:: RFUNCT_3 semantic presentation
begin
definition
let r be real number ;
func max+ r -> Real equals :: RFUNCT_3:def 1
max (r,0);
correctness
coherence
max (r,0) is Real;
by XREAL_0:def_1;
func max- r -> Real equals :: RFUNCT_3:def 2
max ((- r),0);
correctness
coherence
max ((- r),0) is Real;
by XREAL_0:def_1;
end;
:: deftheorem defines max+ RFUNCT_3:def_1_:_
for r being real number holds max+ r = max (r,0);
:: deftheorem defines max- RFUNCT_3:def_2_:_
for r being real number holds max- r = max ((- r),0);
theorem Th1: :: RFUNCT_3:1
for r being real number holds r = (max+ r) - (max- r)
proof
let r be real number ; ::_thesis: r = (max+ r) - (max- r)
now__::_thesis:_(_(_0_<=_r_&_r_=_(max+_r)_-_(max-_r)_)_or_(_r_<_0_&_r_=_(max+_r)_-_(max-_r)_)_)
percases ( 0 <= r or r < 0 ) ;
caseA1: 0 <= r ; ::_thesis: r = (max+ r) - (max- r)
then max- r = 0 by XXREAL_0:def_10;
hence r = (max+ r) - (max- r) by A1, XXREAL_0:def_10; ::_thesis: verum
end;
case r < 0 ; ::_thesis: r = (max+ r) - (max- r)
then ( max+ r = 0 & max- r = - r ) by XXREAL_0:def_10;
hence r = (max+ r) - (max- r) ; ::_thesis: verum
end;
end;
end;
hence r = (max+ r) - (max- r) ; ::_thesis: verum
end;
theorem Th2: :: RFUNCT_3:2
for r being real number holds abs r = (max+ r) + (max- r)
proof
let r be real number ; ::_thesis: abs r = (max+ r) + (max- r)
now__::_thesis:_(_(_0_<=_r_&_abs_r_=_(max+_r)_+_(max-_r)_)_or_(_r_<_0_&_abs_r_=_(max+_r)_+_(max-_r)_)_)
percases ( 0 <= r or r < 0 ) ;
caseA1: 0 <= r ; ::_thesis: abs r = (max+ r) + (max- r)
then ( max+ r = r & max- r = 0 ) by XXREAL_0:def_10;
hence abs r = (max+ r) + (max- r) by A1, ABSVALUE:def_1; ::_thesis: verum
end;
caseA2: r < 0 ; ::_thesis: abs r = (max+ r) + (max- r)
then ( max+ r = 0 & max- r = - r ) by XXREAL_0:def_10;
hence abs r = (max+ r) + (max- r) by A2, ABSVALUE:def_1; ::_thesis: verum
end;
end;
end;
hence abs r = (max+ r) + (max- r) ; ::_thesis: verum
end;
theorem Th3: :: RFUNCT_3:3
for r being real number holds 2 * (max+ r) = r + (abs r)
proof
let r be real number ; ::_thesis: 2 * (max+ r) = r + (abs r)
thus r + (abs r) = ((max+ r) - (max- r)) + (abs r) by Th1
.= ((max+ r) - (max- r)) + ((max+ r) + (max- r)) by Th2
.= 2 * (max+ r) ; ::_thesis: verum
end;
theorem Th4: :: RFUNCT_3:4
for r, s being real number st 0 <= r holds
max+ (r * s) = r * (max+ s)
proof
let r, s be real number ; ::_thesis: ( 0 <= r implies max+ (r * s) = r * (max+ s) )
assume A1: 0 <= r ; ::_thesis: max+ (r * s) = r * (max+ s)
now__::_thesis:_(_(_0_<=_s_&_max+_(r_*_s)_=_r_*_(max+_s)_)_or_(_s_<_0_&_max+_(r_*_s)_=_r_*_(max+_s)_)_)
percases ( 0 <= s or s < 0 ) ;
caseA2: 0 <= s ; ::_thesis: max+ (r * s) = r * (max+ s)
then max+ (r * s) = r * s by A1, XXREAL_0:def_10;
hence max+ (r * s) = r * (max+ s) by A2, XXREAL_0:def_10; ::_thesis: verum
end;
caseA3: s < 0 ; ::_thesis: max+ (r * s) = r * (max+ s)
then max+ s = 0 by XXREAL_0:def_10;
hence max+ (r * s) = r * (max+ s) by A1, A3, XXREAL_0:def_10; ::_thesis: verum
end;
end;
end;
hence max+ (r * s) = r * (max+ s) ; ::_thesis: verum
end;
theorem Th5: :: RFUNCT_3:5
for r, s being real number holds max+ (r + s) <= (max+ r) + (max+ s)
proof
let r, s be real number ; ::_thesis: max+ (r + s) <= (max+ r) + (max+ s)
A1: ( 0 <= max (r,0) & 0 <= max (s,0) ) by XXREAL_0:25;
A2: ( r <= max (r,0) & s <= max (s,0) ) by XXREAL_0:25;
now__::_thesis:_(_(_0_<=_r_+_s_&_max+_(r_+_s)_<=_(max+_r)_+_(max+_s)_)_or_(_r_+_s_<_0_&_max+_(r_+_s)_<=_(max+_r)_+_(max+_s)_)_)
percases ( 0 <= r + s or r + s < 0 ) ;
case 0 <= r + s ; ::_thesis: max+ (r + s) <= (max+ r) + (max+ s)
then max+ (r + s) = r + s by XXREAL_0:def_10;
hence max+ (r + s) <= (max+ r) + (max+ s) by A2, XREAL_1:7; ::_thesis: verum
end;
case r + s < 0 ; ::_thesis: max+ (r + s) <= (max+ r) + (max+ s)
then max+ (r + s) = 0 + 0 by XXREAL_0:def_10;
hence max+ (r + s) <= (max+ r) + (max+ s) by A1; ::_thesis: verum
end;
end;
end;
hence max+ (r + s) <= (max+ r) + (max+ s) ; ::_thesis: verum
end;
Lm1: for n being Element of NAT
for D being non empty set
for f being FinSequence of D st len f <= n holds
f | n = f
proof
let n be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st len f <= n holds
f | n = f
let D be non empty set ; ::_thesis: for f being FinSequence of D st len f <= n holds
f | n = f
let f be FinSequence of D; ::_thesis: ( len f <= n implies f | n = f )
A1: dom f = Seg (len f) by FINSEQ_1:def_3;
assume len f <= n ; ::_thesis: f | n = f
then ( f | n = f | (Seg n) & dom f c= Seg n ) by A1, FINSEQ_1:5, FINSEQ_1:def_15;
hence f | n = f by RELAT_1:68; ::_thesis: verum
end;
Lm2: for f being Function
for x being set st not x in rng f holds
f " {x} = {}
proof
let f be Function; ::_thesis: for x being set st not x in rng f holds
f " {x} = {}
let x be set ; ::_thesis: ( not x in rng f implies f " {x} = {} )
assume A1: not x in rng f ; ::_thesis: f " {x} = {}
rng f misses {x}
proof
set y = the Element of (rng f) /\ {x};
assume (rng f) /\ {x} <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction
then ( the Element of (rng f) /\ {x} in rng f & the Element of (rng f) /\ {x} in {x} ) by XBOOLE_0:def_4;
hence contradiction by A1, TARSKI:def_1; ::_thesis: verum
end;
hence f " {x} = {} by RELAT_1:138; ::_thesis: verum
end;
begin
theorem Th6: :: RFUNCT_3:6
for D being non empty set
for F being PartFunc of D,REAL
for r, s being real number st r <> 0 holds
F " {(s / r)} = (r (#) F) " {s}
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r, s being real number st r <> 0 holds
F " {(s / r)} = (r (#) F) " {s}
let F be PartFunc of D,REAL; ::_thesis: for r, s being real number st r <> 0 holds
F " {(s / r)} = (r (#) F) " {s}
let r, s be real number ; ::_thesis: ( r <> 0 implies F " {(s / r)} = (r (#) F) " {s} )
assume A1: r <> 0 ; ::_thesis: F " {(s / r)} = (r (#) F) " {s}
thus F " {(s / r)} c= (r (#) F) " {s} :: according to XBOOLE_0:def_10 ::_thesis: (r (#) F) " {s} c= F " {(s / r)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(s / r)} or x in (r (#) F) " {s} )
assume A2: x in F " {(s / r)} ; ::_thesis: x in (r (#) F) " {s}
then reconsider d = x as Element of D ;
d in dom F by A2, FUNCT_1:def_7;
then A3: d in dom (r (#) F) by VALUED_1:def_5;
F . d in {(s / r)} by A2, FUNCT_1:def_7;
then F . d = s / r by TARSKI:def_1;
then r * (F . d) = s by A1, XCMPLX_1:87;
then (r (#) F) . d = s by A3, VALUED_1:def_5;
then (r (#) F) . d in {s} by TARSKI:def_1;
hence x in (r (#) F) " {s} by A3, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (r (#) F) " {s} or x in F " {(s / r)} )
assume A4: x in (r (#) F) " {s} ; ::_thesis: x in F " {(s / r)}
then reconsider d = x as Element of D ;
A5: d in dom (r (#) F) by A4, FUNCT_1:def_7;
(r (#) F) . d in {s} by A4, FUNCT_1:def_7;
then (r (#) F) . d = s by TARSKI:def_1;
then r * (F . d) = s by A5, VALUED_1:def_5;
then F . d = s / r by A1, XCMPLX_1:89;
then A6: F . d in {(s / r)} by TARSKI:def_1;
d in dom F by A5, VALUED_1:def_5;
hence x in F " {(s / r)} by A6, FUNCT_1:def_7; ::_thesis: verum
end;
theorem Th7: :: RFUNCT_3:7
for D being non empty set
for F being PartFunc of D,REAL holds (0 (#) F) " {0} = dom F
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds (0 (#) F) " {0} = dom F
let F be PartFunc of D,REAL; ::_thesis: (0 (#) F) " {0} = dom F
thus (0 (#) F) " {0} c= dom F :: according to XBOOLE_0:def_10 ::_thesis: dom F c= (0 (#) F) " {0}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (0 (#) F) " {0} or x in dom F )
assume A1: x in (0 (#) F) " {0} ; ::_thesis: x in dom F
then reconsider d = x as Element of D ;
d in dom (0 (#) F) by A1, FUNCT_1:def_7;
hence x in dom F by VALUED_1:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom F or x in (0 (#) F) " {0} )
assume A2: x in dom F ; ::_thesis: x in (0 (#) F) " {0}
then reconsider d = x as Element of D ;
A3: d in dom (0 (#) F) by A2, VALUED_1:def_5;
then (0 (#) F) . d = 0 * (F . d) by VALUED_1:def_5
.= 0 ;
then (0 (#) F) . d in {0} by TARSKI:def_1;
hence x in (0 (#) F) " {0} by A3, FUNCT_1:def_7; ::_thesis: verum
end;
theorem Th8: :: RFUNCT_3:8
for D being non empty set
for F being PartFunc of D,REAL
for r being Real st 0 < r holds
(abs F) " {r} = F " {(- r),r}
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r being Real st 0 < r holds
(abs F) " {r} = F " {(- r),r}
let F be PartFunc of D,REAL; ::_thesis: for r being Real st 0 < r holds
(abs F) " {r} = F " {(- r),r}
let r be Real; ::_thesis: ( 0 < r implies (abs F) " {r} = F " {(- r),r} )
assume A1: 0 < r ; ::_thesis: (abs F) " {r} = F " {(- r),r}
A2: dom (abs F) = dom F by VALUED_1:def_11;
thus (abs F) " {r} c= F " {(- r),r} :: according to XBOOLE_0:def_10 ::_thesis: F " {(- r),r} c= (abs F) " {r}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (abs F) " {r} or x in F " {(- r),r} )
assume A3: x in (abs F) " {r} ; ::_thesis: x in F " {(- r),r}
then reconsider rr = x as Element of D ;
(abs F) . rr in {r} by A3, FUNCT_1:def_7;
then abs (F . rr) in {r} by VALUED_1:18;
then A4: abs (F . rr) = r by TARSKI:def_1;
A5: rr in dom (abs F) by A3, FUNCT_1:def_7;
now__::_thesis:_(_(_0_<=_F_._rr_&_x_in_F_"_{(-_r),r}_)_or_(_F_._rr_<_0_&_x_in_F_"_{(-_r),r}_)_)
percases ( 0 <= F . rr or F . rr < 0 ) ;
case 0 <= F . rr ; ::_thesis: x in F " {(- r),r}
then F . rr = r by A4, ABSVALUE:def_1;
then F . rr in {(- r),r} by TARSKI:def_2;
hence x in F " {(- r),r} by A2, A5, FUNCT_1:def_7; ::_thesis: verum
end;
case F . rr < 0 ; ::_thesis: x in F " {(- r),r}
then - (F . rr) = r by A4, ABSVALUE:def_1;
then F . rr in {(- r),r} by TARSKI:def_2;
hence x in F " {(- r),r} by A2, A5, FUNCT_1:def_7; ::_thesis: verum
end;
end;
end;
hence x in F " {(- r),r} ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(- r),r} or x in (abs F) " {r} )
assume A6: x in F " {(- r),r} ; ::_thesis: x in (abs F) " {r}
then reconsider rr = x as Element of D ;
A7: rr in dom F by A6, FUNCT_1:def_7;
A8: F . rr in {(- r),r} by A6, FUNCT_1:def_7;
now__::_thesis:_(_(_F_._rr_=_-_r_&_x_in_(abs_F)_"_{r}_)_or_(_F_._rr_=_r_&_x_in_(abs_F)_"_{r}_)_)
percases ( F . rr = - r or F . rr = r ) by A8, TARSKI:def_2;
case F . rr = - r ; ::_thesis: x in (abs F) " {r}
then r = abs (- (F . rr)) by A1, ABSVALUE:def_1
.= abs (F . rr) by COMPLEX1:52
.= (abs F) . rr by VALUED_1:18 ;
then (abs F) . rr in {r} by TARSKI:def_1;
hence x in (abs F) " {r} by A2, A7, FUNCT_1:def_7; ::_thesis: verum
end;
case F . rr = r ; ::_thesis: x in (abs F) " {r}
then r = abs (F . rr) by A1, ABSVALUE:def_1
.= (abs F) . rr by VALUED_1:18 ;
then (abs F) . rr in {r} by TARSKI:def_1;
