17: (proj k1) . (h /. j) = h1 . k by Def3;
A18: dom (y0 | (Seg (((len G) + 1) -' j))) = (dom y0) /\ (Seg (((len G) + 1) -' j)) by RELAT_1:61;
A19: the carrier of (product G) = product (carr G) by Th10;
percases ( not k in Seg (((len G) + 1) -' j) or k in Seg (((len G) + 1) -' j) ) ;
supposeA20: not k in Seg (((len G) + 1) -' j) ; ::_thesis: F1 . k = F2 . k
then not k in dom (y0 | (Seg (((len G) + 1) -' j))) by A18, XBOOLE_0:def_4;
then (proj k1) . (h /. j) = Z . k by A17, A6, FUNCT_4:11;
then A21: (proj k1) . (h /. j) = (proj k1) . (0. (product G)) by A2, Def3;
( not 1 <= k or not k <= ((len G) + 1) -' j ) by A20, FINSEQ_1:1;
then not k in dom ({(((len G) + 1) -' j)} --> (xy . (((len G) + 1) -' j))) by A13, TARSKI:def_1;
then ((x + (h /. (j + 1))) +* ({(((len G) + 1) -' j)} --> (xy . (((len G) + 1) -' j)))) . k1 = (x + (h /. (j + 1))) . k1 by FUNCT_4:11;
then A22: F2 . k = (proj k1) . (x + (h /. (j + 1))) by A19, A11, Def3;
A23: (proj k1) . (h /. (j + 1)) = h2 . k by Def3;
((len G) + 1) -' (j + 1) <= ((len G) + 1) -' j by NAT_1:11, NAT_D:41;
then Seg (((len G) + 1) -' (j + 1)) c= Seg (((len G) + 1) -' j) by FINSEQ_1:5;
then not k in Seg (((len G) + 1) -' (j + 1)) by A20;
then not k in (dom y0) /\ (Seg (((len G) + 1) -' (j + 1))) by XBOOLE_0:def_4;
then not k in dom (y0 | (Seg (((len G) + 1) -' (j + 1)))) by RELAT_1:61;
then (Z +* (y0 | (Seg (((len G) + 1) -' (j + 1))))) . k = Z . k by FUNCT_4:11;
then (proj k1) . (h /. (j + 1)) = (proj k1) . (0. (product G)) by A2, A23, A7, Def3;
hence F1 . k = F2 . k by A21, A15, A22, Th35; ::_thesis: verum
end;
supposeA24: k in Seg (((len G) + 1) -' j) ; ::_thesis: F1 . k = F2 . k
then A25: k in dom (y0 | (Seg (((len G) + 1) -' j))) by A18, A14, A16, A12, XBOOLE_0:def_4;
then (proj k1) . (h /. j) = (y0 | (Seg (((len G) + 1) -' j))) . k by A17, A6, FUNCT_4:13;
then (proj k1) . (h /. j) = y0 . k by A25, FUNCT_1:47;
then A26: (proj k1) . (h /. j) = (proj k1) . y by A1, Def3;
then A27: F1 . k = (proj k1) . (x + y) by A15, Th35;
percases ( k = ((len G) + 1) -' j or k <> ((len G) + 1) -' j ) ;
supposeA28: k = ((len G) + 1) -' j ; ::_thesis: F1 . k = F2 . k
A29: k in {k} by TARSKI:def_1;
then k in dom ({k} --> (xy . k)) by FUNCOP_1:13;
then ((x + (h /. (j + 1))) +* ({k} --> (xy . k))) . k1 = ({k} --> (xy . k)) . k by FUNCT_4:13;
then F2 . k = xy . k by A11, A29, A28, FUNCOP_1:7;
hence F1 . k = F2 . k by A27, Def3; ::_thesis: verum
end;
supposeA30: k <> ((len G) + 1) -' j ; ::_thesis: F1 . k = F2 . k
then not k in dom ({(((len G) + 1) -' j)} --> (xy . (((len G) + 1) -' j))) by TARSKI:def_1;
then F2 . k = (x + (h /. (j + 1))) . k by A11, FUNCT_4:11;
then A31: F2 . k = (proj k1) . (x + (h /. (j + 1))) by A19, Def3;
k <= ((len G) + 1) -' j by A24, FINSEQ_1:1;
then k < ((len G) + 1) -' j by A30, XXREAL_0:1;
then k <= (((len G) + 1) -' j) -' 1 by NAT_D:49;
then k <= ((len G) + 1) -' (j + 1) by NAT_2:30;
then k in Seg (((len G) + 1) -' (j + 1)) by A13, FINSEQ_1:1;
then A32: k in dom (y0 | (Seg (((len G) + 1) -' (j + 1)))) by A14, A16, A12, RELAT_1:57;
(proj k1) . (h /. (j + 1)) = h2 . k by Def3;
then (proj k1) . (h /. (j + 1)) = (y0 | (Seg (((len G) + 1) -' (j + 1)))) . k1 by A7, A32, FUNCT_4:13;
then (proj k1) . (h /. (j + 1)) = y0 . k by A32, FUNCT_1:47;
then (proj k1) . (h /. (j + 1)) = (proj k1) . y by A1, Def3;
hence F1 . k = F2 . k by A26, A15, A31, Th35; ::_thesis: verum
end;
end;
end;
end;
end;
hence x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) by A12, FINSEQ_1:def_17; ::_thesis: verum
end;
theorem Th55: :: NDIFF_5:55
for G being non-trivial RealNormSpace-Sequence
for h being FinSequence of (product G)
for y, x being Point of (product G)
for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for h being FinSequence of (product G)
for y, x being Point of (product G)
for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y
let h be FinSequence of (product G); ::_thesis: for y, x being Point of (product G)
for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y
let y, x be Point of (product G); ::_thesis: for y0, Z being Element of product (carr G)
for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y
let y0, Z be Element of product (carr G); ::_thesis: for j being Element of NAT st y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) holds
((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y
let j be Element of NAT ; ::_thesis: ( y = y0 & Z = 0. (product G) & len h = (len G) + 1 & 1 <= j & j <= len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) implies ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y )
assume that
A1: y = y0 and
A2: Z = 0. (product G) and
A3: ( len h = (len G) + 1 & 1 <= j & j <= len G ) and
A4: for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ; ::_thesis: ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y
x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) by A1, A2, A3, A4, Th54;
then (proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. j)) = (proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y) by Th46;
then A5: ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . ((x + (h /. j)) - (x + (h /. (j + 1)))) by Th37;
(x + (h /. j)) - (x + (h /. (j + 1))) = (((h /. j) + x) - x) - (h /. (j + 1)) by RLVECT_1:27
.= ((h /. j) + (x - x)) - (h /. (j + 1)) by RLVECT_1:28
.= ((h /. j) + (0. (product G))) - (h /. (j + 1)) by RLVECT_1:15
.= (h /. j) - (h /. (j + 1)) by RLVECT_1:4 ;