hence x in (abs F) " {r} by A2, A7, FUNCT_1:def_7; ::_thesis: verum
end;
end;
end;
hence x in (abs F) " {r} ; ::_thesis: verum
end;
theorem Th9: :: RFUNCT_3:9
for D being non empty set
for F being PartFunc of D,REAL holds (abs F) " {0} = F " {0}
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds (abs F) " {0} = F " {0}
let F be PartFunc of D,REAL; ::_thesis: (abs F) " {0} = F " {0}
A1: dom (abs F) = dom F by VALUED_1:def_11;
thus (abs F) " {0} c= F " {0} :: according to XBOOLE_0:def_10 ::_thesis: F " {0} c= (abs F) " {0}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (abs F) " {0} or x in F " {0} )
assume A2: x in (abs F) " {0} ; ::_thesis: x in F " {0}
then reconsider r = x as Element of D ;
(abs F) . r in {0} by A2, FUNCT_1:def_7;
then abs (F . r) in {0} by VALUED_1:18;
then abs (F . r) = 0 by TARSKI:def_1;
then F . r = 0 by ABSVALUE:2;
then A3: F . r in {0} by TARSKI:def_1;
r in dom (abs F) by A2, FUNCT_1:def_7;
hence x in F " {0} by A1, A3, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {0} or x in (abs F) " {0} )
assume A4: x in F " {0} ; ::_thesis: x in (abs F) " {0}
then reconsider r = x as Element of D ;
F . r in {0} by A4, FUNCT_1:def_7;
then F . r = 0 by TARSKI:def_1;
then abs (F . r) = 0 by ABSVALUE:2;
then (abs F) . r = 0 by VALUED_1:18;
then A5: (abs F) . r in {0} by TARSKI:def_1;
r in dom F by A4, FUNCT_1:def_7;
hence x in (abs F) " {0} by A1, A5, FUNCT_1:def_7; ::_thesis: verum
end;
theorem Th10: :: RFUNCT_3:10
for D being non empty set
for F being PartFunc of D,REAL
for r being Real st r < 0 holds
(abs F) " {r} = {}
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r being Real st r < 0 holds
(abs F) " {r} = {}
let F be PartFunc of D,REAL; ::_thesis: for r being Real st r < 0 holds
(abs F) " {r} = {}
let r be Real; ::_thesis: ( r < 0 implies (abs F) " {r} = {} )
assume A1: r < 0 ; ::_thesis: (abs F) " {r} = {}
set x = the Element of (abs F) " {r};
assume A2: (abs F) " {r} <> {} ; ::_thesis: contradiction
then reconsider x = the Element of (abs F) " {r} as Element of D by TARSKI:def_3;
(abs F) . x in {r} by A2, FUNCT_1:def_7;
then abs (F . x) in {r} by VALUED_1:18;
then abs (F . x) = r by TARSKI:def_1;
hence contradiction by A1, COMPLEX1:46; ::_thesis: verum
end;
theorem Th11: :: RFUNCT_3:11
for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL
for r being Real st r <> 0 holds
( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent )
proof
let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for G being PartFunc of C,REAL
for r being Real st r <> 0 holds
( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent )
let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL
for r being Real st r <> 0 holds
( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent )
let G be PartFunc of C,REAL; ::_thesis: for r being Real st r <> 0 holds
( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent )
let r be Real; ::_thesis: ( r <> 0 implies ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent ) )
assume A1: r <> 0 ; ::_thesis: ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent )
A2: ( rng (r (#) F) c= REAL & rng (r (#) G) c= REAL ) ;
thus ( F,G are_fiberwise_equipotent implies r (#) F,r (#) G are_fiberwise_equipotent ) ::_thesis: ( r (#) F,r (#) G are_fiberwise_equipotent implies F,G are_fiberwise_equipotent )
proof
assume A3: F,G are_fiberwise_equipotent ; ::_thesis: r (#) F,r (#) G are_fiberwise_equipotent
now__::_thesis:_for_x_being_Real_holds_card_(Coim_((r_(#)_F),x))_=_card_(Coim_((r_(#)_G),x))
let x be Real; ::_thesis: card (Coim ((r (#) F),x)) = card (Coim ((r (#) G),x))
( Coim (F,(x / r)) = Coim ((r (#) F),x) & Coim (G,(x / r)) = Coim ((r (#) G),x) ) by A1, Th6;
hence card (Coim ((r (#) F),x)) = card (Coim ((r (#) G),x)) by A3, CLASSES1:def_9; ::_thesis: verum
end;
hence r (#) F,r (#) G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum
end;
assume A4: r (#) F,r (#) G are_fiberwise_equipotent ; ::_thesis: F,G are_fiberwise_equipotent
A5: now__::_thesis:_for_x_being_Real_holds_card_(Coim_(F,x))_=_card_(Coim_(G,x))
let x be Real; ::_thesis: card (Coim (F,x)) = card (Coim (G,x))
A6: G " {((r * x) / r)} = Coim ((r (#) G),(r * x)) by A1, Th6;
( (r * x) / r = x & F " {((r * x) / r)} = Coim ((r (#) F),(r * x)) ) by A1, Th6, XCMPLX_1:89;
hence card (Coim (F,x)) = card (Coim (G,x)) by A4, A6, CLASSES1:def_9; ::_thesis: verum
end;
( rng F c= REAL & rng G c= REAL ) ;
hence F,G are_fiberwise_equipotent by A5, CLASSES1:79; ::_thesis: verum
end;
theorem :: RFUNCT_3:12
for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL holds
( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent )
proof
let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for G being PartFunc of C,REAL holds
( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent )
let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL holds
( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent )
let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent )
- F = (- 1) (#) F ;
hence ( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent ) by Th11; ::_thesis: verum
end;
theorem :: RFUNCT_3:13
for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
abs F, abs G are_fiberwise_equipotent
proof
let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
abs F, abs G are_fiberwise_equipotent
let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
abs F, abs G are_fiberwise_equipotent
let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent implies abs F, abs G are_fiberwise_equipotent )
assume A1: F,G are_fiberwise_equipotent ; ::_thesis: abs F, abs G are_fiberwise_equipotent
A2: now__::_thesis:_for_r_being_Real_holds_card_(Coim_((abs_F),r))_=_card_(Coim_((abs_G),r))
let r be Real; ::_thesis: card (Coim ((abs F),r)) = card (Coim ((abs G),r))
now__::_thesis:_(_(_0_<_r_&_card_((abs_F)_"_{r})_=_card_((abs_G)_"_{r})_)_or_(_0_=_r_&_card_((abs_F)_"_{r})_=_card_((abs_G)_"_{r})_)_or_(_r_<_0_&_card_((abs_F)_"_{r})_=_card_((abs_G)_"_{r})_)_)
percases ( 0 < r or 0 = r or r < 0 ) ;
case 0 < r ; ::_thesis: card ((abs F) " {r}) = card ((abs G) " {r})
then ( (abs F) " {r} = F " {(- r),r} & (abs G) " {r} = G " {(- r),r} ) by Th8;
hence card ((abs F) " {r}) = card ((abs G) " {r}) by A1, CLASSES1:78; ::_thesis: verum
end;
case 0 = r ; ::_thesis: card ((abs F) " {r}) = card ((abs G) " {r})
then ( (abs F) " {r} = F " {r} & (abs G) " {r} = G " {r} ) by Th9;
hence card ((abs F) " {r}) = card ((abs G) " {r}) by A1, CLASSES1:78; ::_thesis: verum
end;
caseA3: r < 0 ; ::_thesis: card ((abs F) " {r}) = card ((abs G) " {r})
then (abs F) " {r} = {} by Th10;
hence card ((abs F) " {r}) = card ((abs G) " {r}) by A3, Th10; ::_thesis: verum
end;
end;
end;
hence card (Coim ((abs F),r)) = card (Coim ((abs G),r)) ; ::_thesis: verum
end;
( rng (abs F) c= REAL & rng (abs G) c= REAL ) ;
hence abs F, abs G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum
end;
definition
let X, Y be set ;
mode PartFunc-set of X,Y -> set means :Def3: :: RFUNCT_3:def 3
for x being Element of it holds x is PartFunc of X,Y;
existence
ex b1 being set st
for x being Element of b1 holds x is PartFunc of X,Y
proof
reconsider h = {} as PartFunc of X,Y by RELSET_1:12;
take {h} ; ::_thesis: for x being Element of {h} holds x is PartFunc of X,Y
thus for x being Element of {h} holds x is PartFunc of X,Y by TARSKI:def_1; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines PartFunc-set RFUNCT_3:def_3_:_
for X, Y being set
for b3 being set holds
( b3 is PartFunc-set of X,Y iff for x being Element of b3 holds x is PartFunc of X,Y );
registration
let X, Y be set ;
cluster non empty for PartFunc-set of X,Y;
existence
not for b1 being PartFunc-set of X,Y holds b1 is empty
proof
reconsider h = {} as PartFunc of X,Y by RELSET_1:12;
{h} is PartFunc-set of X,Y
proof
let x be Element of {h}; :: according to RFUNCT_3:def_3 ::_thesis: x is PartFunc of X,Y
thus x is PartFunc of X,Y by TARSKI:def_1; ::_thesis: verum
end;
hence not for b1 being PartFunc-set of X,Y holds b1 is empty ; ::_thesis: verum
end;
end;
definition
let X, Y be set ;
mode PFUNC_DOMAIN of X,Y is non empty PartFunc-set of X,Y;
end;
definition
let X, Y be set ;
:: original: PFuncs
redefine func PFuncs (X,Y) -> PartFunc-set of X,Y;
coherence
PFuncs (X,Y) is PartFunc-set of X,Y
proof
for x being Element of PFuncs (X,Y) holds x is PartFunc of X,Y by PARTFUN1:47;
hence PFuncs (X,Y) is PartFunc-set of X,Y by Def3; ::_thesis: verum
end;
let P be non empty PartFunc-set of X,Y;
:: original: Element
redefine mode Element of P -> PartFunc of X,Y;
coherence
for b1 being Element of P holds b1 is PartFunc of X,Y by Def3;
end;
definition
let D, C be non empty set ;
let X be Subset of D;
let c be Element of C;
:: original: -->
redefine funcX --> c -> Element of PFuncs (D,C);
coherence
X --> c is Element of PFuncs (D,C)
proof
X --> c is PartFunc of D,C ;
hence X --> c is Element of PFuncs (D,C) by PARTFUN1:45; ::_thesis: verum
end;
end;
registration
let D be non empty set ;
let E be real-membered set ;
cluster -> real-valued for Element of PFuncs (D,E);
coherence
for b1 being Element of PFuncs (D,E) holds b1 is real-valued ;
end;
definition
let D be non empty set ;
let E be real-membered set ;
let F1, F2 be Element of PFuncs (D,E);
:: original: +
redefine funcF1 + F2 -> Element of PFuncs (D,REAL);
coherence
F1 + F2 is Element of PFuncs (D,REAL)
proof
reconsider F1 = F1, F2 = F2 as PartFunc of D,E ;
F1 + F2 is PartFunc of D,REAL ;
hence F1 + F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
:: original: -
redefine funcF1 - F2 -> Element of PFuncs (D,REAL);
coherence
F1 - F2 is Element of PFuncs (D,REAL)
proof
reconsider F1 = F1, F2 = F2 as PartFunc of D,E ;
F1 - F2 is PartFunc of D,REAL ;
hence F1 - F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
:: original: (#)
redefine funcF1 (#) F2 -> Element of PFuncs (D,REAL);
coherence
F1 (#) F2 is Element of PFuncs (D,REAL)
proof
reconsider F1 = F1, F2 = F2 as PartFunc of D,E ;
F1 (#) F2 is PartFunc of D,REAL ;
hence F1 (#) F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
:: original: /
redefine funcF1 / F2 -> Element of PFuncs (D,REAL);
coherence
F1 / F2 is Element of PFuncs (D,REAL)
proof
reconsider F1 = F1, F2 = F2 as PartFunc of D,E ;
F1 / F2 is PartFunc of D,REAL ;
hence F1 / F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
end;
definition
let D be non empty set ;
let E be real-membered set ;
let F be Element of PFuncs (D,E);
:: original: |.