then A6: ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = ((proj (modetrans (G,(((len G) + 1) -' j)))) . (h /. j)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (h /. (j + 1))) by A5, Th37;
y0 is Element of the carrier of (product G) by Th10;
then A7: dom y0 = Seg (len G) by FINSEQ_1:89;
j in Seg (len G) by A3;
then ((len G) -' j) + 1 in Seg (len G) by NAT_2:6;
then A8: ((len G) + 1) -' j in Seg (len G) by A3, NAT_D:38;
A9: j < (len G) + 1 by A3, NAT_1:13;
then ((len G) + 1) -' j in Seg (((len G) + 1) -' j) by FINSEQ_1:3, NAT_D:36;
then A10: ((len G) + 1) -' j in dom (y0 | (Seg (((len G) + 1) -' j))) by A7, A8, RELAT_1:57;
((len G) + 1) -' j = (((len G) + 1) -' (j + 1)) + 1 by A9, NAT_2:7;
then A11: ((len G) + 1) -' (j + 1) < ((len G) + 1) -' j by NAT_1:13;
dom (y0 | (Seg (((len G) + 1) -' (j + 1)))) c= Seg (((len G) + 1) -' (j + 1)) by RELAT_1:58;
then A12: not ((len G) + 1) -' j in dom (y0 | (Seg (((len G) + 1) -' (j + 1)))) by A11, FINSEQ_1:1;
reconsider h1 = h /. j as Element of product (carr G) by Th10;
reconsider h2 = h /. (j + 1) as Element of product (carr G) by Th10;
j in Seg (len h) by A3, A9;
then j in dom h by FINSEQ_1:def_3;
then A13: h /. j = Z +* (y0 | (Seg (((len G) + 1) -' j))) by A4;
((len G) + 1) -' j in dom G by A8, FINSEQ_1:def_3;
then A14: modetrans (G,(((len G) + 1) -' j)) = ((len G) + 1) -' j by Def5;
then A15: (proj (modetrans (G,(((len G) + 1) -' j)))) . (h /. j) = h1 . (((len G) + 1) -' j) by Def3
.= (y0 | (Seg (((len G) + 1) -' j))) . (((len G) + 1) -' j) by A10, A13, FUNCT_4:13
.= y0 . (((len G) + 1) -' j) by A10, FUNCT_1:47
.= (proj (modetrans (G,(((len G) + 1) -' j)))) . y by A1, A14, Def3 ;
( 1 <= j + 1 & j + 1 <= len h ) by A3, NAT_1:12, XREAL_1:6;
then j + 1 in Seg (len h) ;
then j + 1 in dom h by FINSEQ_1:def_3;
then A16: h /. (j + 1) = Z +* (y0 | (Seg (((len G) + 1) -' (j + 1)))) by A4;
(proj (modetrans (G,(((len G) + 1) -' j)))) . (h /. (j + 1)) = h2 . (((len G) + 1) -' j) by A14, Def3
.= Z . (((len G) + 1) -' j) by A16, A12, FUNCT_4:11
.= (proj (modetrans (G,(((len G) + 1) -' j)))) . (0. (product G)) by A14, A2, Def3 ;
hence ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . (y - (0. (product G))) by A6, A15, Th37
.= (proj (modetrans (G,(((len G) + 1) -' j)))) . y by RLVECT_1:13 ;
::_thesis: verum
end;
theorem Th56: :: NDIFF_5:56
for G being non-trivial RealNormSpace-Sequence
for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for X being Subset of (product G)
for x being Point of (product G) st X is open & x in X & ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for S being non trivial RealNormSpace
for f being PartFunc of (product G),S
for X being Subset of (product G)
for x being Point of (product G) st X is open & x in X & ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) )
let S be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),S
for X being Subset of (product G)
for x being Point of (product G) st X is open & x in X & ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) )
let f be PartFunc of (product G),S; ::_thesis: for X being Subset of (product G)
for x being Point of (product G) st X is open & x in X & ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) )
let X be Subset of (product G); ::_thesis: for x being Point of (product G) st X is open & x in X & ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) )
let x be Point of (product G); ::_thesis: ( X is open & x in X & ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) implies ( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) ) )
assume A1: ( X is open & x in X & ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) ) ; ::_thesis: ( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) )
set m = len G;
A2: dom G = Seg (len G) by FINSEQ_1:def_3;
reconsider Z0 = 0. (product G) as Element of product (carr G) by Th10;
reconsider x0 = x as Element of product (carr G) by Th10;
reconsider x1 = x as len G -element FinSequence ;
reconsider Z1 = 0. (product G) as len G -element FinSequence ;
consider L being Lipschitzian LinearOperator of (product G),S such that
A3: for h being Point of (product G) ex w being FinSequence of S st
( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w ) by Lm5;
A4: for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
proof
let h be Point of (product G); ::_thesis: ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
consider w being FinSequence of S such that
A5: ( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w ) by A3;
take w ; ::_thesis: ( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
thus dom w = dom G by A5, FINSEQ_1:def_3; ::_thesis: ( ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w )
thus ( ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & L . h = Sum w ) by A5, A2; ::_thesis: verum
end;
consider d0 being Real such that
A6: d0 > 0 and
A7: { y where y is Element of (product G) : ||.(y - x).|| < d0 } c= X by A1, NDIFF_1:3;
set N = { y where y is Element of (product G) : ||.(y - x).|| < d0 } ;
{ y where y is Element of (product G) : ||.(y - x).|| < d0 } c= the carrier of (product G) by A7, XBOOLE_1:1;
then A8: { y where y is Element of (product G) : ||.(y - x).|| < d0 } is Neighbourhood of x by A6, NFCONT_1:def_1;
A9: 1 <= len G by NAT_1:14;
then len G in dom G by A2;
then f is_partial_differentiable_on X, len G by A1;
then X c= dom f by Def8;
then A10: { y where y is Element of (product G) : ||.(y - x).|| < d0 } c= dom f by A7, XBOOLE_1:1;