redefine func abs F -> Element of PFuncs (D,REAL);
coherence
|.F.| is Element of PFuncs (D,REAL)
proof
reconsider F = F as PartFunc of D,E ;
abs F is PartFunc of D,REAL ;
hence |.F.| is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
:: original: -
redefine func - F -> Element of PFuncs (D,REAL);
coherence
- F is Element of PFuncs (D,REAL)
proof
reconsider F = F as PartFunc of D,E ;
- F is PartFunc of D,REAL ;
hence - F is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
:: original: ^
redefine funcF ^ -> Element of PFuncs (D,REAL);
coherence
F ^ is Element of PFuncs (D,REAL)
proof
reconsider F = F as PartFunc of D,E ;
F ^ is PartFunc of D,REAL ;
hence F ^ is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
end;
definition
let D be non empty set ;
let E be real-membered set ;
let F be Element of PFuncs (D,E);
let r be real number ;
:: original: (#)
redefine funcr (#) F -> Element of PFuncs (D,REAL);
coherence
r (#) F is Element of PFuncs (D,REAL)
proof
reconsider F = F as PartFunc of D,E ;
r (#) F is PartFunc of D,REAL ;
hence r (#) F is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
end;
definition
let D be non empty set ;
func addpfunc D -> BinOp of (PFuncs (D,REAL)) means :Def4: :: RFUNCT_3:def 4
for F1, F2 being Element of PFuncs (D,REAL) holds it . (F1,F2) = F1 + F2;
existence
ex b1 being BinOp of (PFuncs (D,REAL)) st
for F1, F2 being Element of PFuncs (D,REAL) holds b1 . (F1,F2) = F1 + F2
proof
deffunc H1( Element of PFuncs (D,REAL), Element of PFuncs (D,REAL)) -> Element of PFuncs (D,REAL) = $1 + $2;
ex o being BinOp of (PFuncs (D,REAL)) st
for a, b being Element of PFuncs (D,REAL) holds o . (a,b) = H1(a,b) from BINOP_1:sch_4();
hence ex b1 being BinOp of (PFuncs (D,REAL)) st
for F1, F2 being Element of PFuncs (D,REAL) holds b1 . (F1,F2) = F1 + F2 ; ::_thesis: verum
end;
uniqueness
for b1, b2 being BinOp of (PFuncs (D,REAL)) st ( for F1, F2 being Element of PFuncs (D,REAL) holds b1 . (F1,F2) = F1 + F2 ) & ( for F1, F2 being Element of PFuncs (D,REAL) holds b2 . (F1,F2) = F1 + F2 ) holds
b1 = b2
proof
let o1, o2 be BinOp of (PFuncs (D,REAL)); ::_thesis: ( ( for F1, F2 being Element of PFuncs (D,REAL) holds o1 . (F1,F2) = F1 + F2 ) & ( for F1, F2 being Element of PFuncs (D,REAL) holds o2 . (F1,F2) = F1 + F2 ) implies o1 = o2 )
assume that
A1: for f1, f2 being Element of PFuncs (D,REAL) holds o1 . (f1,f2) = f1 + f2 and
A2: for f1, f2 being Element of PFuncs (D,REAL) holds o2 . (f1,f2) = f1 + f2 ; ::_thesis: o1 = o2
now__::_thesis:_for_f1,_f2_being_Element_of_PFuncs_(D,REAL)_holds_o1_._(f1,f2)_=_o2_._(f1,f2)
let f1, f2 be Element of PFuncs (D,REAL); ::_thesis: o1 . (f1,f2) = o2 . (f1,f2)
o1 . (f1,f2) = f1 + f2 by A1;
hence o1 . (f1,f2) = o2 . (f1,f2) by A2; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:2; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines addpfunc RFUNCT_3:def_4_:_
for D being non empty set
for b2 being BinOp of (PFuncs (D,REAL)) holds
( b2 = addpfunc D iff for F1, F2 being Element of PFuncs (D,REAL) holds b2 . (F1,F2) = F1 + F2 );
theorem Th14: :: RFUNCT_3:14
for D being non empty set holds addpfunc D is commutative
proof
let D be non empty set ; ::_thesis: addpfunc D is commutative
let F1, F2 be Element of PFuncs (D,REAL); :: according to BINOP_1:def_2 ::_thesis: (addpfunc D) . (F1,F2) = (addpfunc D) . (F2,F1)
set o = addpfunc D;
thus (addpfunc D) . (F1,F2) = F2 + F1 by Def4
.= (addpfunc D) . (F2,F1) by Def4 ; ::_thesis: verum
end;
theorem Th15: :: RFUNCT_3:15
for D being non empty set holds addpfunc D is associative
proof
let D be non empty set ; ::_thesis: addpfunc D is associative
let F1, F2, F3 be Element of PFuncs (D,REAL); :: according to BINOP_1:def_3 ::_thesis: (addpfunc D) . (F1,((addpfunc D) . (F2,F3))) = (addpfunc D) . (((addpfunc D) . (F1,F2)),F3)
set o = addpfunc D;
thus (addpfunc D) . (F1,((addpfunc D) . (F2,F3))) = (addpfunc D) . (F1,(F2 + F3)) by Def4
.= F1 + (F2 + F3) by Def4
.= (F1 + F2) + F3 by RFUNCT_1:8
.= ((addpfunc D) . (F1,F2)) + F3 by Def4
.= (addpfunc D) . (((addpfunc D) . (F1,F2)),F3) by Def4 ; ::_thesis: verum
end;
theorem Th16: :: RFUNCT_3:16
for D being non empty set holds ([#] D) --> 0 is_a_unity_wrt addpfunc D
proof
let D be non empty set ; ::_thesis: ([#] D) --> 0 is_a_unity_wrt addpfunc D
set F = ([#] D) --> 0;
A1: dom (([#] D) --> 0) = D by FUNCOP_1:13;
A2: now__::_thesis:_for_G_being_Element_of_PFuncs_(D,REAL)_holds_(addpfunc_D)_._(G,(([#]_D)_-->_0))_=_G
let G be Element of PFuncs (D,REAL); ::_thesis: (addpfunc D) . (G,(([#] D) --> 0)) = G
A3: now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_(G_+_(([#]_D)_-->_0))_holds_
(G_+_(([#]_D)_-->_0))_._d_=_G_._d
let d be Element of D; ::_thesis: ( d in dom (G + (([#] D) --> 0)) implies (G + (([#] D) --> 0)) . d = G . d )
assume d in dom (G + (([#] D) --> 0)) ; ::_thesis: (G + (([#] D) --> 0)) . d = G . d
hence (G + (([#] D) --> 0)) . d = (G . d) + ((([#] D) --> 0) . d) by VALUED_1:def_1
.= (G . d) + 0 by FUNCOP_1:7
.= G . d ;
::_thesis: verum
end;
(dom G) /\ D = dom G by XBOOLE_1:28;
then dom (G + (([#] D) --> 0)) = dom G by A1, VALUED_1:def_1;
then G + (([#] D) --> 0) = G by A3, PARTFUN1:5;
hence (addpfunc D) . (G,(([#] D) --> 0)) = G by Def4; ::_thesis: verum
end;
addpfunc D is commutative by Th14;
hence ([#] D) --> 0 is_a_unity_wrt addpfunc D by A2, BINOP_1:5; ::_thesis: verum
end;
theorem Th17: :: RFUNCT_3:17
for D being non empty set holds the_unity_wrt (addpfunc D) = ([#] D) --> 0
proof
let D be non empty set ; ::_thesis: the_unity_wrt (addpfunc D) = ([#] D) --> 0
([#] D) --> 0 is_a_unity_wrt addpfunc D by Th16;
hence the_unity_wrt (addpfunc D) = ([#] D) --> 0 by BINOP_1:def_8; ::_thesis: verum
end;
theorem Th18: :: RFUNCT_3:18
for D being non empty set holds addpfunc D is having_a_unity
proof
let D be non empty set ; ::_thesis: addpfunc D is having_a_unity
take ([#] D) --> 0 ; :: according to SETWISEO:def_2 ::_thesis: ([#] D) --> 0 is_a_unity_wrt addpfunc D
thus ([#] D) --> 0 is_a_unity_wrt addpfunc D by Th16; ::_thesis: verum
end;
definition
let D be non empty set ;
let f be FinSequence of PFuncs (D,REAL);
func Sum f -> Element of PFuncs (D,REAL) equals :: RFUNCT_3:def 5
(addpfunc D) $$ f;
correctness
coherence
(addpfunc D) $$ f is Element of PFuncs (D,REAL);
;
end;
:: deftheorem defines Sum RFUNCT_3:def_5_:_
for D being non empty set
for f being FinSequence of PFuncs (D,REAL) holds Sum f = (addpfunc D) $$ f;
theorem Th19: :: RFUNCT_3:19
for D being non empty set holds Sum (<*> (PFuncs (D,REAL))) = ([#] D) --> 0
proof
let D be non empty set ; ::_thesis: Sum (<*> (PFuncs (D,REAL))) = ([#] D) --> 0
set o = addpfunc D;
set o0 = ([#] D) --> 0;
the_unity_wrt (addpfunc D) = ([#] D) --> 0 by Th17;
hence Sum (<*> (PFuncs (D,REAL))) = ([#] D) --> 0 by Th18, FINSOP_1:10; ::_thesis: verum
end;
theorem Th20: :: RFUNCT_3:20
for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for G being Element of PFuncs (D,REAL) holds Sum (f ^ <*G*>) = (Sum f) + G
proof
let D be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D,REAL)
for G being Element of PFuncs (D,REAL) holds Sum (f ^ <*G*>) = (Sum f) + G
let f be FinSequence of PFuncs (D,REAL); ::_thesis: for G being Element of PFuncs (D,REAL) holds Sum (f ^ <*G*>) = (Sum f) + G
let G be Element of PFuncs (D,REAL); ::_thesis: Sum (f ^ <*G*>) = (Sum f) + G
set o = addpfunc D;
thus Sum (f ^ <*G*>) = (addpfunc D) . (((addpfunc D) $$ f),G) by Th18, FINSOP_1:4
.= (Sum f) + G by Def4 ; ::_thesis: verum
end;
theorem Th21: :: RFUNCT_3:21
for D being non empty set
for f1, f2 being FinSequence of PFuncs (D,REAL) holds Sum (f1 ^ f2) = (Sum f1) + (Sum f2)
proof
let D be non empty set ; ::_thesis: for f1, f2 being FinSequence of PFuncs (D,REAL) holds Sum (f1 ^ f2) = (Sum f1) + (Sum f2)
let f1, f2 be FinSequence of PFuncs (D,REAL); ::_thesis: Sum (f1 ^ f2) = (Sum f1) + (Sum f2)
set o = addpfunc D;
addpfunc D is associative by Th15;
hence Sum (f1 ^ f2) = (addpfunc D) . ((Sum f1),(Sum f2)) by Th18, FINSOP_1:5
.= (Sum f1) + (Sum f2) by Def4 ;
::_thesis: verum
end;
theorem :: RFUNCT_3:22
for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for G being Element of PFuncs (D,REAL) holds Sum (<*G*> ^ f) = G + (Sum f)
proof
let D be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D,REAL)
for G being Element of PFuncs (D,REAL) holds Sum (<*G*> ^ f) = G + (Sum f)
let f be FinSequence of PFuncs (D,REAL); ::_thesis: for G being Element of PFuncs (D,REAL) holds Sum (<*G*> ^ f) = G + (Sum f)
let G be Element of PFuncs (D,REAL); ::_thesis: Sum (<*G*> ^ f) = G + (Sum f)
thus Sum (<*G*> ^ f) = (Sum <*G*>) + (Sum f) by Th21
.= G + (Sum f) by FINSOP_1:11 ; ::_thesis: verum
end;
theorem Th23: :: RFUNCT_3:23
for D being non empty set
for G1, G2 being Element of PFuncs (D,REAL) holds Sum <*G1,G2*> = G1 + G2
proof
let D be non empty set ; ::_thesis: for G1, G2 being Element of PFuncs (D,REAL) holds Sum <*G1,G2*> = G1 + G2
let G1, G2 be Element of PFuncs (D,REAL); ::_thesis: Sum <*G1,G2*> = G1 + G2
thus Sum <*G1,G2*> = Sum (<*G1*> ^ <*G2*>) by FINSEQ_1:def_9
.= (Sum <*G1*>) + G2 by Th20
.= G1 + G2 by FINSOP_1:11 ; ::_thesis: verum
end;
theorem :: RFUNCT_3:24
for D being non empty set
for G1, G2, G3 being Element of PFuncs (D,REAL) holds Sum <*G1,G2,G3*> = (G1 + G2) + G3
proof
let D be non empty set ; ::_thesis: for G1, G2, G3 being Element of PFuncs (D,REAL) holds Sum <*G1,G2,G3*> = (G1 + G2) + G3
let G1, G2, G3 be Element of PFuncs (D,REAL); ::_thesis: Sum <*G1,G2,G3*> = (G1 + G2) + G3
thus Sum <*G1,G2,G3*> = Sum (<*G1,G2*> ^ <*G3*>) by FINSEQ_1:43
.= (Sum <*G1,G2*>) + G3 by Th20
.= (G1 + G2) + G3 by Th23 ; ::_thesis: verum
end;
theorem :: RFUNCT_3:25
for D being non empty set
for f, g being FinSequence of PFuncs (D,REAL) st f,g are_fiberwise_equipotent holds
Sum f = Sum g
proof
let D be non empty set ; ::_thesis: for f, g being FinSequence of PFuncs (D,REAL) st f,g are_fiberwise_equipotent holds
Sum f = Sum g
defpred S1[ Element of NAT ] means for f, g being FinSequence of PFuncs (D,REAL) st f,g are_fiberwise_equipotent & len f = $1 holds
Sum f = Sum g;
let f, g be FinSequence of PFuncs (D,REAL); ::_thesis: ( f,g are_fiberwise_equipotent implies Sum f = Sum g )
assume A1: f,g are_fiberwise_equipotent ; ::_thesis: Sum f = Sum g
A2: len f = len f ;
A3: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A4: S1[n] ; ::_thesis: S1[n + 1]
let f, g be FinSequence of PFuncs (D,REAL); ::_thesis: ( f,g are_fiberwise_equipotent & len f = n + 1 implies Sum f = Sum g )
assume that
A5: f,g are_fiberwise_equipotent and
A6: len f = n + 1 ; ::_thesis: Sum f = Sum g
0 + 1 <= n + 1 by NAT_1:13;
then A7: n + 1 in dom f by A6, FINSEQ_3:25;
then reconsider a = f . (n + 1) as Element of PFuncs (D,REAL) by FINSEQ_2:11;
rng f = rng g by A5, CLASSES1:75;
then a in rng g by A7, FUNCT_1:def_3;
then consider m being Nat such that
A8: m in dom g and
A9: g . m = a by FINSEQ_2:10;
A10: g = (g | m) ^ (g /^ m) by RFINSEQ:8;
set gg = g /^ m;
set gm = g | m;
m <= len g by A8, FINSEQ_3:25;
then A11: len (g | m) = m by FINSEQ_1:59;
set fn = f | n;
A12: f = (f | n) ^ <*a*> by A6, RFINSEQ:7;
A13: 1 <= m by A8, FINSEQ_3:25;
then max (0,(m - 1)) = m - 1 by FINSEQ_2:4;
then reconsider m1 = m - 1 as Element of NAT by FINSEQ_2:5;
A14: m = m1 + 1 ;
then m1 <= m by NAT_1:11;
then A15: Seg m1 c= Seg m by FINSEQ_1:5;
m in Seg m by A13, FINSEQ_1:1;
then (g | m) . m = a by A8, A9, RFINSEQ:6;
then A16: g | m = ((g | m) | m1) ^ <*a*> by A11, A14, RFINSEQ:7;
A17: (g | m) | m1 = (g | m) | (Seg m1) by FINSEQ_1:def_15
.= (g | (Seg m)) | (Seg m1) by FINSEQ_1:def_15
.= g | ((Seg m) /\ (Seg m1)) by RELAT_1:71
.= g | (Seg m1) by A15, XBOOLE_1:28
.= g | m1 by FINSEQ_1:def_15 ;
now__::_thesis:_for_x_being_set_holds_card_(Coim_((f_|_n),x))_=_card_(Coim_(((g_|_m1)_^_(g_/^_m)),x))
let x be set ; ::_thesis: card (Coim ((f | n),x)) = card (Coim (((g | m1) ^ (g /^ m)),x))
card (Coim (f,x)) = card (Coim (g,x)) by A5, CLASSES1:def_9;
then (card ((f | n) " {x})) + (card (<*a*> " {x})) = card ((((g | m1) ^ <*a*>) ^ (g /^ m)) " {x}) by A10, A16, A17, A12, FINSEQ_3:57
.= (card (((g | m1) ^ <*a*>) " {x})) + (card ((g /^ m) " {x})) by FINSEQ_3:57
.= ((card ((g | m1) " {x})) + (card (<*a*> " {x}))) + (card ((g /^ m) " {x})) by FINSEQ_3:57
.= ((card ((g | m1) " {x})) + (card ((g /^ m) " {x}))) + (card (<*a*> " {x}))
.= (card (((g | m1) ^ (g /^ m)) " {x})) + (card (<*a*> " {x})) by FINSEQ_3:57 ;
hence card (Coim ((f | n),x)) = card (Coim (((g | m1) ^ (g /^ m)),x)) ; ::_thesis: verum
end;
then A18: f | n,(g | m1) ^ (g /^ m) are_fiberwise_equipotent by CLASSES1:def_9;
len (f | n) = n by A6, FINSEQ_1:59, NAT_1:11;
then Sum (f | n) = Sum ((g | m1) ^ (g /^ m)) by A4, A18;
hence Sum f = (Sum ((g | m1) ^ (g /^ m))) + (Sum <*a*>) by A12, Th21
.= ((Sum (g | m1)) + (Sum (g /^ m))) + (Sum <*a*>) by Th21
.= ((Sum (g | m1)) + (Sum <*a*>)) + (Sum (g /^ m)) by RFUNCT_1:8
.= (Sum (g | m)) + (Sum (g /^ m)) by A16, A17, Th21
.= Sum g by A10, Th21 ;
::_thesis: verum
end;
A19: S1[ 0 ]
proof
let f, g be FinSequence of PFuncs (D,REAL); ::_thesis: ( f,g are_fiberwise_equipotent & len f = 0 implies Sum f = Sum g )
assume ( f,g are_fiberwise_equipotent & len f = 0 ) ; ::_thesis: Sum f = Sum g
then A20: ( len g = 0 & f = <*> (PFuncs (D,REAL)) ) by RFINSEQ:3;
then g = <*> (PFuncs (D,REAL)) ;
hence Sum f = Sum g by A20; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A19, A3);
hence Sum f = Sum g by A1, A2; ::_thesis: verum
end;
definition
let D be non empty set ;
let f be FinSequence;
func CHI (f,D) -> FinSequence of PFuncs (D,REAL) means :Def6: :: RFUNCT_3:def 6
( len it = len f & ( for n being Element of NAT st n in dom it holds
it . n = chi ((f . n),D) ) );
existence
ex b1 being FinSequence of PFuncs (D,REAL) st
( len b1 = len f & ( for n being Element of NAT st n in dom b1 holds
b1 . n = chi ((f . n),D) ) )
proof
deffunc H1( Nat) -> Element of K19(K20(D,REAL)) = chi ((f . $1),D);
consider p being FinSequence such that
A1: len p = len f and
A2: for n being Nat st n in dom p holds
p . n = H1(n) from FINSEQ_1:sch_2();
rng p c= PFuncs (D,REAL)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng p or x in PFuncs (D,REAL) )
assume x in rng p ; ::_thesis: x in PFuncs (D,REAL)
then consider n being Nat such that
A3: ( n in dom p & p . n = x ) by FINSEQ_2:10;
x = chi ((f . n),D) by A2, A3;
hence x in PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum
end;
then reconsider p = p as FinSequence of PFuncs (D,REAL) by FINSEQ_1:def_4;
take p ; ::_thesis: ( len p = len f & ( for n being Element of NAT st n in dom p holds
p . n = chi ((f . n),D) ) )
thus len p = len f by A1; ::_thesis: for n being Element of NAT st n in dom p holds
p . n = chi ((f . n),D)
let n be Element of NAT ; ::_thesis: ( n in dom p implies p . n = chi ((f . n),D) )
assume n in dom p ; ::_thesis: p . n = chi ((f . n),D)
hence p . n = chi ((f . n),D) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence of PFuncs (D,REAL) st len b1 = len f & ( for n being Element of NAT st n in dom b1 holds
b1 . n = chi ((f . n),D) ) & len b2 = len f & ( for n being Element of NAT st n in dom b2 holds
b2 . n = chi ((f . n),D) ) holds
b1 = b2
proof
let p1, p2 be FinSequence of PFuncs (D,REAL); ::_thesis: ( len p1 = len f & ( for n being Element of NAT st n in dom p1 holds
p1 . n = chi ((f . n),D) ) & len p2 = len f & ( for n being Element of NAT st n in dom p2 holds
p2 . n = chi ((f . n),D) ) implies p1 = p2 )
assume that
A4: len p1 = len f and
A5: for n being Element of NAT st n in dom p1 holds
p1 . n = chi ((f . n),D) and
A6: len p2 = len f and
A7: for n being Element of NAT st n in dom p2 holds
p2 . n = chi ((f . n),D) ; ::_thesis: p1 = p2
A8: ( dom p1 = Seg (len p1) & dom p2 = Seg (len p2) ) by FINSEQ_1:def_3;
now__::_thesis:_for_n_being_Nat_st_n_in_dom_p1_holds_
p1_._n_=_p2_._n
let n be Nat; ::_thesis: ( n in dom p1 implies p1 . n = p2 . n )
assume A9: n in dom p1 ; ::_thesis: p1 . n = p2 . n
then p1 . n = chi ((f . n),D) by A5;
hence p1 . n = p2 . n by A4, A6, A7, A8, A9; ::_thesis: verum
end;
hence p1 = p2 by A4, A6, FINSEQ_2:9; ::_thesis: verum
end;
end;
:: deftheorem Def6 defines CHI RFUNCT_3:def_6_:_
for D being non empty set
for f being FinSequence
for b3 being FinSequence of PFuncs (D,REAL) holds
( b3 = CHI (f,D) iff ( len b3 = len f & ( for n being Element of NAT st n in dom b3 holds
b3 . n = chi ((f . n),D) ) ) );
definition
let D be non empty set ;
let f be FinSequence of PFuncs (D,REAL);
let R be FinSequence of REAL ;
funcR (#) f -> FinSequence of PFuncs (D,REAL) means :Def7: :: RFUNCT_3:def 7
( len it = min ((len R),(len f)) & ( for n being Element of NAT st n in dom it holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
it . n = r (#) F ) );
existence
ex b1 being FinSequence of PFuncs (D,REAL) st
( len b1 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b1 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b1 . n = r (#) F ) )
proof
defpred S1[ Nat, set ] means for F being PartFunc of D,REAL
for r being Real st r = R . $1 & F = f . $1 holds
$2 = r (#) F;
set m = min ((len R),(len f));
A1: min ((len R),(len f)) <= len f by XXREAL_0:17;
A2: for n being Nat st n in Seg (min ((len R),(len f))) holds
ex x being Element of PFuncs (D,REAL) st S1[n,x]
proof
let n be Nat; ::_thesis: ( n in Seg (min ((len R),(len f))) implies ex x being Element of PFuncs (D,REAL) st S1[n,x] )
reconsider r = R . n as Real ;
assume A3: n in Seg (min ((len R),(len f))) ; ::_thesis: ex x being Element of PFuncs (D,REAL) st S1[n,x]
then n <= min ((len R),(len f)) by FINSEQ_1:1;
then A4: n <= len f by A1, XXREAL_0:2;
1 <= n by A3, FINSEQ_1:1;
then n in dom f by A4, FINSEQ_3:25;
then reconsider F = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11;
reconsider a = r (#) F as Element of PFuncs (D,REAL) ;
take a ; ::_thesis: S1[n,a]
thus S1[n,a] ; ::_thesis: verum
end;
consider p being FinSequence of PFuncs (D,REAL) such that
A5: ( dom p = Seg (min ((len R),(len f))) & ( for n being Nat st n in Seg (min ((len R),(len f))) holds
S1[n,p . n] ) ) from FINSEQ_1:sch_5(A2);
take p ; ::_thesis: ( len p = min ((len R),(len f)) & ( for n being Element of NAT st n in dom p holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p . n = r (#) F ) )
thus len p = min ((len R),(len f)) by A5, FINSEQ_1:def_3; ::_thesis: for n being Element of NAT st n in dom p holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p . n = r (#) F
let n be Element of NAT ; ::_thesis: ( n in dom p implies for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p . n = r (#) F )
assume n in dom p ; ::_thesis: for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p . n = r (#) F
hence for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p . n = r (#) F by A5; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence of PFuncs (D,REAL) st len b1 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b1 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b1 . n = r (#) F ) & len b2 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b2 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b2 . n = r (#) F ) holds
b1 = b2
proof
set m = min ((len R),(len f));
let p1, p2 be FinSequence of PFuncs (D,REAL); ::_thesis: ( len p1 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom p1 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p1 . n = r (#) F ) & len p2 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom p2 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p2 . n = r (#) F ) implies p1 = p2 )
assume that
A6: len p1 = min ((len R),(len f)) and
A7: for n being Element of NAT st n in dom p1 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p1 . n = r (#) F and
A8: len p2 = min ((len R),(len f)) and
A9: for n being Element of NAT st n in dom p2 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
p2 . n = r (#) F ; ::_thesis: p1 = p2
A10: dom p1 = Seg (min ((len R),(len f))) by A6, FINSEQ_1:def_3;
A11: ( dom p1 = Seg (len p1) & dom p2 = Seg (len p2) ) by FINSEQ_1:def_3;
A12: min ((len R),(len f)) <= len f by XXREAL_0:17;
now__::_thesis:_for_n_being_Nat_st_n_in_dom_p1_holds_
p1_._n_=_p2_._n
let n be Nat; ::_thesis: ( n in dom p1 implies p1 . n = p2 . n )
reconsider r = R . n as Real ;
assume A13: n in dom p1 ; ::_thesis: p1 . n = p2 . n
then n <= min ((len R),(len f)) by A10, FINSEQ_1:1;
then A14: n <= len f by A12, XXREAL_0:2;
1 <= n by A10, A13, FINSEQ_1:1;
then n in dom f by A14, FINSEQ_3:25;
then reconsider F = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11;
p1 . n = r (#) F by A7, A13;
hence p1 . n = p2 . n by A6, A8, A9, A11, A13; ::_thesis: verum
end;
hence p1 = p2 by A6, A8, FINSEQ_2:9; ::_thesis: verum
end;
end;
:: deftheorem Def7 defines (#) RFUNCT_3:def_7_:_
for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for R being FinSequence of REAL
for b4 being FinSequence of PFuncs (D,REAL) holds
( b4 = R (#) f iff ( len b4 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b4 holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b4 . n = r (#) F ) ) );
definition
let D be non empty set ;
let f be FinSequence of PFuncs (D,REAL);
let d be Element of D;
funcf # d -> FinSequence of REAL means :Def8: :: RFUNCT_3:def 8
( len it = len f & ( for n being Element of NAT st n in dom it holds
it . n = (f . n) . d ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = len f & ( for n being Element of NAT st n in dom b1 holds
b1 . n = (f . n) . d ) )
proof
defpred S1[ Nat, set ] means $2 = (f . $1) . d;
A1: for n being Nat st n in Seg (len f) holds
ex x being Element of REAL st S1[n,x]
proof
let n be Nat; ::_thesis: ( n in Seg (len f) implies ex x being Element of REAL st S1[n,x] )
assume n in Seg (len f) ; ::_thesis: ex x being Element of REAL st S1[n,x]
then n in dom f by FINSEQ_1:def_3;
then reconsider G = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11;
take G . d ; ::_thesis: S1[n,G . d]
thus S1[n,G . d] ; ::_thesis: verum
end;
consider p being FinSequence of REAL such that
A2: dom p = Seg (len f) and
A3: for n being Nat st n in Seg (len f) holds
S1[n,p . n] from FINSEQ_1:sch_5(A1);
take p ; ::_thesis: ( len p = len f & ( for n being Element of NAT st n in dom p holds
p . n = (f . n) . d ) )
thus len p = len f by A2, FINSEQ_1:def_3; ::_thesis: for n being Element of NAT st n in dom p holds
p . n = (f . n) . d
thus for n being Element of NAT st n in dom p holds
p . n = (f . n) . d by A2, A3; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence of REAL st len b1 = len f & ( for n being Element of NAT st n in dom b1 holds
b1 . n = (f . n) . d ) & len b2 = len f & ( for n being Element of NAT st n in dom b2 holds
b2 . n = (f . n) . d ) holds
b1 = b2
proof
let p1, p2 be FinSequence of REAL ; ::_thesis: ( len p1 = len f & ( for n being Element of NAT st n in dom p1 holds
p1 . n = (f . n) . d ) & len p2 = len f & ( for n being Element of NAT st n in dom p2 holds
p2 . n = (f . n) . d ) implies p1 = p2 )
assume that
A4: len p1 = len f and
A5: for n being Element of NAT st n in dom p1 holds
p1 . n = (f . n) . d and
A6: len p2 = len f and
A7: for n being Element of NAT st n in dom p2 holds
p2 . n = (f . n) . d ; ::_thesis: p1 = p2
A8: dom p1 = Seg (len p1) by FINSEQ_1:def_3;
A9: dom p2 = Seg (len p2) by FINSEQ_1:def_3;
now__::_thesis:_for_n_being_Nat_st_n_in_dom_p1_holds_
p1_._n_=_p2_._n
let n be Nat; ::_thesis: ( n in dom p1 implies p1 . n = p2 . n )
assume A10: n in dom p1 ; ::_thesis: p1 . n = p2 . n
then p1 . n = (f . n) . d by A5;
hence p1 . n = p2 . n by A4, A6, A7, A8, A9, A10; ::_thesis: verum
end;
hence p1 = p2 by A4, A6, FINSEQ_2:9; ::_thesis: verum
end;
end;
:: deftheorem Def8 defines # RFUNCT_3:def_8_:_
for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for d being Element of D
for b4 being FinSequence of REAL holds
( b4 = f # d iff ( len b4 = len f & ( for n being Element of NAT st n in dom b4 holds
b4 . n = (f . n) . d ) ) );
definition
let D, C be non empty set ;
let f be FinSequence of PFuncs (D,C);
let d be Element of D;
predd is_common_for_dom f means :Def9: :: RFUNCT_3:def 9
for n being Element of NAT st n in dom f holds
d in dom (f . n);
end;
:: deftheorem Def9 defines is_common_for_dom RFUNCT_3:def_9_:_
for D, C being non empty set
for f being FinSequence of PFuncs (D,C)
for d being Element of D holds
( d is_common_for_dom f iff for n being Element of NAT st n in dom f holds
d in dom (f . n) );
theorem Th26: :: RFUNCT_3:26
for D, C being non empty set
for f being FinSequence of PFuncs (D,C)
for d being Element of D
for n being Element of NAT st d is_common_for_dom f & n <> 0 holds
d is_common_for_dom f | n
proof
let D1, D2 be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D1,D2)
for d being Element of D1
for n being Element of NAT st d is_common_for_dom f & n <> 0 holds
d is_common_for_dom f | n
let f be FinSequence of PFuncs (D1,D2); ::_thesis: for d being Element of D1
for n being Element of NAT st d is_common_for_dom f & n <> 0 holds
d is_common_for_dom f | n
let d1 be Element of D1; ::_thesis: for n being Element of NAT st d1 is_common_for_dom f & n <> 0 holds
d1 is_common_for_dom f | n
let n be Element of NAT ; ::_thesis: ( d1 is_common_for_dom f & n <> 0 implies d1 is_common_for_dom f | n )
assume that
A1: d1 is_common_for_dom f and
A2: n <> 0 ; ::_thesis: d1 is_common_for_dom f | n
let m be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( m in dom (f | n) implies d1 in dom ((f | n) . m) )
assume A3: m in dom (f | n) ; ::_thesis: d1 in dom ((f | n) . m)
set G = (f | n) . m;
now__::_thesis:_(_(_n_>=_len_f_&_d1_in_dom_((f_|_n)_._m)_)_or_(_n_<_len_f_&_d1_in_dom_((f_|_n)_._m)_)_)
percases ( n >= len f or n < len f ) ;
case n >= len f ; ::_thesis: d1 in dom ((f | n) . m)
then f | n = f by Lm1;
hence d1 in dom ((f | n) . m) by A1, A3, Def9; ::_thesis: verum
end;
caseA4: n < len f ; ::_thesis: d1 in dom ((f | n) . m)
0 + 1 <= n by A2, NAT_1:13;