deffunc H1( Element of (product G)) -> Element of the carrier of S = ((f /. (x + $1)) - (f /. x)) - (L . $1);
consider R being Function of the carrier of (product G), the carrier of S such that
A11: for h being Element of the carrier of (product G) holds R . h = H1(h) from FUNCT_2:sch_4();
now__::_thesis:_for_r0_being_Real_st_r0_>_0_holds_
ex_d_being_Element_of_REAL_st_
(_0_<_d_&_(_for_y_being_Point_of_(product_G)_st_y_<>_0._(product_G)_&_||.y.||_<_d_holds_
(||.y.||_")_*_||.(R_/._y).||_<_r0_)_)
let r0 be Real; ::_thesis: ( r0 > 0 implies ex d being Element of REAL st
( 0 < d & ( for y being Point of (product G) st y <> 0. (product G) & ||.y.|| < d holds
(||.y.|| ") * ||.(R /. y).|| < r0 ) ) )
assume A12: r0 > 0 ; ::_thesis: ex d being Element of REAL st
( 0 < d & ( for y being Point of (product G) st y <> 0. (product G) & ||.y.|| < d holds
(||.y.|| ") * ||.(R /. y).|| < r0 ) )
set r1 = r0 / 2;
set r = (r0 / 2) / (len G);
defpred S1[ Nat, Element of REAL ] means ex k being Element of NAT st
( $1 = k & 0 < $2 & ( for q being Element of (product G) st q in X & ||.(q - x).|| < $2 holds
||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G) ) );
A13: for k0 being Nat st k0 in Seg (len G) holds
ex d being Element of REAL st S1[k0,d]
proof
let k0 be Nat; ::_thesis: ( k0 in Seg (len G) implies ex d being Element of REAL st S1[k0,d] )
assume A14: k0 in Seg (len G) ; ::_thesis: ex d being Element of REAL st S1[k0,d]
reconsider k = k0 as Element of NAT by ORDINAL1:def_12;
f `partial| (X,k) is_continuous_on X by A2, A14, A1;
then consider d being Real such that
A15: ( 0 < d & ( for q being Point of (product G) st q in X & ||.(q - x).|| < d holds
||.(((f `partial| (X,k)) /. q) - ((f `partial| (X,k)) /. x)).|| < (r0 / 2) / (len G) ) ) by A12, A1, NFCONT_1:19;
take d ; ::_thesis: S1[k0,d]
for q being Point of (product G) st q in X & ||.(q - x).|| < d holds
||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G)
proof
let q be Point of (product G); ::_thesis: ( q in X & ||.(q - x).|| < d implies ||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G) )
assume A16: ( q in X & ||.(q - x).|| < d ) ; ::_thesis: ||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G)
then A17: ||.(((f `partial| (X,k)) /. q) - ((f `partial| (X,k)) /. x)).|| < (r0 / 2) / (len G) by A15;
A18: f is_partial_differentiable_on X,k by A1, A14, A2;
then (f `partial| (X,k)) /. q = partdiff (f,q,k) by A16, A14, A2, Def9;
hence ||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G) by A17, A18, A1, A14, A2, Def9; ::_thesis: verum
end;
hence ex k being Element of NAT st
( k0 = k & 0 < d & ( for q being Element of (product G) st q in X & ||.(q - x).|| < d holds
||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G) ) ) by A15; ::_thesis: verum
end;
consider Dseq being FinSequence of REAL such that
A19: ( dom Dseq = Seg (len G) & ( for i being Nat st i in Seg (len G) holds
S1[i,Dseq . i] ) ) from FINSEQ_1:sch_5(A13);
len G in Seg (len G) by A9;
then reconsider rDseq = rng Dseq as non empty ext-real-membered set by A19, FUNCT_1:3;
reconsider rDseq = rDseq as non empty ext-real-membered left_end right_end set ;
A20: min rDseq in rng Dseq by XXREAL_2:def_7;
then reconsider d1 = min rDseq as Real ;
set d = min (d0,d1);
A21: ( min (d0,d1) <= d0 & min (d0,d1) <= d1 ) by XXREAL_0:17;
consider i1 being set such that
A22: ( i1 in dom Dseq & d1 = Dseq . i1 ) by A20, FUNCT_1:def_3;
reconsider i1 = i1 as Nat by A22;
A23: ex k being Element of NAT st
( i1 = k & 0 < Dseq . i1 & ( for q being Element of (product G) st q in X & ||.(q - x).|| < Dseq . i1 holds
||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G) ) ) by A19, A22;
A24: now__::_thesis:_for_q_being_Element_of_(product_G)_st_||.(q_-_x).||_<_min_(d0,d1)_holds_
q_in_X
let q be Element of (product G); ::_thesis: ( ||.(q - x).|| < min (d0,d1) implies q in X )
assume ||.(q - x).|| < min (d0,d1) ; ::_thesis: q in X
then ||.(q - x).|| < d0 by A21, XXREAL_0:2;
then q in { y where y is Element of (product G) : ||.(y - x).|| < d0 } ;
hence q in X by A7; ::_thesis: verum
end;
A25: now__::_thesis:_for_q_being_Element_of_(product_G)
for_i_being_Element_of_NAT_st_||.(q_-_x).||_<_min_(d0,d1)_&_i_in_Seg_(len_G)_holds_
||.((partdiff_(f,q,i))_-_(partdiff_(f,x,i))).||_<_(r0_/_2)_/_(len_G)
let q be Element of (product G); ::_thesis: for i being Element of NAT st ||.(q - x).|| < min (d0,d1) & i in Seg (len G) holds
||.((partdiff (f,q,i)) - (partdiff (f,x,i))).|| < (r0 / 2) / (len G)
let i be Element of NAT ; ::_thesis: ( ||.(q - x).|| < min (d0,d1) & i in Seg (len G) implies ||.((partdiff (f,q,i)) - (partdiff (f,x,i))).|| < (r0 / 2) / (len G) )
assume A26: ( ||.(q - x).|| < min (d0,d1) & i in Seg (len G) ) ; ::_thesis: ||.((partdiff (f,q,i)) - (partdiff (f,x,i))).|| < (r0 / 2) / (len G)
reconsider i0 = i as Nat ;
consider k being Element of NAT such that
A27: ( i0 = k & 0 < Dseq . i0 & ( for q being Element of (product G) st q in X & ||.(q - x).|| < Dseq . i0 holds
||.((partdiff (f,q,k)) - (partdiff (f,x,k))).|| < (r0 / 2) / (len G) ) ) by A19, A26;
Dseq . i0 in rng Dseq by A19, A26, FUNCT_1:3;
then d1 <= Dseq . i0 by XXREAL_2:def_7;
then min (d0,d1) <= Dseq . i0 by A21, XXREAL_0:2;
then ||.(q - x).|| < Dseq . i0 by A26, XXREAL_0:2;
hence ||.((partdiff (f,q,i)) - (partdiff (f,x,i))).|| < (r0 / 2) / (len G) by A24, A26, A27; ::_thesis: verum
end;
take d = min (d0,d1); ::_thesis: ( 0 < d & ( for y being Point of (product G) st y <> 0. (product G) & ||.y.|| < d holds
(||.y.|| ") * ||.(R /. y).|| < r0 ) )