then A5: n in dom f by A4, FINSEQ_3:25;
( dom (f | n) = Seg (len (f | n)) & len (f | n) = n ) by A4, FINSEQ_1:59, FINSEQ_1:def_3;
then ( (f | n) . m = f . m & m in dom f ) by A3, A5, RFINSEQ:6;
hence d1 in dom ((f | n) . m) by A1, Def9; ::_thesis: verum
end;
end;
end;
hence d1 in dom ((f | n) . m) ; ::_thesis: verum
end;
theorem :: RFUNCT_3:27
for D, C being non empty set
for f being FinSequence of PFuncs (D,C)
for d being Element of D
for n being Element of NAT st d is_common_for_dom f holds
d is_common_for_dom f /^ n
proof
let D1, D2 be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D1,D2)
for d being Element of D1
for n being Element of NAT st d is_common_for_dom f holds
d is_common_for_dom f /^ n
let f be FinSequence of PFuncs (D1,D2); ::_thesis: for d being Element of D1
for n being Element of NAT st d is_common_for_dom f holds
d is_common_for_dom f /^ n
let d1 be Element of D1; ::_thesis: for n being Element of NAT st d1 is_common_for_dom f holds
d1 is_common_for_dom f /^ n
let n be Element of NAT ; ::_thesis: ( d1 is_common_for_dom f implies d1 is_common_for_dom f /^ n )
assume A1: d1 is_common_for_dom f ; ::_thesis: d1 is_common_for_dom f /^ n
let m be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( m in dom (f /^ n) implies d1 in dom ((f /^ n) . m) )
set fn = f /^ n;
assume A2: m in dom (f /^ n) ; ::_thesis: d1 in dom ((f /^ n) . m)
set G = (f /^ n) . m;
now__::_thesis:_(_(_len_f_<_n_&_d1_in_dom_((f_/^_n)_._m)_)_or_(_n_<=_len_f_&_d1_in_dom_((f_/^_n)_._m)_)_)
percases ( len f < n or n <= len f ) ;
case len f < n ; ::_thesis: d1 in dom ((f /^ n) . m)
hence d1 in dom ((f /^ n) . m) by A2, RELAT_1:38, RFINSEQ:def_1; ::_thesis: verum
end;
caseA3: n <= len f ; ::_thesis: d1 in dom ((f /^ n) . m)
( 1 <= m & m <= m + n ) by A2, FINSEQ_3:25, NAT_1:11;
then A4: 1 <= m + n by XXREAL_0:2;
A5: m <= len (f /^ n) by A2, FINSEQ_3:25;
len (f /^ n) = (len f) - n by A3, RFINSEQ:def_1;
then m + n <= len f by A5, XREAL_1:19;
then A6: m + n in dom f by A4, FINSEQ_3:25;
(f /^ n) . m = f . (m + n) by A2, A3, RFINSEQ:def_1;
hence d1 in dom ((f /^ n) . m) by A1, A6, Def9; ::_thesis: verum
end;
end;
end;
hence d1 in dom ((f /^ n) . m) ; ::_thesis: verum
end;
theorem Th28: :: RFUNCT_3:28
for D being non empty set
for d being Element of D
for f being FinSequence of PFuncs (D,REAL) st len f <> 0 holds
( d is_common_for_dom f iff d in dom (Sum f) )
proof
let D be non empty set ; ::_thesis: for d being Element of D
for f being FinSequence of PFuncs (D,REAL) st len f <> 0 holds
( d is_common_for_dom f iff d in dom (Sum f) )
let d be Element of D; ::_thesis: for f being FinSequence of PFuncs (D,REAL) st len f <> 0 holds
( d is_common_for_dom f iff d in dom (Sum f) )
defpred S1[ Element of NAT ] means for f being FinSequence of PFuncs (D,REAL) st len f = $1 & len f <> 0 holds
( d is_common_for_dom f iff d in dom (Sum f) );
let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f <> 0 implies ( d is_common_for_dom f iff d in dom (Sum f) ) )
assume A1: len f <> 0 ; ::_thesis: ( d is_common_for_dom f iff d in dom (Sum f) )
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f = n + 1 & len f <> 0 implies ( d is_common_for_dom f iff d in dom (Sum f) ) )
assume that
A4: len f = n + 1 and
len f <> 0 ; ::_thesis: ( d is_common_for_dom f iff d in dom (Sum f) )
A5: dom f = Seg (len f) by FINSEQ_1:def_3;
now__::_thesis:_(_(_n_=_0_&_(_d_is_common_for_dom_f_implies_d_in_dom_(Sum_f)_)_&_(_d_in_dom_(Sum_f)_implies_d_is_common_for_dom_f_)_)_or_(_n_<>_0_&_(_d_is_common_for_dom_f_implies_d_in_dom_(Sum_f)_)_&_(_d_in_dom_(Sum_f)_implies_d_is_common_for_dom_f_)_)_)
percases ( n = 0 or n <> 0 ) ;
caseA6: n = 0 ; ::_thesis: ( ( d is_common_for_dom f implies d in dom (Sum f) ) & ( d in dom (Sum f) implies d is_common_for_dom f ) )
then A7: 1 in dom f by A4, FINSEQ_3:25;
then reconsider G = f . 1 as Element of PFuncs (D,REAL) by FINSEQ_2:11;
f = <*G*> by A4, A6, FINSEQ_1:40;
then A8: Sum f = G by FINSOP_1:11;
hence ( d is_common_for_dom f implies d in dom (Sum f) ) by A7, Def9; ::_thesis: ( d in dom (Sum f) implies d is_common_for_dom f )
assume d in dom (Sum f) ; ::_thesis: d is_common_for_dom f
then for m being Element of NAT st m in dom f holds
d in dom (f . m) by A4, A5, A6, A8, FINSEQ_1:2, TARSKI:def_1;
hence d is_common_for_dom f by Def9; ::_thesis: verum
end;
caseA9: n <> 0 ; ::_thesis: ( ( d is_common_for_dom f implies d in dom (Sum f) ) & ( d in dom (Sum f) implies d is_common_for_dom f ) )
A10: n <= n + 1 by NAT_1:11;
0 + 1 <= n by A9, NAT_1:13;
then A11: n in dom f by A4, A10, FINSEQ_3:25;
0 + 1 <= n + 1 by NAT_1:13;
then A12: n + 1 in dom f by A4, FINSEQ_3:25;
then reconsider G = f . (n + 1) as Element of PFuncs (D,REAL) by FINSEQ_2:11;
set fn = f | n;
A13: len (f | n) = n by A4, FINSEQ_1:59, NAT_1:11;
f = (f | n) ^ <*G*> by A4, RFINSEQ:7;
then A14: Sum f = (Sum (f | n)) + G by Th20;
thus ( d is_common_for_dom f implies d in dom (Sum f) ) ::_thesis: ( d in dom (Sum f) implies d is_common_for_dom f )
proof
assume A15: d is_common_for_dom f ; ::_thesis: d in dom (Sum f)
then d is_common_for_dom f | n by A9, Th26;
then A16: d in dom (Sum (f | n)) by A3, A9, A13;
d in dom G by A12, A15, Def9;
then d in (dom (Sum (f | n))) /\ (dom G) by A16, XBOOLE_0:def_4;
hence d in dom (Sum f) by A14, VALUED_1:def_1; ::_thesis: verum
end;
assume d in dom (Sum f) ; ::_thesis: d is_common_for_dom f
then A17: d in (dom (Sum (f | n))) /\ (dom G) by A14, VALUED_1:def_1;
then d in dom (Sum (f | n)) by XBOOLE_0:def_4;
then A18: d is_common_for_dom f | n by A3, A9, A13;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_f_holds_
d_in_dom_(f_._m)
let m be Element of NAT ; ::_thesis: ( m in dom f implies d in dom (f . m) )
assume A19: m in dom f ; ::_thesis: d in dom (f . m)
set F = f . m;
A20: m <= len f by A19, FINSEQ_3:25;
A21: 1 <= m by A19, FINSEQ_3:25;
now__::_thesis:_(_(_m_=_len_f_&_d_in_dom_(f_._m)_)_or_(_m_<>_len_f_&_d_in_dom_(f_._m)_)_)
percases ( m = len f or m <> len f ) ;
case m = len f ; ::_thesis: d in dom (f . m)
hence d in dom (f . m) by A4, A17, XBOOLE_0:def_4; ::_thesis: verum
end;
case m <> len f ; ::_thesis: d in dom (f . m)
then m < len f by A20, XXREAL_0:1;
then m <= n by A4, NAT_1:13;
then A22: m in Seg n by A21, FINSEQ_1:1;
then ( dom (f | n) = Seg (len (f | n)) & f . m = (f | n) . m ) by A11, FINSEQ_1:def_3, RFINSEQ:6;
hence d in dom (f . m) by A13, A18, A22, Def9; ::_thesis: verum
end;
end;
end;
hence d in dom (f . m) ; ::_thesis: verum
end;
hence d is_common_for_dom f by Def9; ::_thesis: verum
end;
end;
end;
hence ( d is_common_for_dom f iff d in dom (Sum f) ) ; ::_thesis: verum
end;
A23: S1[ 0 ] ;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A23, A2);
hence ( d is_common_for_dom f iff d in dom (Sum f) ) by A1; ::_thesis: verum
end;
theorem Th29: :: RFUNCT_3:29
for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for d being Element of D
for n being Element of NAT holds (f | n) # d = (f # d) | n
proof
let D1 be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D1,REAL)
for d being Element of D1
for n being Element of NAT holds (f | n) # d = (f # d) | n
let f be FinSequence of PFuncs (D1,REAL); ::_thesis: for d being Element of D1
for n being Element of NAT holds (f | n) # d = (f # d) | n
let d1 be Element of D1; ::_thesis: for n being Element of NAT holds (f | n) # d1 = (f # d1) | n
let n be Element of NAT ; ::_thesis: (f | n) # d1 = (f # d1) | n
A1: len (f # d1) = len f by Def8;
A2: len ((f | n) # d1) = len (f | n) by Def8;
now__::_thesis:_(_(_len_f_<=_n_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_or_(_n_<_len_f_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_)
percases ( len f <= n or n < len f ) ;
caseA3: len f <= n ; ::_thesis: (f | n) # d1 = (f # d1) | n
then f | n = f by Lm1;
hence (f | n) # d1 = (f # d1) | n by A1, A3, Lm1; ::_thesis: verum
end;
caseA4: n < len f ; ::_thesis: (f | n) # d1 = (f # d1) | n
then A5: len (f | n) = n by FINSEQ_1:59;
A6: len ((f # d1) | n) = n by A1, A4, FINSEQ_1:59;
A7: ( dom f = Seg (len f) & dom (f # d1) = Seg (len (f # d1)) ) by FINSEQ_1:def_3;
A8: dom ((f | n) # d1) = Seg (len ((f | n) # d1)) by FINSEQ_1:def_3;
now__::_thesis:_(_(_n_=_0_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_or_(_n_<>_0_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_)
percases ( n = 0 or n <> 0 ) ;
caseA9: n = 0 ; ::_thesis: (f | n) # d1 = (f # d1) | n
then (f # d1) | n = <*> REAL ;
hence (f | n) # d1 = (f # d1) | n by A2, A9; ::_thesis: verum
end;
caseA10: n <> 0 ; ::_thesis: (f | n) # d1 = (f # d1) | n
A11: dom ((f # d1) | n) = Seg (len (f | n)) by A5, A6, FINSEQ_1:def_3;
0 + 1 <= n by A10, NAT_1:13;
then A12: n in dom f by A4, FINSEQ_3:25;
now__::_thesis:_for_m_being_Nat_st_m_in_dom_((f_#_d1)_|_n)_holds_
((f_#_d1)_|_n)_._m_=_((f_|_n)_#_d1)_._m
let m be Nat; ::_thesis: ( m in dom ((f # d1) | n) implies ((f # d1) | n) . m = ((f | n) # d1) . m )
assume A13: m in dom ((f # d1) | n) ; ::_thesis: ((f # d1) | n) . m = ((f | n) # d1) . m
then A14: m in dom (f # d1) by A1, A5, A7, A12, A11, RFINSEQ:6;
then reconsider G = f . m as Element of PFuncs (D1,REAL) by A1, A7, FINSEQ_2:11;
((f # d1) | n) . m = (f # d1) . m by A1, A5, A7, A12, A11, A13, RFINSEQ:6;
then A15: ((f # d1) | n) . m = G . d1 by A14, Def8;
(f | n) . m = G by A5, A12, A11, A13, RFINSEQ:6;
hence ((f # d1) | n) . m = ((f | n) # d1) . m by A2, A8, A11, A13, A15, Def8; ::_thesis: verum
end;
hence (f | n) # d1 = (f # d1) | n by A2, A5, A6, FINSEQ_2:9; ::_thesis: verum
end;
end;
end;
hence (f | n) # d1 = (f # d1) | n ; ::_thesis: verum
end;
end;
end;
hence (f | n) # d1 = (f # d1) | n ; ::_thesis: verum
end;
theorem Th30: :: RFUNCT_3:30
for D being non empty set
for f being FinSequence
for d being Element of D holds d is_common_for_dom CHI (f,D)
proof
let D be non empty set ; ::_thesis: for f being FinSequence
for d being Element of D holds d is_common_for_dom CHI (f,D)
let f be FinSequence; ::_thesis: for d being Element of D holds d is_common_for_dom CHI (f,D)
let d be Element of D; ::_thesis: d is_common_for_dom CHI (f,D)
let n be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( n in dom (CHI (f,D)) implies d in dom ((CHI (f,D)) . n) )
assume n in dom (CHI (f,D)) ; ::_thesis: d in dom ((CHI (f,D)) . n)
then (CHI (f,D)) . n = chi ((f . n),D) by Def6;
then dom ((CHI (f,D)) . n) = D by RFUNCT_1:61;
hence d in dom ((CHI (f,D)) . n) ; ::_thesis: verum
end;
theorem Th31: :: RFUNCT_3:31
for D being non empty set
for d being Element of D
for f being FinSequence of PFuncs (D,REAL)
for R being FinSequence of REAL st d is_common_for_dom f holds
d is_common_for_dom R (#) f
proof
let D be non empty set ; ::_thesis: for d being Element of D
for f being FinSequence of PFuncs (D,REAL)
for R being FinSequence of REAL st d is_common_for_dom f holds
d is_common_for_dom R (#) f
let d be Element of D; ::_thesis: for f being FinSequence of PFuncs (D,REAL)
for R being FinSequence of REAL st d is_common_for_dom f holds
d is_common_for_dom R (#) f
let f be FinSequence of PFuncs (D,REAL); ::_thesis: for R being FinSequence of REAL st d is_common_for_dom f holds
d is_common_for_dom R (#) f
let R be FinSequence of REAL ; ::_thesis: ( d is_common_for_dom f implies d is_common_for_dom R (#) f )
assume A1: d is_common_for_dom f ; ::_thesis: d is_common_for_dom R (#) f
set m = min ((len R),(len f));
let n be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( n in dom (R (#) f) implies d in dom ((R (#) f) . n) )
assume A2: n in dom (R (#) f) ; ::_thesis: d in dom ((R (#) f) . n)
set G = (R (#) f) . n;
len (R (#) f) = min ((len R),(len f)) by Def7;
then ( min ((len R),(len f)) <= len f & n <= min ((len R),(len f)) ) by A2, FINSEQ_3:25, XXREAL_0:17;
then A3: n <= len f by XXREAL_0:2;
1 <= n by A2, FINSEQ_3:25;
then A4: n in dom f by A3, FINSEQ_3:25;
then reconsider F = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11;
A5: d in dom F by A1, A4, Def9;
reconsider r = R . n as Real ;
(R (#) f) . n = r (#) F by A2, Def7;
hence d in dom ((R (#) f) . n) by A5, VALUED_1:def_5; ::_thesis: verum
end;
theorem :: RFUNCT_3:32
for D being non empty set
for f being FinSequence
for R being FinSequence of REAL
for d being Element of D holds d is_common_for_dom R (#) (CHI (f,D)) by Th30, Th31;
theorem :: RFUNCT_3:33
for D being non empty set
for d being Element of D
for f being FinSequence of PFuncs (D,REAL) st d is_common_for_dom f holds
(Sum f) . d = Sum (f # d)
proof
let D be non empty set ; ::_thesis: for d being Element of D
for f being FinSequence of PFuncs (D,REAL) st d is_common_for_dom f holds
(Sum f) . d = Sum (f # d)
let d be Element of D; ::_thesis: for f being FinSequence of PFuncs (D,REAL) st d is_common_for_dom f holds
(Sum f) . d = Sum (f # d)
defpred S1[ Element of NAT ] means for f being FinSequence of PFuncs (D,REAL) st len f = $1 & d is_common_for_dom f holds
(Sum f) . d = Sum (f # d);
let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( d is_common_for_dom f implies (Sum f) . d = Sum (f # d) )
assume A1: d is_common_for_dom f ; ::_thesis: (Sum f) . d = Sum (f # d)
A2: len f = len f ;
A3: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A4: S1[n] ; ::_thesis: S1[n + 1]
let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f = n + 1 & d is_common_for_dom f implies (Sum f) . d = Sum (f # d) )
assume that
A5: len f = n + 1 and
A6: d is_common_for_dom f ; ::_thesis: (Sum f) . d = Sum (f # d)
set fn = f | n;
A7: len (f | n) = n by A5, FINSEQ_1:59, NAT_1:11;
0 + 1 <= n + 1 by NAT_1:13;
then A8: n + 1 in dom f by A5, FINSEQ_3:25;
then reconsider G = f . (n + 1) as Element of PFuncs (D,REAL) by FINSEQ_2:11;
A9: ( dom f = Seg (len f) & dom (f # d) = Seg (len (f # d)) ) by FINSEQ_1:def_3;
f = (f | n) ^ <*G*> by A5, RFINSEQ:7;
then A10: Sum f = (Sum (f | n)) + G by Th20;
A11: len (f # d) = len f by Def8;
A12: d in dom G by A6, A8, Def9;
now__::_thesis:_(_(_n_=_0_&_(Sum_f)_._d_=_Sum_(f_#_d)_)_or_(_n_<>_0_&_(Sum_f)_._d_=_Sum_(f_#_d)_)_)
percases ( n = 0 or n <> 0 ) ;
caseA13: n = 0 ; ::_thesis: (Sum f) . d = Sum (f # d)
then A14: len (f # d) = 1 by A5, Def8;
then A15: 1 in dom (f # d) by FINSEQ_3:25;
A16: now__::_thesis:_for_m_being_Nat_st_m_in_Seg_1_holds_
(f_#_d)_._m_=_<*(G_._d)*>_._m
let m be Nat; ::_thesis: ( m in Seg 1 implies (f # d) . m = <*(G . d)*> . m )
assume m in Seg 1 ; ::_thesis: (f # d) . m = <*(G . d)*> . m
then A17: m = 1 by FINSEQ_1:2, TARSKI:def_1;
hence (f # d) . m = G . d by A13, A15, Def8
.= <*(G . d)*> . m by A17, FINSEQ_1:40 ;
::_thesis: verum
end;
( len <*(G . d)*> = 1 & dom (f # d) = Seg 1 ) by A14, FINSEQ_1:40, FINSEQ_1:def_3;
then A18: f # d = <*(G . d)*> by A14, A16, FINSEQ_2:9;
f = <*G*> by A5, A13, FINSEQ_1:40;
hence (Sum f) . d = G . d by FINSOP_1:11
.= Sum (f # d) by A18, FINSOP_1:11 ;
::_thesis: verum
end;
caseA19: n <> 0 ; ::_thesis: (Sum f) . d = Sum (f # d)
A20: (f # d) . (n + 1) = G . d by A9, A11, A8, Def8;
d is_common_for_dom f | n by A6, A19, Th26;
then d in dom (Sum (f | n)) by A7, A19, Th28;
then d in (dom (Sum (f | n))) /\ (dom G) by A12, XBOOLE_0:def_4;
then d in dom ((Sum (f | n)) + G) by VALUED_1:def_1;
hence (Sum f) . d = ((Sum (f | n)) . d) + (G . d) by A10, VALUED_1:def_1
.= (Sum ((f | n) # d)) + (G . d) by A4, A6, A7, A19, Th26
.= (Sum ((f # d) | n)) + (G . d) by Th29
.= Sum (((f # d) | n) ^ <*(G . d)*>) by RVSUM_1:74
.= Sum (f # d) by A5, A11, A20, RFINSEQ:7 ;
::_thesis: verum
end;
end;
end;
hence (Sum f) . d = Sum (f # d) ; ::_thesis: verum
end;
A21: S1[ 0 ]
proof
let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f = 0 & d is_common_for_dom f implies (Sum f) . d = Sum (f # d) )
assume that
A22: len f = 0 and
d is_common_for_dom f ; ::_thesis: (Sum f) . d = Sum (f # d)
f = <*> (PFuncs (D,REAL)) by A22;
then A23: (Sum f) . d = (([#] D) --> 0) . d by Th19
.= 0 by FUNCOP_1:7 ;
len (f # d) = 0 by A22, Def8;
then f # d = <*> (PFuncs (D,REAL)) ;
hence (Sum f) . d = Sum (f # d) by A23, RVSUM_1:72; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A21, A3);
hence (Sum f) . d = Sum (f # d) by A1, A2; ::_thesis: verum
end;
definition
let D be non empty set ;
let F be PartFunc of D,REAL;
func max+ F -> PartFunc of D,REAL means :Def10: :: RFUNCT_3:def 10
( dom it = dom F & ( for d being Element of D st d in dom it holds
it . d = max+ (F . d) ) );
existence
ex b1 being PartFunc of D,REAL st
( dom b1 = dom F & ( for d being Element of D st d in dom b1 holds
b1 . d = max+ (F . d) ) )
proof
deffunc H1( set ) -> Real = max+ (F . $1);
defpred S1[ set ] means $1 in dom F;
consider f being PartFunc of D,REAL such that
A1: for d being Element of D holds
( d in dom f iff S1[d] ) and
A2: for d being Element of D st d in dom f holds
f . d = H1(d) from SEQ_1:sch_3();
take f ; ::_thesis: ( dom f = dom F & ( for d being Element of D st d in dom f holds
f . d = max+ (F . d) ) )
thus dom f = dom F ::_thesis: for d being Element of D st d in dom f holds
f . d = max+ (F . d)
proof
thus dom f c= dom F :: according to XBOOLE_0:def_10 ::_thesis: dom F c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in dom F )
assume x in dom f ; ::_thesis: x in dom F
hence x in dom F by A1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom F or x in dom f )
assume x in dom F ; ::_thesis: x in dom f
hence x in dom f by A1; ::_thesis: verum
end;
thus for d being Element of D st d in dom f holds
f . d = max+ (F . d) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds
b1 . d = max+ (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds
b2 . d = max+ (F . d) ) holds
b1 = b2
proof
deffunc H1( set ) -> Real = max+ (F . $1);
for f, g being PartFunc of D,REAL st dom f = dom F & ( for c being Element of D st c in dom f holds
f . c = H1(c) ) & dom g = dom F & ( for c being Element of D st c in dom g holds
g . c = H1(c) ) holds
f = g from SEQ_1:sch_4();
hence for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds
b1 . d = max+ (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds
b2 . d = max+ (F . d) ) holds
b1 = b2 ; ::_thesis: verum
end;
func max- F -> PartFunc of D,REAL means :Def11: :: RFUNCT_3:def 11
( dom it = dom F & ( for d being Element of D st d in dom it holds
it . d = max- (F . d) ) );
existence
ex b1 being PartFunc of D,REAL st
( dom b1 = dom F & ( for d being Element of D st d in dom b1 holds
b1 . d = max- (F . d) ) )
proof
deffunc H1( set ) -> Real = max- (F . $1);
defpred S1[ set ] means $1 in dom F;
consider f being PartFunc of D,REAL such that
A3: for d being Element of D holds
( d in dom f iff S1[d] ) and
A4: for d being Element of D st d in dom f holds
f . d = H1(d) from SEQ_1:sch_3();
take f ; ::_thesis: ( dom f = dom F & ( for d being Element of D st d in dom f holds
f . d = max- (F . d) ) )
thus dom f = dom F ::_thesis: for d being Element of D st d in dom f holds
f . d = max- (F . d)
proof
thus dom f c= dom F :: according to XBOOLE_0:def_10 ::_thesis: dom F c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in dom F )
assume x in dom f ; ::_thesis: x in dom F
hence x in dom F by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom F or x in dom f )
assume x in dom F ; ::_thesis: x in dom f
hence x in dom f by A3; ::_thesis: verum
end;
thus for d being Element of D st d in dom f holds
f . d = max- (F . d) by A4; ::_thesis: verum
end;
uniqueness
for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds
b1 . d = max- (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds
b2 . d = max- (F . d) ) holds
b1 = b2
proof
deffunc H1( set ) -> Real = max- (F . $1);
for f, g being PartFunc of D,REAL st dom f = dom F & ( for c being Element of D st c in dom f holds
f . c = H1(c) ) & dom g = dom F & ( for c being Element of D st c in dom g holds
g . c = H1(c) ) holds
f = g from SEQ_1:sch_4();
hence for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds
b1 . d = max- (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds
b2 . d = max- (F . d) ) holds
b1 = b2 ; ::_thesis: verum
end;
end;
:: deftheorem Def10 defines max+ RFUNCT_3:def_10_:_
for D being non empty set
for F, b3 being PartFunc of D,REAL holds
( b3 = max+ F iff ( dom b3 = dom F & ( for d being Element of D st d in dom b3 holds
b3 . d = max+ (F . d) ) ) );
:: deftheorem Def11 defines max- RFUNCT_3:def_11_:_
for D being non empty set
for F, b3 being PartFunc of D,REAL holds
( b3 = max- F iff ( dom b3 = dom F & ( for d being Element of D st d in dom b3 holds
b3 . d = max- (F . d) ) ) );
theorem :: RFUNCT_3:34
for D being non empty set
for F being PartFunc of D,REAL holds
( F = (max+ F) - (max- F) & abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) )
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds
( F = (max+ F) - (max- F) & abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) )
let F be PartFunc of D,REAL; ::_thesis: ( F = (max+ F) - (max- F) & abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) )
A1: dom F = (dom F) /\ (dom F) ;
A2: dom (max+ F) = dom F by Def10;
A3: dom (max- F) = dom F by Def11;
dom (- (max- F)) = dom (max- F) by VALUED_1:def_5;
then A4: dom F = dom ((max+ F) + (- (max- F))) by A2, A3, A1, VALUED_1:def_1;
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_
((max+_F)_-_(max-_F))_._d_=_F_._d
let d be Element of D; ::_thesis: ( d in dom F implies ((max+ F) - (max- F)) . d = F . d )
assume A5: d in dom F ; ::_thesis: ((max+ F) - (max- F)) . d = F . d
hence ((max+ F) - (max- F)) . d = ((max+ F) . d) - ((max- F) . d) by A4, VALUED_1:13
.= (max+ (F . d)) - ((max- F) . d) by A2, A5, Def10
.= (max+ (F . d)) - (max- (F . d)) by A3, A5, Def11
.= F . d by Th1 ;
::_thesis: verum
end;
hence F = (max+ F) - (max- F) by A4, PARTFUN1:5; ::_thesis: ( abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) )
A6: dom (abs F) = dom F by VALUED_1:def_11;
then A7: dom (abs F) = dom ((max+ F) + (max- F)) by A2, A3, A1, VALUED_1:def_1;
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_(abs_F)_holds_
((max+_F)_+_(max-_F))_._d_=_(abs_F)_._d
let d be Element of D; ::_thesis: ( d in dom (abs F) implies ((max+ F) + (max- F)) . d = (abs F) . d )
assume A8: d in dom (abs F) ; ::_thesis: ((max+ F) + (max- F)) . d = (abs F) . d
hence ((max+ F) + (max- F)) . d = ((max+ F) . d) + ((max- F) . d) by A7, VALUED_1:def_1
.= (max+ (F . d)) + ((max- F) . d) by A2, A6, A8, Def10
.= (max+ (F . d)) + (max- (F . d)) by A3, A6, A8, Def11
.= abs (F . d) by Th2
.= (abs F) . d by VALUED_1:18 ;
::_thesis: verum
end;
hence abs F = (max+ F) + (max- F) by A7, PARTFUN1:5; ::_thesis: 2 (#) (max+ F) = F + (abs F)
A9: dom (2 (#) (max+ F)) = dom (max+ F) by VALUED_1:def_5;
then A10: dom (2 (#) (max+ F)) = dom (F + (abs F)) by A2, A6, A1, VALUED_1:def_1;
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_
(2_(#)_(max+_F))_._d_=_(F_+_(abs_F))_._d
let d be Element of D; ::_thesis: ( d in dom F implies (2 (#) (max+ F)) . d = (F + (abs F)) . d )
assume A11: d in dom F ; ::_thesis: (2 (#) (max+ F)) . d = (F + (abs F)) . d
hence (2 (#) (max+ F)) . d = 2 * ((max+ F) . d) by A2, A9, VALUED_1:def_5
.= 2 * (max+ (F . d)) by A2, A11, Def10
.= (F . d) + (abs (F . d)) by Th3
.= (F . d) + ((abs F) . d) by VALUED_1:18
.= (F + (abs F)) . d by A2, A9, A10, A11, VALUED_1:def_1 ;
::_thesis: verum
end;
hence 2 (#) (max+ F) = F + (abs F) by A2, A9, A10, PARTFUN1:5; ::_thesis: verum
end;
theorem Th35: :: RFUNCT_3:35
for D being non empty set
for F being PartFunc of D,REAL
for r being Real st 0 < r holds
F " {r} = (max+ F) " {r}
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r being Real st 0 < r holds
F " {r} = (max+ F) " {r}
let F be PartFunc of D,REAL; ::_thesis: for r being Real st 0 < r holds
F " {r} = (max+ F) " {r}
let r be Real; ::_thesis: ( 0 < r implies F " {r} = (max+ F) " {r} )
A1: dom (max+ F) = dom F by Def10;
assume A2: 0 < r ; ::_thesis: F " {r} = (max+ F) " {r}
thus F " {r} c= (max+ F) " {r} :: according to XBOOLE_0:def_10 ::_thesis: (max+ F) " {r} c= F " {r}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {r} or x in (max+ F) " {r} )
assume A3: x in F " {r} ; ::_thesis: x in (max+ F) " {r}
then reconsider d = x as Element of D ;
F . d in {r} by A3, FUNCT_1:def_7;
then A4: F . d = r by TARSKI:def_1;
A5: d in dom F by A3, FUNCT_1:def_7;
then (max+ F) . d = max+ (F . d) by A1, Def10
.= r by A2, A4, XXREAL_0:def_10 ;
then (max+ F) . d in {r} by TARSKI:def_1;
hence x in (max+ F) " {r} by A1, A5, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max+ F) " {r} or x in F " {r} )
assume A6: x in (max+ F) " {r} ; ::_thesis: x in F " {r}
then reconsider d = x as Element of D ;
(max+ F) . d in {r} by A6, FUNCT_1:def_7;
then A7: (max+ F) . d = r by TARSKI:def_1;
A8: d in dom F by A1, A6, FUNCT_1:def_7;
then (max+ F) . d = max+ (F . d) by A1, Def10
.= max ((F . d),0) ;
then F . d = r by A2, A7, XXREAL_0:16;
then F . d in {r} by TARSKI:def_1;
hence x in F " {r} by A8, FUNCT_1:def_7; ::_thesis: verum
end;
theorem Th36: :: RFUNCT_3:36
for D being non empty set
for F being PartFunc of D,REAL holds F " (left_closed_halfline 0) = (max+ F) " {0}
proof
set li = left_closed_halfline 0;
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds F " (left_closed_halfline 0) = (max+ F) " {0}
let F be PartFunc of D,REAL; ::_thesis: F " (left_closed_halfline 0) = (max+ F) " {0}
A1: dom (max+ F) = dom F by Def10;
A2: left_closed_halfline 0 = { s where s is Real : s <= 0 } by XXREAL_1:231;
thus F " (left_closed_halfline 0) c= (max+ F) " {0} :: according to XBOOLE_0:def_10 ::_thesis: (max+ F) " {0} c= F " (left_closed_halfline 0)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " (left_closed_halfline 0) or x in (max+ F) " {0} )
assume A3: x in F " (left_closed_halfline 0) ; ::_thesis: x in (max+ F) " {0}
then reconsider d = x as Element of D ;
F . d in left_closed_halfline 0 by A3, FUNCT_1:def_7;
then ex s being Real st
( s = F . d & s <= 0 ) by A2;
then A4: max ((F . d),0) = 0 by XXREAL_0:def_10;
A5: d in dom F by A3, FUNCT_1:def_7;