thus 0 < d by A6, A22, A23, XXREAL_0:21; ::_thesis: for y being Point of (product G) st y <> 0. (product G) & ||.y.|| < d holds
(||.y.|| ") * ||.(R /. y).|| < r0
thus for y being Point of (product G) st y <> 0. (product G) & ||.y.|| < d holds
(||.y.|| ") * ||.(R /. y).|| < r0 ::_thesis: verum
proof
let y be Point of (product G); ::_thesis: ( y <> 0. (product G) & ||.y.|| < d implies (||.y.|| ") * ||.(R /. y).|| < r0 )
assume A28: ( y <> 0. (product G) & ||.y.|| < d ) ; ::_thesis: (||.y.|| ") * ||.(R /. y).|| < r0
then A29: 0 <> ||.y.|| by NORMSP_0:def_5;
set z = R /. y;
consider h being FinSequence of (product G), g being FinSequence of S, Z, y0 being Element of product (carr G) such that
A30: ( y0 = y & Z = 0. (product G) & len h = (len G) + 1 & len g = len G & ( for i being Nat st i in dom h holds
h /. i = Z +* (y0 | (Seg (((len G) + 1) -' i))) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Point of (product G) st i in dom h & h /. i = hi holds
||.hi.|| <= ||.y.|| ) & (f /. (x + y)) - (f /. x) = Sum g ) by Th45;
consider w being FinSequence of S such that
A31: ( dom w = Seg (len G) & ( for i being Element of NAT st i in Seg (len G) holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . y) ) & L . y = Sum w ) by A3;
A32: ( dom (idseq (len G)) = Seg (len G) & rng (idseq (len G)) = Seg (len G) ) by FUNCT_2:def_1, FUNCT_2:def_3;
then A33: ( dom (Rev (idseq (len G))) = Seg (len G) & rng (Rev (idseq (len G))) = Seg (len G) ) by FINSEQ_5:57;
then reconsider Ri = Rev (idseq (len G)) as Function of (Seg (len G)),(Seg (len G)) by FUNCT_2:1;
( Ri is one-to-one & Ri is onto ) by A33, FUNCT_2:def_3;
then reconsider Ri = Rev (idseq (len G)) as Permutation of (dom w) by A31;
A34: ( len (idseq (len G)) = len G & len w = len G ) by A31, A32, FINSEQ_1:def_3;
dom (w * Ri) = dom Ri by A33, RELAT_1:27;
then A35: dom (w * Ri) = dom (Rev w) by A33, A31, FINSEQ_5:57;
reconsider wRi = w * Ri as FinSequence of S by FINSEQ_2:47;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_(Rev_w)_holds_
(Rev_w)_._k_=_wRi_._k
let k be Nat; ::_thesis: ( k in dom (Rev w) implies (Rev w) . k = wRi . k )
assume A36: k in dom (Rev w) ; ::_thesis: (Rev w) . k = wRi . k
then A37: k in dom (Rev (idseq (len G))) by A33, A31, FINSEQ_5:57;
then A38: ( 1 <= k & k <= len G ) by A33, FINSEQ_1:1;
then reconsider mk = (len G) - k as Nat by NAT_1:21;
reconsider zr0 = 0 as Nat ;
0 <= mk ;
then A39: zr0 + 1 <= ((len G) - k) + 1 by XREAL_1:6;
k - 1 >= 1 - 1 by A38, XREAL_1:9;
then (len G) - (k - 1) <= len G by XREAL_1:43;
then A40: mk + 1 in Seg (len G) by A39;
(Rev w) . k = w . (((len (idseq (len G))) - k) + 1) by A34, A36, FINSEQ_5:def_3
.= w . ((idseq (len G)) . (((len (idseq (len G))) - k) + 1)) by A40, A34, FINSEQ_2:49
.= w . ((Rev (idseq (len G))) . k) by A37, FINSEQ_5:def_3 ;
hence (Rev w) . k = wRi . k by A36, A35, FUNCT_1:12; ::_thesis: verum
end;
then A41: Sum (Rev w) = Sum w by A35, FINSEQ_1:13, RLVECT_2:7;
deffunc H2( Nat) -> Element of the carrier of S = (g /. $1) - ((Rev w) /. $1);
consider gw being FinSequence of S such that
A42: ( len gw = len G & ( for j being Nat st j in dom gw holds
gw . j = H2(j) ) ) from FINSEQ_2:sch_1();
A43: now__::_thesis:_for_j_being_Element_of_NAT_st_j_in_dom_g_holds_
gw_._j_=_(g_/._j)_-_((Rev_w)_/._j)
let j be Element of NAT ; ::_thesis: ( j in dom g implies gw . j = (g /. j) - ((Rev w) /. j) )
assume j in dom g ; ::_thesis: gw . j = (g /. j) - ((Rev w) /. j)
then j in Seg (len G) by A30, FINSEQ_1:def_3;
then j in dom gw by A42, FINSEQ_1:def_3;
hence gw . j = (g /. j) - ((Rev w) /. j) by A42; ::_thesis: verum
end;
len (Rev w) = len g by A30, A34, FINSEQ_5:def_3;
then Sum gw = (Sum g) - (Sum (Rev w)) by A30, A42, A43, RLVECT_2:5;
then A44: R /. y = Sum gw by A11, A30, A31, A41;
A45: for j being Element of NAT st j in dom gw holds
||.(gw /. j).|| <= ||.y.|| * ((r0 / 2) / (len G))
proof
let j be Element of NAT ; ::_thesis: ( j in dom gw implies ||.(gw /. j).|| <= ||.y.|| * ((r0 / 2) / (len G)) )
assume A46: j in dom gw ; ::_thesis: ||.(gw /. j).|| <= ||.y.|| * ((r0 / 2) / (len G))
then A47: j in Seg (len G) by A42, FINSEQ_1:def_3;
then A48: j in dom g by A30, FINSEQ_1:def_3;
then A49: g /. j = (f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1)))) by A30;
A50: ( 1 <= j & j <= len G ) by A47, FINSEQ_1:1;
then A51: ( (len G) + 1 <= (len G) + j & j + 1 <= (len G) + 1 ) by XREAL_1:6;
then ( ((len G) + 1) - j <= len G & 1 <= ((len G) + 1) - j ) by XREAL_1:19, XREAL_1:20;
then ( ((len G) + 1) -' j <= len G & 1 <= ((len G) + 1) -' j ) by A50, NAT_D:37;
then A52: ((len G) + 1) -' j in Seg (len G) ;
then f is_partial_differentiable_on X,((len G) + 1) -' j by A1, A2;
then A53: ( X c= dom f & ( for x being Element of (product G) st x in X holds
f is_partial_differentiable_in x,((len G) + 1) -' j ) ) by Th24, A1;
w /. (((len G) + 1) -' j) = w . (((len G) + 1) -' j) by A31, A52, PARTFUN1:def_6;
then A54: w /. (((len G) + 1) -' j) = (partdiff (f,x,(((len G) + 1) -' j))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . y) by A52, A31;
A55: now__::_thesis:_for_j_being_Element_of_NAT_st_1_<=_j_&_j_<=_(len_G)_+_1_holds_
||.((x_+_(h_/._j))_-_x).||_<_d
let j be Element of NAT ; ::_thesis: ( 1 <= j & j <= (len G) + 1 implies ||.((x + (h /. j)) - x).|| < d )
reconsider hj = h /. j as Element of (product G) ;
assume ( 1 <= j & j <= (len G) + 1 ) ; ::_thesis: ||.((x + (h /. j)) - x).|| < d
then j in dom h by A30, FINSEQ_3:25;
then A56: ||.hj.|| <= ||.y.|| by A30;