then (max+ F) . d = max+ (F . d) by A1, Def10
.= max ((F . d),0) ;
then (max+ F) . d in {0} by A4, TARSKI:def_1;
hence x in (max+ F) " {0} by A1, A5, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max+ F) " {0} or x in F " (left_closed_halfline 0) )
assume A6: x in (max+ F) " {0} ; ::_thesis: x in F " (left_closed_halfline 0)
then reconsider d = x as Element of D ;
(max+ F) . d in {0} by A6, FUNCT_1:def_7;
then A7: (max+ F) . d = 0 by TARSKI:def_1;
A8: d in dom F by A1, A6, FUNCT_1:def_7;
then (max+ F) . d = max+ (F . d) by A1, Def10
.= max ((F . d),0) ;
then F . d <= 0 by A7, XXREAL_0:def_10;
then F . d in left_closed_halfline 0 by A2;
hence x in F " (left_closed_halfline 0) by A8, FUNCT_1:def_7; ::_thesis: verum
end;
theorem Th37: :: RFUNCT_3:37
for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D holds 0 <= (max+ F) . d
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for d being Element of D holds 0 <= (max+ F) . d
let F be PartFunc of D,REAL; ::_thesis: for d being Element of D holds 0 <= (max+ F) . d
let d be Element of D; ::_thesis: 0 <= (max+ F) . d
A1: dom F = dom (max+ F) by Def10;
percases ( d in dom F or not d in dom F ) ;
suppose d in dom F ; ::_thesis: 0 <= (max+ F) . d
then (max+ F) . d = max+ (F . d) by A1, Def10
.= max ((F . d),0) ;
hence 0 <= (max+ F) . d by XXREAL_0:25; ::_thesis: verum
end;
suppose not d in dom F ; ::_thesis: 0 <= (max+ F) . d
hence 0 <= (max+ F) . d by A1, FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
theorem Th38: :: RFUNCT_3:38
for D being non empty set
for F being PartFunc of D,REAL
for r being Real st 0 < r holds
F " {(- r)} = (max- F) " {r}
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r being Real st 0 < r holds
F " {(- r)} = (max- F) " {r}
let F be PartFunc of D,REAL; ::_thesis: for r being Real st 0 < r holds
F " {(- r)} = (max- F) " {r}
let r be Real; ::_thesis: ( 0 < r implies F " {(- r)} = (max- F) " {r} )
A1: dom (max- F) = dom F by Def11;
assume A2: 0 < r ; ::_thesis: F " {(- r)} = (max- F) " {r}
thus F " {(- r)} c= (max- F) " {r} :: according to XBOOLE_0:def_10 ::_thesis: (max- F) " {r} c= F " {(- r)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(- r)} or x in (max- F) " {r} )
assume A3: x in F " {(- r)} ; ::_thesis: x in (max- F) " {r}
then reconsider d = x as Element of D ;
F . d in {(- r)} by A3, FUNCT_1:def_7;
then A4: F . d = - r by TARSKI:def_1;
A5: d in dom F by A3, FUNCT_1:def_7;
then (max- F) . d = max- (F . d) by A1, Def11
.= r by A2, A4, XXREAL_0:def_10 ;
then (max- F) . d in {r} by TARSKI:def_1;
hence x in (max- F) " {r} by A1, A5, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max- F) " {r} or x in F " {(- r)} )
assume A6: x in (max- F) " {r} ; ::_thesis: x in F " {(- r)}
then reconsider d = x as Element of D ;
(max- F) . d in {r} by A6, FUNCT_1:def_7;
then A7: (max- F) . d = r by TARSKI:def_1;
A8: d in dom F by A1, A6, FUNCT_1:def_7;
then (max- F) . d = max- (F . d) by A1, Def11
.= max ((- (F . d)),0) ;
then - (F . d) = r by A2, A7, XXREAL_0:16;
then F . d in {(- r)} by TARSKI:def_1;
hence x in F " {(- r)} by A8, FUNCT_1:def_7; ::_thesis: verum
end;
theorem Th39: :: RFUNCT_3:39
for D being non empty set
for F being PartFunc of D,REAL holds F " (right_closed_halfline 0) = (max- F) " {0}
proof
set li = right_closed_halfline 0;
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds F " (right_closed_halfline 0) = (max- F) " {0}
let F be PartFunc of D,REAL; ::_thesis: F " (right_closed_halfline 0) = (max- F) " {0}
A1: dom (max- F) = dom F by Def11;
A2: right_closed_halfline 0 = { s where s is Real : 0 <= s } by XXREAL_1:232;
thus F " (right_closed_halfline 0) c= (max- F) " {0} :: according to XBOOLE_0:def_10 ::_thesis: (max- F) " {0} c= F " (right_closed_halfline 0)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " (right_closed_halfline 0) or x in (max- F) " {0} )
assume A3: x in F " (right_closed_halfline 0) ; ::_thesis: x in (max- F) " {0}
then reconsider d = x as Element of D ;
F . d in right_closed_halfline 0 by A3, FUNCT_1:def_7;
then ex s being Real st
( s = F . d & 0 <= s ) by A2;
then A4: max ((- (F . d)),0) = 0 by XXREAL_0:def_10;
A5: d in dom F by A3, FUNCT_1:def_7;
then (max- F) . d = max- (F . d) by A1, Def11
.= max ((- (F . d)),0) ;
then (max- F) . d in {0} by A4, TARSKI:def_1;
hence x in (max- F) " {0} by A1, A5, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max- F) " {0} or x in F " (right_closed_halfline 0) )
assume A6: x in (max- F) " {0} ; ::_thesis: x in F " (right_closed_halfline 0)
then reconsider d = x as Element of D ;
(max- F) . d in {0} by A6, FUNCT_1:def_7;
then A7: (max- F) . d = 0 by TARSKI:def_1;
A8: d in dom F by A1, A6, FUNCT_1:def_7;
then (max- F) . d = max- (F . d) by A1, Def11
.= max ((- (F . d)),0) ;
then - (F . d) <= - 0 by A7, XXREAL_0:def_10;
then 0 <= F . d by XREAL_1:24;
then F . d in right_closed_halfline 0 by A2;
hence x in F " (right_closed_halfline 0) by A8, FUNCT_1:def_7; ::_thesis: verum
end;
theorem Th40: :: RFUNCT_3:40
for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D holds 0 <= (max- F) . d
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for d being Element of D holds 0 <= (max- F) . d
let F be PartFunc of D,REAL; ::_thesis: for d being Element of D holds 0 <= (max- F) . d
let d be Element of D; ::_thesis: 0 <= (max- F) . d
A1: dom F = dom (max- F) by Def11;
percases ( d in dom F or not d in dom F ) ;
suppose d in dom F ; ::_thesis: 0 <= (max- F) . d
then (max- F) . d = max- (F . d) by A1, Def11
.= max ((- (F . d)),0) ;
hence 0 <= (max- F) . d by XXREAL_0:25; ::_thesis: verum
end;
suppose not d in dom F ; ::_thesis: 0 <= (max- F) . d
hence 0 <= (max- F) . d by A1, FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
theorem :: RFUNCT_3:41
for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max+ F, max+ G are_fiberwise_equipotent
proof
set li = left_closed_halfline 0;
let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max+ F, max+ G are_fiberwise_equipotent
let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max+ F, max+ G are_fiberwise_equipotent
let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent implies max+ F, max+ G are_fiberwise_equipotent )
assume A1: F,G are_fiberwise_equipotent ; ::_thesis: max+ F, max+ G are_fiberwise_equipotent
A2: now__::_thesis:_for_r_being_Real_holds_card_(Coim_((max+_F),r))_=_card_(Coim_((max+_G),r))
let r be Real; ::_thesis: card (Coim ((max+ F),r)) = card (Coim ((max+ G),r))
now__::_thesis:_(_(_0_<_r_&_card_(Coim_((max+_F),r))_=_card_(Coim_((max+_G),r))_)_or_(_r_=_0_&_card_((max+_F)_"_{r})_=_card_((max+_G)_"_{r})_)_or_(_r_<_0_&_card_((max+_F)_"_{r})_=_card_((max+_G)_"_{r})_)_)
percases ( 0 < r or r = 0 or r < 0 ) ;
case 0 < r ; ::_thesis: card (Coim ((max+ F),r)) = card (Coim ((max+ G),r))
then ( Coim (F,r) = Coim ((max+ F),r) & Coim (G,r) = Coim ((max+ G),r) ) by Th35;
hence card (Coim ((max+ F),r)) = card (Coim ((max+ G),r)) by A1, CLASSES1:def_9; ::_thesis: verum
end;
caseA3: r = 0 ; ::_thesis: card ((max+ F) " {r}) = card ((max+ G) " {r})
( F " (left_closed_halfline 0) = (max+ F) " {0} & G " (left_closed_halfline 0) = (max+ G) " {0} ) by Th36;
hence card ((max+ F) " {r}) = card ((max+ G) " {r}) by A1, A3, CLASSES1:78; ::_thesis: verum
end;
caseA4: r < 0 ; ::_thesis: card ((max+ F) " {r}) = card ((max+ G) " {r})
now__::_thesis:_not_r_in_rng_(max+_F)
assume r in rng (max+ F) ; ::_thesis: contradiction
then ex d being Element of D st
( d in dom (max+ F) & (max+ F) . d = r ) by PARTFUN1:3;
hence contradiction by A4, Th37; ::_thesis: verum
end;
then A5: (max+ F) " {r} = {} by Lm2;
now__::_thesis:_not_r_in_rng_(max+_G)
assume r in rng (max+ G) ; ::_thesis: contradiction
then ex c being Element of C st
( c in dom (max+ G) & (max+ G) . c = r ) by PARTFUN1:3;
hence contradiction by A4, Th37; ::_thesis: verum
end;
hence card ((max+ F) " {r}) = card ((max+ G) " {r}) by A5, Lm2; ::_thesis: verum
end;
end;
end;
hence card (Coim ((max+ F),r)) = card (Coim ((max+ G),r)) ; ::_thesis: verum
end;
( rng (max+ F) c= REAL & rng (max+ G) c= REAL ) ;
hence max+ F, max+ G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum
end;
theorem :: RFUNCT_3:42
for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max- F, max- G are_fiberwise_equipotent
proof
set li = right_closed_halfline 0;
let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max- F, max- G are_fiberwise_equipotent
let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max- F, max- G are_fiberwise_equipotent
let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent implies max- F, max- G are_fiberwise_equipotent )
assume A1: F,G are_fiberwise_equipotent ; ::_thesis: max- F, max- G are_fiberwise_equipotent
A2: now__::_thesis:_for_r_being_Real_holds_card_(Coim_((max-_F),r))_=_card_(Coim_((max-_G),r))
let r be Real; ::_thesis: card (Coim ((max- F),r)) = card (Coim ((max- G),r))
now__::_thesis:_(_(_0_<_r_&_card_(Coim_((max-_F),r))_=_card_(Coim_((max-_G),r))_)_or_(_r_=_0_&_card_((max-_F)_"_{r})_=_card_((max-_G)_"_{r})_)_or_(_r_<_0_&_card_((max-_F)_"_{r})_=_card_((max-_G)_"_{r})_)_)
percases ( 0 < r or r = 0 or r < 0 ) ;
case 0 < r ; ::_thesis: card (Coim ((max- F),r)) = card (Coim ((max- G),r))
then ( Coim (F,(- r)) = (max- F) " {r} & Coim (G,(- r)) = (max- G) " {r} ) by Th38;
hence card (Coim ((max- F),r)) = card (Coim ((max- G),r)) by A1, CLASSES1:def_9; ::_thesis: verum
end;
caseA3: r = 0 ; ::_thesis: card ((max- F) " {r}) = card ((max- G) " {r})
( F " (right_closed_halfline 0) = (max- F) " {0} & G " (right_closed_halfline 0) = (max- G) " {0} ) by Th39;
hence card ((max- F) " {r}) = card ((max- G) " {r}) by A1, A3, CLASSES1:78; ::_thesis: verum
end;
caseA4: r < 0 ; ::_thesis: card ((max- F) " {r}) = card ((max- G) " {r})
now__::_thesis:_not_r_in_rng_(max-_F)
assume r in rng (max- F) ; ::_thesis: contradiction
then ex d being Element of D st
( d in dom (max- F) & (max- F) . d = r ) by PARTFUN1:3;
hence contradiction by A4, Th40; ::_thesis: verum
end;
then A5: (max- F) " {r} = {} by Lm2;
now__::_thesis:_not_r_in_rng_(max-_G)
assume r in rng (max- G) ; ::_thesis: contradiction
then ex c being Element of C st
( c in dom (max- G) & (max- G) . c = r ) by PARTFUN1:3;
hence contradiction by A4, Th40; ::_thesis: verum
end;
hence card ((max- F) " {r}) = card ((max- G) " {r}) by A5, Lm2; ::_thesis: verum
end;
end;
end;
hence card (Coim ((max- F),r)) = card (Coim ((max- G),r)) ; ::_thesis: verum
end;
( rng (max- F) c= REAL & rng (max- G) c= REAL ) ;
hence max- F, max- G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum
end;
registration
let D be non empty set ;
let F be finite PartFunc of D,REAL;
cluster max+ F -> finite ;
coherence
max+ F is finite
proof
dom F is finite ;
then dom (max+ F) is finite by Def10;
hence max+ F is finite by FINSET_1:10; ::_thesis: verum
end;
cluster max- F -> finite ;
coherence
max- F is finite
proof
dom F is finite ;
then dom (max- F) is finite by Def11;
hence max- F is finite by FINSET_1:10; ::_thesis: verum
end;
end;
theorem :: RFUNCT_3:43
for D, C being non empty set
for F being finite PartFunc of D,REAL
for G being finite PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent holds
F,G are_fiberwise_equipotent
proof
let D, C be non empty set ; ::_thesis: for F being finite PartFunc of D,REAL
for G being finite PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent holds
F,G are_fiberwise_equipotent
let F be finite PartFunc of D,REAL; ::_thesis: for G being finite PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent holds
F,G are_fiberwise_equipotent
let G be finite PartFunc of C,REAL; ::_thesis: ( max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent implies F,G are_fiberwise_equipotent )
assume that
A1: max+ F, max+ G are_fiberwise_equipotent and
A2: max- F, max- G are_fiberwise_equipotent ; ::_thesis: F,G are_fiberwise_equipotent
set lh = left_closed_halfline 0;
set rh = right_closed_halfline 0;
set fp0 = (max+ F) " {0};
set fm0 = (max- F) " {0};
set gp0 = (max+ G) " {0};
set gm0 = (max- G) " {0};
A3: (left_closed_halfline 0) /\ (right_closed_halfline 0) = [.0,0.] by XXREAL_1:272
.= {0} by XXREAL_1:17 ;
F " (rng F) c= F " REAL by RELAT_1:143;
then A4: ( F " REAL c= dom F & dom F c= F " REAL ) by RELAT_1:132, RELAT_1:134;
A5: ( F " (left_closed_halfline 0) = (max+ F) " {0} & F " (right_closed_halfline 0) = (max- F) " {0} ) by Th36, Th39;
G " (rng G) c= G " REAL by RELAT_1:143;
then A6: ( G " REAL c= dom G & dom G c= G " REAL ) by RELAT_1:132, RELAT_1:134;
A7: ( G " (left_closed_halfline 0) = (max+ G) " {0} & G " (right_closed_halfline 0) = (max- G) " {0} ) by Th36, Th39;
reconsider fp0 = (max+ F) " {0}, fm0 = (max- F) " {0}, gp0 = (max+ G) " {0}, gm0 = (max- G) " {0} as finite set ;
A8: (left_closed_halfline 0) \/ (right_closed_halfline 0) = REAL \ ].0,0.[ by XXREAL_1:398
.= REAL \ {} by XXREAL_1:28
.= REAL ;
then fp0 \/ fm0 = F " REAL by A5, RELAT_1:140;