(x + (h /. j)) - x = (h /. j) + (x - x) by RLVECT_1:28
.= (h /. j) + (0. (product G)) by RLVECT_1:15 ;
then (x + (h /. j)) - x = h /. j by RLVECT_1:4;
hence ||.((x + (h /. j)) - x).|| < d by A56, A28, XXREAL_0:2; ::_thesis: verum
end;
len G <= (len G) + 1 by NAT_1:11;
then Seg (len G) c= Seg ((len G) + 1) by FINSEQ_1:5;
then ( 1 <= j & j <= (len G) + 1 ) by A47, FINSEQ_1:1;
then A57: ||.((x + (h /. j)) - x).|| < d by A55;
1 <= j + 1 by NAT_1:11;
then A58: ||.((x + (h /. (j + 1))) - x).|| < d by A51, A55;
A59: x + (h /. j) = (reproj ((modetrans (G,(((len G) + 1) -' j))),(x + (h /. (j + 1))))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) by Th54, A30, A50;
A60: ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + y)) - ((proj (modetrans (G,(((len G) + 1) -' j)))) . (x + (h /. (j + 1)))) = (proj (modetrans (G,(((len G) + 1) -' j)))) . y by Th55, A30, A50;
for z being Point of (product G) st ||.(z - x).|| < d holds
||.((partdiff (f,z,(((len G) + 1) -' j))) - (partdiff (f,x,(((len G) + 1) -' j)))).|| <= (r0 / 2) / (len G) by A25, A52;
then A61: ||.(((f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1))))) - ((partdiff (f,x,(((len G) + 1) -' j))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . y))).|| <= ||.((proj (modetrans (G,(((len G) + 1) -' j)))) . y).|| * ((r0 / 2) / (len G)) by A1, A53, A52, A2, A24, A57, A58, A59, A60, Th53;
A62: ((len G) + 1) -' j = ((len G) + 1) - j by A50, NAT_1:12, XREAL_1:233;
j in Seg (len (Rev w)) by A47, A34, FINSEQ_5:def_3;
then A63: j in dom (Rev w) by FINSEQ_1:def_3;
then A64: (Rev w) /. j = (Rev w) . j by PARTFUN1:def_6
.= w . (((len G) - j) + 1) by A34, A63, FINSEQ_5:def_3
.= w /. (((len G) + 1) -' j) by A62, A52, A31, PARTFUN1:def_6 ;
A65: gw /. j = gw . j by A46, PARTFUN1:def_6
.= ((f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1))))) - ((partdiff (f,x,(((len G) + 1) -' j))) . ((proj (modetrans (G,(((len G) + 1) -' j)))) . y)) by A54, A49, A64, A48, A43 ;
||.((proj (modetrans (G,(((len G) + 1) -' j)))) . y).|| * ((r0 / 2) / (len G)) <= ||.y.|| * ((r0 / 2) / (len G)) by A12, Th31, XREAL_1:64;
hence ||.(gw /. j).|| <= ||.y.|| * ((r0 / 2) / (len G)) by A65, A61, XXREAL_0:2; ::_thesis: verum
end;
defpred S2[ set , set ] means $2 = ||.(gw /. $1).||;
A66: for k being Nat st k in Seg (len G) holds
ex x being Element of REAL st S2[k,x] ;
consider yseq being FinSequence of REAL such that
A67: ( dom yseq = Seg (len G) & ( for i being Nat st i in Seg (len G) holds
S2[i,yseq . i] ) ) from FINSEQ_1:sch_5(A66);
A68: len gw = len yseq by A42, A67, FINSEQ_1:def_3;
A69: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_gw_holds_
yseq_._i_=_||.(gw_/._i).||
let i be Element of NAT ; ::_thesis: ( i in dom gw implies yseq . i = ||.(gw /. i).|| )
assume i in dom gw ; ::_thesis: yseq . i = ||.(gw /. i).||
then i in Seg (len G) by A42, FINSEQ_1:def_3;
hence yseq . i = ||.(gw /. i).|| by A67; ::_thesis: verum
end;
reconsider yseq = yseq as Element of REAL (len G) by A68, A42, FINSEQ_2:92;
A70: ||.(Sum gw).|| <= Sum yseq by A69, A68, Th7;
for j being Nat st j in Seg (len G) holds
yseq . j <= ((len G) |-> (||.y.|| * ((r0 / 2) / (len G)))) . j
proof
let j be Nat; ::_thesis: ( j in Seg (len G) implies yseq . j <= ((len G) |-> (||.y.|| * ((r0 / 2) / (len G)))) . j )
assume A71: j in Seg (len G) ; ::_thesis: yseq . j <= ((len G) |-> (||.y.|| * ((r0 / 2) / (len G)))) . j
then j in dom gw by A42, FINSEQ_1:def_3;
then A72: ||.(gw /. j).|| <= ||.y.|| * ((r0 / 2) / (len G)) by A45;
yseq . j = ||.(gw /. j).|| by A67, A71;
hence yseq . j <= ((len G) |-> (||.y.|| * ((r0 / 2) / (len G)))) . j by A71, A72, FINSEQ_2:57; ::_thesis: verum
end;
then Sum yseq <= Sum ((len G) |-> (||.y.|| * ((r0 / 2) / (len G)))) by RVSUM_1:82;
then Sum yseq <= (len G) * (||.y.|| * ((r0 / 2) / (len G))) by RVSUM_1:80;
then ||.(R /. y).|| <= (len G) * (||.y.|| * ((r0 / 2) / (len G))) by A44, A70, XXREAL_0:2;
then ||.(R /. y).|| * (||.y.|| ") <= (((len G) * ||.y.||) * ((r0 / 2) / (len G))) * (||.y.|| ") by XREAL_1:64;
then ||.(R /. y).|| * (||.y.|| ") <= (len G) * ((((r0 / 2) / (len G)) * ||.y.||) * (||.y.|| ")) ;
then (||.y.|| ") * ||.(R /. y).|| <= (len G) * ((r0 / 2) / (len G)) by A29, XCMPLX_1:203;
then A73: (||.y.|| ") * ||.(R /. y).|| <= r0 / 2 by XCMPLX_1:87;
r0 / 2 < r0 by A12, XREAL_1:216;
hence (||.y.|| ") * ||.(R /. y).|| < r0 by A73, XXREAL_0:2; ::_thesis: verum
end;
end;
then reconsider R = R as RestFunc of (product G),S by NDIFF_1:23;
reconsider L = L as Point of (R_NormSpace_of_BoundedLinearOperators ((product G),S)) by LOPBAN_1:def_9;
A74: for y being Point of (product G) st y in { y where y is Element of (product G) : ||.(y - x).|| < d0 } holds
(f /. y) - (f /. x) = (L . (y - x)) + (R /. (y - x))
proof
let y be Point of (product G); ::_thesis: ( y in { y where y is Element of (product G) : ||.(y - x).|| < d0 } implies (f /. y) - (f /. x) = (L . (y - x)) + (R /. (y - x)) )
assume y in { y where y is Element of (product G) : ||.(y - x).|| < d0 } ; ::_thesis: (f /. y) - (f /. x) = (L . (y - x)) + (R /. (y - x))
y - x in the carrier of (product G) ;
then y - x in dom R by PARTFUN1:def_2;
then R /. (y - x) = R . (y - x) by PARTFUN1:def_6;
then R /. (y - x) = ((f /. (x + (y - x))) - (f /. x)) - (L . (y - x)) by A11;
hence (L . (y - x)) + (R /. (y - x)) = ((f /. (x + (y - x))) - (f /. x)) - ((L . (y - x)) - (L . (y - x))) by RLVECT_1:29
.= ((f /. (x + (y - x))) - (f /. x)) - (0. S) by RLVECT_1:5
.= (f /. (x + (y - x))) - (f /. x) by RLVECT_1:13
.= (f /. (y - (x - x))) - (f /. x) by RLVECT_1:29
.= (f /. (y - (0. (product G)))) - (f /. x) by RLVECT_1:5
.= (f /. y) - (f /. x) by RLVECT_1:13 ;
::_thesis: verum
end;
then f is_differentiable_in x by A10, A8, NDIFF_1:def_6;
then diff (f,x) = L by A8, A10, A74, NDIFF_1:def_7;
hence ( f is_differentiable_in x & ( for h being Point of (product G) ex w being FinSequence of S st
( dom w = dom G & ( for i being set st i in dom G holds
w . i = (partdiff (f,x,i)) . ((proj (modetrans (G,i))) . h) ) & (diff (f,x)) . h = Sum w ) ) ) by A4, A74, A10, A8, NDIFF_1:def_6; ::_thesis: verum
end;
theorem :: NDIFF_5:57
for G being non-trivial RealNormSpace-Sequence
for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for X being Subset of (product G) st X is open holds
( ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
proof
let G be non-trivial RealNormSpace-Sequence; ::_thesis: for F being non trivial RealNormSpace
for f being PartFunc of (product G),F
for X being Subset of (product G) st X is open holds
( ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
let F be non trivial RealNormSpace; ::_thesis: for f being PartFunc of (product G),F
for X being Subset of (product G) st X is open holds
( ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
let f be PartFunc of (product G),F; ::_thesis: for X being Subset of (product G) st X is open holds
( ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
let X be Subset of (product G); ::_thesis: ( X is open implies ( ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) ) )
assume A1: X is open ; ::_thesis: ( ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
set m = len G;
A2: dom G = Seg (len G) by FINSEQ_1:def_3;
hereby ::_thesis: ( f is_differentiable_on X & f `| X is_continuous_on X implies for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume A3: for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & f `| X is_continuous_on X )
A4: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_G_holds_
(_X_c=_dom_(f_`partial|_(X,i))_&_(_for_y0_being_Point_of_(product_G)
for_r_being_Real_st_y0_in_X_&_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_y1_being_Point_of_(product_G)_st_y1_in_X_&_||.(y1_-_y0).||_<_s_holds_
||.(((f_`partial|_(X,i))_/._y1)_-_((f_`partial|_(X,i))_/._y0)).||_<_r_)_)_)_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len G implies ( X c= dom (f `partial| (X,i)) & ( for y0 being Point of (product G)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) ) ) )
assume ( 1 <= i & i <= len G ) ; ::_thesis: ( X c= dom (f `partial| (X,i)) & ( for y0 being Point of (product G)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) ) )
then i in Seg (len G) ;
then f `partial| (X,i) is_continuous_on X by A3, A2;
hence ( X c= dom (f `partial| (X,i)) & ( for y0 being Point of (product G)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) ) ) by NFCONT_1:19; ::_thesis: verum
end;
A5: 1 <= len G by NAT_1:14;
then len G in dom G by A2;
then f is_partial_differentiable_on X, len G by A3;
then A6: X c= dom f by Def8;
for x being Point of (product G) st x in X holds
f is_differentiable_in x by A1, A3, Th56;
hence A7: f is_differentiable_on X by A1, A6, NDIFF_1:31; ::_thesis: f `| X is_continuous_on X
then A8: dom (f `| X) = X by NDIFF_1:def_9;
for y0 being Point of (product G)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
proof
let y0 be Point of (product G); ::_thesis: for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
let r be Real; ::_thesis: ( y0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) ) )
assume A9: ( y0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
defpred S1[ Nat, Real] means for i being Element of NAT st i = $1 holds
( 0 < $2 & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < $2 holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r / (2 * (len G)) ) );
A10: now__::_thesis:_for_i_being_Nat_st_i_in_Seg_(len_G)_holds_
ex_s_being_Element_of_REAL_st_S1[i,s]
let i be Nat; ::_thesis: ( i in Seg (len G) implies ex s being Element of REAL st S1[i,s] )
reconsider j = i as Element of NAT by ORDINAL1:def_12;
assume i in Seg (len G) ; ::_thesis: ex s being Element of REAL st S1[i,s]
then ( 1 <= j & j <= len G ) by FINSEQ_1:1;
then consider s being Real such that
A11: ( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,j)) /. y1) - ((f `partial| (X,j)) /. y0)).|| < r / (2 * (len G)) ) ) by A9, A4;
reconsider s = s as Element of REAL ;
take s = s; ::_thesis: S1[i,s]
thus S1[i,s] by A11; ::_thesis: verum
end;
consider S being FinSequence of REAL such that
A12: ( dom S = Seg (len G) & ( for i being Nat st i in Seg (len G) holds
S1[i,S . i] ) ) from FINSEQ_1:sch_5(A10);
take s = min S; ::_thesis: ( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
A13: len S = len G by A12, FINSEQ_1:def_3;
then min_p S in dom S by RFINSEQ2:def_2;
hence s > 0 by A12; ::_thesis: for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r
let y1 be Point of (product G); ::_thesis: ( y1 in X & ||.(y1 - y0).|| < s implies ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r )
assume A14: ( y1 in X & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r
reconsider DD = (diff (f,y1)) - (diff (f,y0)) as Lipschitzian LinearOperator of (product G),F by LOPBAN_1:def_9;
A15: upper_bound (PreNorms DD) = ||.((diff (f,y1)) - (diff (f,y0))).|| by LOPBAN_1:30;
now__::_thesis:_for_mt_being_real_number_st_mt_in_PreNorms_DD_holds_
mt_<=_r_/_2
let mt be real number ; ::_thesis: ( mt in PreNorms DD implies mt <= r / 2 )
assume mt in PreNorms DD ; ::_thesis: mt <= r / 2
then consider t being VECTOR of (product G) such that
A16: ( mt = ||.(DD . t).|| & ||.t.|| <= 1 ) ;
consider w0 being FinSequence of F such that
A17: ( dom w0 = dom G & ( for i being set st i in dom G holds
w0 . i = (partdiff (f,y0,i)) . ((proj (modetrans (G,i))) . t) ) & (diff (f,y0)) . t = Sum w0 ) by A1, A3, Th56, A9;
reconsider Sw0 = Sum w0 as Point of F ;
consider w1 being FinSequence of F such that
A18: ( dom w1 = dom G & ( for i being set st i in dom G holds
w1 . i = (partdiff (f,y1,i)) . ((proj (modetrans (G,i))) . t) ) & (diff (f,y1)) . t = Sum w1 ) by A1, A3, Th56, A14;
reconsider Sw1 = Sum w1 as Point of F ;
deffunc H1( set ) -> Element of the carrier of F = (w1 /. $1) - (w0 /. $1);
consider w2 being FinSequence of F such that
A19: ( len w2 = len G & ( for i being Nat st i in dom w2 holds
w2 . i = H1(i) ) ) from FINSEQ_2:sch_1();
A20: ( len w1 = len G & len w0 = len G ) by A2, A17, A18, FINSEQ_1:def_3;
now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_w1_holds_
w2_._i_=_H1(i)
let i be Element of NAT ; ::_thesis: ( i in dom w1 implies w2 . i = H1(i) )
assume i in dom w1 ; ::_thesis: w2 . i = H1(i)
then i in dom w2 by A19, A2, A18, FINSEQ_1:def_3;
hence w2 . i = H1(i) by A19; ::_thesis: verum
end;
then Sum w2 = (Sum w1) - (Sum w0) by A19, A20, RLVECT_2:5;
then A21: mt = ||.(Sum w2).|| by A16, A18, A17, LOPBAN_1:40;
deffunc H2( Nat) -> Element of REAL = ||.(w2 /. $1).||;
consider ys being FinSequence of REAL such that
A22: ( len ys = len G & ( for j being Nat st j in dom ys holds
ys . j = H2(j) ) ) from FINSEQ_2:sch_1();
A23: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_w2_holds_
ys_._i_=_||.(w2_/._i).||
let i be Element of NAT ; ::_thesis: ( i in dom w2 implies ys . i = ||.(w2 /. i).|| )
assume i in dom w2 ; ::_thesis: ys . i = ||.(w2 /. i).||
then i in Seg (len G) by A19, FINSEQ_1:def_3;
then i in dom ys by A22, FINSEQ_1:def_3;
hence ys . i = ||.(w2 /. i).|| by A22; ::_thesis: verum
end;
then A24: ||.(Sum w2).|| <= Sum ys by A19, A22, Th7;
reconsider rm = r / (2 * (len G)) as Element of REAL ;
deffunc H3( Nat) -> Element of REAL = rm;
consider rs being FinSequence of REAL such that
A25: ( len rs = len G & ( for j being Nat st j in dom rs holds
rs . j = H3(j) ) ) from FINSEQ_2:sch_1();
A26: dom rs = Seg (len G) by A25, FINSEQ_1:def_3;
now__::_thesis:_for_a_being_set_st_a_in_rng_rs_holds_
a_in_{rm}
let a be set ; ::_thesis: ( a in rng rs implies a in {rm} )
assume a in rng rs ; ::_thesis: a in {rm}
then consider b being set such that
A27: ( b in dom rs & a = rs . b ) by FUNCT_1:def_3;
reconsider b = b as Nat by A27;
rs . b = rm by A27, A25;
hence a in {rm} by A27, TARSKI:def_1; ::_thesis: verum
end;
then A28: rng rs c= {rm} by TARSKI:def_3;
now__::_thesis:_for_a_being_set_st_a_in_{rm}_holds_
a_in_rng_rs
let a be set ; ::_thesis: ( a in {rm} implies a in rng rs )
assume a in {rm} ; ::_thesis: a in rng rs
then A29: a = rm by TARSKI:def_1;
A30: 1 in dom rs by A5, A26;
then a = rs . 1 by A29, A25;
hence a in rng rs by A30, FUNCT_1:3; ::_thesis: verum
end;
then {rm} c= rng rs by TARSKI:def_3;
then {rm} = rng rs by A28, XBOOLE_0:def_10;
then rs = (len G) |-> (r / (2 * (len G))) by A26, FUNCOP_1:9;
then Sum rs = (len G) * (r / (2 * (len G))) by RVSUM_1:80
.= (len G) * ((r / 2) / (len G)) by XCMPLX_1:78 ;
then A31: Sum rs = r / 2 by XCMPLX_1:87;
now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_ys_holds_
ys_._i_<=_rs_._i
let i be Element of NAT ; ::_thesis: ( i in dom ys implies ys . i <= rs . i )
assume i in dom ys ; ::_thesis: ys . i <= rs . i
then A32: i in Seg (len G) by A22, FINSEQ_1:def_3;
then A33: ( i in dom w2 & i in dom w1 & i in dom w0 ) by A17, A18, A19, FINSEQ_1:def_3;
then A34: ( ys . i = ||.(w2 /. i).|| & w2 /. i = w2 . i ) by A23, PARTFUN1:def_6;
A35: i in dom rs by A25, A32, FINSEQ_1:def_3;
reconsider p1 = partdiff (f,y1,i), p0 = partdiff (f,y0,i) as Lipschitzian LinearOperator of (G . (modetrans (G,i))),F by LOPBAN_1:def_9;
reconsider P1 = p1 . ((proj (modetrans (G,i))) . t) as VECTOR of F ;
reconsider P0 = p0 . ((proj (modetrans (G,i))) . t) as VECTOR of F ;
( w0 /. i = w0 . i & w1 /. i = w1 . i ) by A33, PARTFUN1:def_6;
then ( w0 /. i = P0 & w1 /. i = P1 ) by A2, A17, A18, A32;
then A36: w2 . i = P1 - P0 by A33, A19;
( 1 <= i & i <= len S ) by A13, A32, FINSEQ_1:1;
then A37: ( s <= S . i & f is_partial_differentiable_on X,i ) by A2, A32, A3, RFINSEQ2:2;
then ||.(y1 - y0).|| < S . i by A14, XXREAL_0:2;
then ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r / (2 * (len G)) by A12, A32, A14;
then ||.((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)).|| < r / (2 * (len G)) by Def9, A14, A37, A2, A32;
then A38: ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| < r / (2 * (len G)) by Def9, A9, A37, A2, A32;
reconsider PP = (partdiff (f,y1,i)) - (partdiff (f,y0,i)) as Lipschitzian LinearOperator of (G . (modetrans (G,i))),F by LOPBAN_1:def_9;
A39: upper_bound (PreNorms PP) = ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| by LOPBAN_1:30;
reconsider pt = (proj (modetrans (G,i))) . t as VECTOR of (G . (modetrans (G,i))) ;
A40: PP . pt = P1 - P0 by LOPBAN_1:40;
||.pt.|| <= ||.t.|| by Th31;
then ||.pt.|| <= 1 by A16, XXREAL_0:2;
then ( ||.(PP . pt).|| in PreNorms PP & not PreNorms PP is empty & PreNorms PP is bounded_above ) by LOPBAN_1:27;
then ||.(PP . pt).|| <= upper_bound (PreNorms PP) by SEQ_4:def_1;
then ||.(P1 - P0).|| <= r / (2 * (len G)) by A40, A38, A39, XXREAL_0:2;
hence ys . i <= rs . i by A34, A25, A35, A36; ::_thesis: verum
end;
then Sum ys <= r / 2 by A31, A25, A22, INTEGRA5:3;
hence mt <= r / 2 by A21, A24, XXREAL_0:2; ::_thesis: verum
end;
then ( ||.((diff (f,y1)) - (diff (f,y0))).|| <= r / 2 & r / 2 < r ) by A15, A9, SEQ_4:45, XREAL_1:216;
then ||.((diff (f,y1)) - (diff (f,y0))).|| < r by XXREAL_0:2;
then ||.((diff (f,y1)) - ((f `| X) /. y0)).|| < r by A7, A9, NDIFF_1:def_9;
hence ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r by A7, A14, NDIFF_1:def_9; ::_thesis: verum
end;
hence f `| X is_continuous_on X by A8, NFCONT_1:19; ::_thesis: verum
end;
assume A41: ( f is_differentiable_on X & f `| X is_continuous_on X ) ; ::_thesis: for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then A42: ( X c= dom f & ( for x being Point of (product G) st x in X holds
f is_differentiable_in x ) ) by A1, NDIFF_1:31;
thus for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ::_thesis: verum
proof
let i be set ; ::_thesis: ( i in dom G implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume A43: i in dom G ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then reconsider i0 = i as Element of NAT ;
now__::_thesis:_for_x_being_Point_of_(product_G)_st_x_in_X_holds_
(_f_is_partial_differentiable_in_x,i_&_partdiff_(f,x,i)_=_(diff_(f,x))_*_(reproj_((modetrans_(G,i)),(0._(product_G))))_)
let x be Point of (product G); ::_thesis: ( x in X implies ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) ) )
assume x in X ; ::_thesis: ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) )
then f is_differentiable_in x by A41, A1, NDIFF_1:31;
hence ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj ((modetrans (G,i)),(0. (product G)))) ) by Th41; ::_thesis: verum
end;
then for x being Point of (product G) st x in X holds
f is_partial_differentiable_in x,i ;
hence A44: f is_partial_differentiable_on X,i by A1, Th32, A42; ::_thesis: f `partial| (X,i) is_continuous_on X
then A45: dom (f `partial| (X,i)) = X by Def9, A43;
for y0 being Point of (product G)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
proof
let y0 be Point of (product G); ::_thesis: for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
let r be Real; ::_thesis: ( y0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) )
assume A46: ( y0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
then consider s being Real such that
A47: ( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) ) by A41, NFCONT_1:19;
take s ; ::_thesis: ( 0 < s & ( for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
thus 0 < s by A47; ::_thesis: for y1 being Point of (product G) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r
let y1 be Point of (product G); ::_thesis: ( y1 in X & ||.(y1 - y0).|| < s implies ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r )
assume A48: ( y1 in X & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r
then ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r by A47;
then ||.((diff (f,y1)) - ((f `| X) /. y0)).|| < r by A48, A41, NDIFF_1:def_9;
then A49: ||.((diff (f,y1)) - (diff (f,y0))).|| < r by A46, A41, NDIFF_1:def_9;
( f is_differentiable_in y1 & f is_differentiable_in y0 ) by A41, A1, A48, A46, NDIFF_1:31;
then A50: ( partdiff (f,y1,i) = (diff (f,y1)) * (reproj ((modetrans (G,i)),(0. (product G)))) & partdiff (f,y0,i) = (diff (f,y0)) * (reproj ((modetrans (G,i)),(0. (product G)))) ) by Th41;
reconsider PP = (partdiff (f,y1,i)) - (partdiff (f,y0,i)) as Lipschitzian LinearOperator of (G . (modetrans (G,i))),F by LOPBAN_1:def_9;
reconsider DD = (diff (f,y1)) - (diff (f,y0)) as Lipschitzian LinearOperator of (product G),F by LOPBAN_1:def_9;
A51: ( upper_bound (PreNorms PP) = ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| & upper_bound (PreNorms DD) = ||.((diff (f,y1)) - (diff (f,y0))).|| ) by LOPBAN_1:30;
A52: ( PreNorms PP is bounded_above & PreNorms DD is bounded_above ) by LOPBAN_1:28;
now__::_thesis:_for_a_being_set_st_a_in_PreNorms_PP_holds_
a_in_PreNorms_DD
let a be set ; ::_thesis: ( a in PreNorms PP implies a in PreNorms DD )
assume a in PreNorms PP ; ::_thesis: a in PreNorms DD
then consider t being VECTOR of (G . (modetrans (G,i))) such that
A53: ( a = ||.(PP . t).|| & ||.t.|| <= 1 ) ;
A54: dom (reproj ((modetrans (G,i)),(0. (product G)))) = the carrier of (G . (modetrans (G,i))) by FUNCT_2:def_1;
reconsider tm = (reproj ((modetrans (G,i)),(0. (product G)))) . t as Point of (product G) ;
A55: ||.tm.|| <= 1 by A53, Th21;
( (partdiff (f,y1,i)) . t = (diff (f,y1)) . tm & (partdiff (f,y0,i)) . t = (diff (f,y0)) . tm ) by A54, A50, FUNCT_1:13;
then ||.(PP . t).|| = ||.(((diff (f,y1)) . tm) - ((diff (f,y0)) . tm)).|| by LOPBAN_1:40;
then ||.(PP . t).|| = ||.(DD . tm).|| by LOPBAN_1:40;
hence a in PreNorms DD by A53, A55; ::_thesis: verum
end;
then PreNorms PP c= PreNorms DD by TARSKI:def_3;
then ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| <= ||.((diff (f,y1)) - (diff (f,y0))).|| by A52, A51, SEQ_4:48;
then ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| < r by A49, XXREAL_0:2;
then ||.((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)).|| < r by Def9, A46, A44, A43;
hence ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r by Def9, A48, A44, A43; ::_thesis: verum
end;
hence f `partial| (X,i) is_continuous_on X by A45, NFCONT_1:19; ::_thesis: verum
end;
end;