then A9: fp0 \/ fm0 = dom F by A4, XBOOLE_0:def_10;
gp0 \/ gm0 = G " ((left_closed_halfline 0) \/ (right_closed_halfline 0)) by A7, RELAT_1:140;
then A10: gp0 \/ gm0 = dom G by A8, A6, XBOOLE_0:def_10;
card (fp0 \/ fm0) = ((card fp0) + (card fm0)) - (card (fp0 /\ fm0)) by CARD_2:45;
then A11: card (fp0 /\ fm0) = ((card fp0) + (card fm0)) - (card (fp0 \/ fm0)) ;
card (gp0 \/ gm0) = ((card gp0) + (card gm0)) - (card (gp0 /\ gm0)) by CARD_2:45;
then A12: card (gp0 /\ gm0) = ((card gp0) + (card gm0)) - (card (gp0 \/ gm0)) ;
A13: ( dom F = dom (max+ F) & dom G = dom (max+ G) ) by Def10;
A14: now__::_thesis:_for_r_being_Real_holds_card_(Coim_(F,r))_=_card_(Coim_(G,r))
let r be Real; ::_thesis: card (Coim (F,r)) = card (Coim (G,r))
A15: ( card fp0 = card gp0 & card fm0 = card gm0 ) by A1, A2, CLASSES1:78;
now__::_thesis:_(_(_0_<_r_&_card_(Coim_(F,r))_=_card_(Coim_(G,r))_)_or_(_0_=_r_&_card_(F_"_{r})_=_card_(G_"_{r})_)_or_(_r_<_0_&_card_(Coim_(F,r))_=_card_(Coim_(G,r))_)_)
percases ( 0 < r or 0 = r or r < 0 ) ;
case 0 < r ; ::_thesis: card (Coim (F,r)) = card (Coim (G,r))
then ( Coim (F,r) = Coim ((max+ F),r) & Coim (G,r) = Coim ((max+ G),r) ) by Th35;
hence card (Coim (F,r)) = card (Coim (G,r)) by A1, CLASSES1:def_9; ::_thesis: verum
end;
case 0 = r ; ::_thesis: card (F " {r}) = card (G " {r})
then ( F " {r} = (F " (left_closed_halfline 0)) /\ (F " (right_closed_halfline 0)) & G " {r} = (G " (left_closed_halfline 0)) /\ (G " (right_closed_halfline 0)) ) by A3, FUNCT_1:68;
hence card (F " {r}) = card (G " {r}) by A1, A13, A5, A7, A11, A12, A9, A10, A15, CLASSES1:81; ::_thesis: verum
end;
caseA16: r < 0 ; ::_thesis: card (Coim (F,r)) = card (Coim (G,r))
A17: - (- r) = r ;
0 < - r by A16, XREAL_1:58;
then ( Coim (F,r) = Coim ((max- F),(- r)) & Coim (G,r) = Coim ((max- G),(- r)) ) by A17, Th38;
hence card (Coim (F,r)) = card (Coim (G,r)) by A2, CLASSES1:def_9; ::_thesis: verum
end;
end;
end;
hence card (Coim (F,r)) = card (Coim (G,r)) ; ::_thesis: verum
end;
( rng F c= REAL & rng G c= REAL ) ;
hence F,G are_fiberwise_equipotent by A14, CLASSES1:79; ::_thesis: verum
end;
theorem Th44: :: RFUNCT_3:44
for D being non empty set
for F being PartFunc of D,REAL
for X being set holds (max+ F) | X = max+ (F | X)
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set holds (max+ F) | X = max+ (F | X)
let F be PartFunc of D,REAL; ::_thesis: for X being set holds (max+ F) | X = max+ (F | X)
let X be set ; ::_thesis: (max+ F) | X = max+ (F | X)
A1: dom ((max+ F) | X) = (dom (max+ F)) /\ X by RELAT_1:61;
A2: (dom (max+ F)) /\ X = (dom F) /\ X by Def10
.= dom (F | X) by RELAT_1:61 ;
A3: dom (F | X) = dom (max+ (F | X)) by Def10;
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_((max+_F)_|_X)_holds_
((max+_F)_|_X)_._d_=_(max+_(F_|_X))_._d
let d be Element of D; ::_thesis: ( d in dom ((max+ F) | X) implies ((max+ F) | X) . d = (max+ (F | X)) . d )
assume A4: d in dom ((max+ F) | X) ; ::_thesis: ((max+ F) | X) . d = (max+ (F | X)) . d
then A5: d in dom (max+ F) by A1, XBOOLE_0:def_4;
thus ((max+ F) | X) . d = (max+ F) . d by A4, FUNCT_1:47
.= max+ (F . d) by A5, Def10
.= max+ ((F | X) . d) by A1, A2, A4, FUNCT_1:47
.= (max+ (F | X)) . d by A1, A2, A3, A4, Def10 ; ::_thesis: verum
end;
hence (max+ F) | X = max+ (F | X) by A2, A3, PARTFUN1:5, RELAT_1:61; ::_thesis: verum
end;
theorem :: RFUNCT_3:45
for D being non empty set
for F being PartFunc of D,REAL
for X being set holds (max- F) | X = max- (F | X)
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set holds (max- F) | X = max- (F | X)
let F be PartFunc of D,REAL; ::_thesis: for X being set holds (max- F) | X = max- (F | X)
let X be set ; ::_thesis: (max- F) | X = max- (F | X)
A1: dom ((max- F) | X) = (dom (max- F)) /\ X by RELAT_1:61;
A2: (dom (max- F)) /\ X = (dom F) /\ X by Def11
.= dom (F | X) by RELAT_1:61 ;
A3: dom (F | X) = dom (max- (F | X)) by Def11;
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_((max-_F)_|_X)_holds_
((max-_F)_|_X)_._d_=_(max-_(F_|_X))_._d
let d be Element of D; ::_thesis: ( d in dom ((max- F) | X) implies ((max- F) | X) . d = (max- (F | X)) . d )
assume A4: d in dom ((max- F) | X) ; ::_thesis: ((max- F) | X) . d = (max- (F | X)) . d
then A5: d in dom (max- F) by A1, XBOOLE_0:def_4;
thus ((max- F) | X) . d = (max- F) . d by A4, FUNCT_1:47
.= max- (F . d) by A5, Def11
.= max- ((F | X) . d) by A1, A2, A4, FUNCT_1:47
.= (max- (F | X)) . d by A1, A2, A3, A4, Def11 ; ::_thesis: verum
end;
hence (max- F) | X = max- (F | X) by A2, A3, PARTFUN1:5, RELAT_1:61; ::_thesis: verum
end;
theorem Th46: :: RFUNCT_3:46
for D being non empty set
for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds
F . d >= 0 ) holds
max+ F = F
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds
F . d >= 0 ) holds
max+ F = F
let F be PartFunc of D,REAL; ::_thesis: ( ( for d being Element of D st d in dom F holds
F . d >= 0 ) implies max+ F = F )
A1: dom (max+ F) = dom F by Def10;
assume A2: for d being Element of D st d in dom F holds
F . d >= 0 ; ::_thesis: max+ F = F
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_
(max+_F)_._d_=_F_._d
let d be Element of D; ::_thesis: ( d in dom F implies (max+ F) . d = F . d )
assume A3: d in dom F ; ::_thesis: (max+ F) . d = F . d
then A4: F . d >= 0 by A2;
thus (max+ F) . d = max+ (F . d) by A1, A3, Def10
.= F . d by A4, XXREAL_0:def_10 ; ::_thesis: verum
end;
hence max+ F = F by A1, PARTFUN1:5; ::_thesis: verum
end;
theorem :: RFUNCT_3:47
for D being non empty set
for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds
F . d <= 0 ) holds
max- F = - F
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds
F . d <= 0 ) holds
max- F = - F
let F be PartFunc of D,REAL; ::_thesis: ( ( for d being Element of D st d in dom F holds
F . d <= 0 ) implies max- F = - F )
A1: dom (max- F) = dom F by Def11;
assume A2: for d being Element of D st d in dom F holds
F . d <= 0 ; ::_thesis: max- F = - F
A3: now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_
(max-_F)_._d_=_(-_F)_._d
let d be Element of D; ::_thesis: ( d in dom F implies (max- F) . d = (- F) . d )
assume A4: d in dom F ; ::_thesis: (max- F) . d = (- F) . d
then A5: F . d <= 0 by A2;
thus (max- F) . d = max- (F . d) by A1, A4, Def11
.= - (F . d) by A5, XXREAL_0:def_10
.= (- F) . d by VALUED_1:8 ; ::_thesis: verum
end;
dom F = dom (- F) by VALUED_1:8;
hence max- F = - F by A1, A3, PARTFUN1:5; ::_thesis: verum
end;
theorem Th48: :: RFUNCT_3:48
for D being non empty set
for F being PartFunc of D,REAL holds F - 0 = F
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds F - 0 = F
let F be PartFunc of D,REAL; ::_thesis: F - 0 = F
A1: now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_
(F_-_0)_._d_=_F_._d
let d be Element of D; ::_thesis: ( d in dom F implies (F - 0) . d = F . d )
assume d in dom F ; ::_thesis: (F - 0) . d = F . d
hence (F - 0) . d = (F . d) - 0 by VALUED_1:3
.= F . d ;
::_thesis: verum
end;
dom (F - 0) = dom F by VALUED_1:3;
hence F - 0 = F by A1, PARTFUN1:5; ::_thesis: verum
end;
theorem :: RFUNCT_3:49
for D being non empty set
for F being PartFunc of D,REAL
for r being Real
for X being set holds (F | X) - r = (F - r) | X
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r being Real
for X being set holds (F | X) - r = (F - r) | X
let F be PartFunc of D,REAL; ::_thesis: for r being Real
for X being set holds (F | X) - r = (F - r) | X
let r be Real; ::_thesis: for X being set holds (F | X) - r = (F - r) | X
let X be set ; ::_thesis: (F | X) - r = (F - r) | X
A1: dom ((F | X) - r) = dom (F | X) by VALUED_1:3;
A2: dom (F | X) = (dom F) /\ X by RELAT_1:61;
A3: (dom F) /\ X = (dom (F - r)) /\ X by VALUED_1:3
.= dom ((F - r) | X) by RELAT_1:61 ;
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_((F_|_X)_-_r)_holds_
((F_|_X)_-_r)_._d_=_((F_-_r)_|_X)_._d
let d be Element of D; ::_thesis: ( d in dom ((F | X) - r) implies ((F | X) - r) . d = ((F - r) | X) . d )
assume A4: d in dom ((F | X) - r) ; ::_thesis: ((F | X) - r) . d = ((F - r) | X) . d
then A5: d in dom F by A1, A2, XBOOLE_0:def_4;
thus ((F | X) - r) . d = ((F | X) . d) - r by A1, A4, VALUED_1:3
.= (F . d) - r by A1, A4, FUNCT_1:47
.= (F - r) . d by A5, VALUED_1:3
.= ((F - r) | X) . d by A1, A2, A3, A4, FUNCT_1:47 ; ::_thesis: verum
end;
hence (F | X) - r = (F - r) | X by A2, A3, PARTFUN1:5, VALUED_1:3; ::_thesis: verum
end;
theorem Th50: :: RFUNCT_3:50
for D being non empty set
for F being PartFunc of D,REAL
for r, s being Real holds F " {(s + r)} = (F - r) " {s}
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r, s being Real holds F " {(s + r)} = (F - r) " {s}
let F be PartFunc of D,REAL; ::_thesis: for r, s being Real holds F " {(s + r)} = (F - r) " {s}
let r, s be Real; ::_thesis: F " {(s + r)} = (F - r) " {s}
thus F " {(s + r)} c= (F - r) " {s} :: according to XBOOLE_0:def_10 ::_thesis: (F - r) " {s} c= F " {(s + r)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(s + r)} or x in (F - r) " {s} )
assume A1: x in F " {(s + r)} ; ::_thesis: x in (F - r) " {s}
then reconsider d = x as Element of D ;
A2: d in dom F by A1, FUNCT_1:def_7;
F . d in {(s + r)} by A1, FUNCT_1:def_7;
then F . d = s + r by TARSKI:def_1;
then (F . d) - r = s ;
then (F - r) . d = s by A2, VALUED_1:3;
then A3: (F - r) . d in {s} by TARSKI:def_1;
d in dom (F - r) by A2, VALUED_1:3;
hence x in (F - r) " {s} by A3, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (F - r) " {s} or x in F " {(s + r)} )
assume A4: x in (F - r) " {s} ; ::_thesis: x in F " {(s + r)}
then reconsider d = x as Element of D ;
d in dom (F - r) by A4, FUNCT_1:def_7;
then A5: d in dom F by VALUED_1:3;
(F - r) . d in {s} by A4, FUNCT_1:def_7;
then (F - r) . d = s by TARSKI:def_1;
then (F . d) - r = s by A5, VALUED_1:3;
then F . d in {(s + r)} by TARSKI:def_1;
hence x in F " {(s + r)} by A5, FUNCT_1:def_7; ::_thesis: verum
end;
theorem :: RFUNCT_3:51
for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL
for r being Real holds
( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent )
proof
let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for G being PartFunc of C,REAL
for r being Real holds
( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent )
let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL
for r being Real holds
( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent )
let G be PartFunc of C,REAL; ::_thesis: for r being Real holds
( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent )
let r be Real; ::_thesis: ( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent )
A1: ( rng (F - r) c= REAL & rng (G - r) c= REAL ) ;
thus ( F,G are_fiberwise_equipotent implies F - r,G - r are_fiberwise_equipotent ) ::_thesis: ( F - r,G - r are_fiberwise_equipotent implies F,G are_fiberwise_equipotent )
proof
assume A2: F,G are_fiberwise_equipotent ; ::_thesis: F - r,G - r are_fiberwise_equipotent
now__::_thesis:_for_s_being_Real_holds_card_(Coim_((F_-_r),s))_=_card_(Coim_((G_-_r),s))
let s be Real; ::_thesis: card (Coim ((F - r),s)) = card (Coim ((G - r),s))
thus card (Coim ((F - r),s)) = card (Coim (F,(s + r))) by Th50
.= card (Coim (G,(s + r))) by A2, CLASSES1:def_9
.= card (Coim ((G - r),s)) by Th50 ; ::_thesis: verum
end;
hence F - r,G - r are_fiberwise_equipotent by A1, CLASSES1:79; ::_thesis: verum
end;
assume A3: F - r,G - r are_fiberwise_equipotent ; ::_thesis: F,G are_fiberwise_equipotent
A4: now__::_thesis:_for_s_being_Real_holds_card_(Coim_(F,s))_=_card_(Coim_(G,s))
let s be Real; ::_thesis: card (Coim (F,s)) = card (Coim (G,s))
A5: s = (s - r) + r ;
hence card (Coim (F,s)) = card (Coim ((F - r),(s - r))) by Th50
.= card (Coim ((G - r),(s - r))) by A3, CLASSES1:def_9
.= card (Coim (G,s)) by A5, Th50 ;
::_thesis: verum
end;
( rng F c= REAL & rng G c= REAL ) ;
hence F,G are_fiberwise_equipotent by A4, CLASSES1:79; ::_thesis: verum
end;
definition
let F be PartFunc of REAL,REAL;
let X be set ;
predF is_convex_on X means :Def12: :: RFUNCT_3:def 12
( X c= dom F & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) );
end;
:: deftheorem Def12 defines is_convex_on RFUNCT_3:def_12_:_
for F being PartFunc of REAL,REAL
for X being set holds
( F is_convex_on X iff ( X c= dom F & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) );
theorem Th52: :: RFUNCT_3:52
for a, b being Real
for F being PartFunc of REAL,REAL holds
( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in [.a,b.] & s in [.a,b.] holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) )
proof
let a, b be Real; ::_thesis: for F being PartFunc of REAL,REAL holds
( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in [.a,b.] & s in [.a,b.] holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convex_on [.a,b.] iff ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in [.a,b.] & s in [.a,b.] holds
f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) ) ) )
set ab = { r where r is Real : ( a <= r & r <= b ) } ;
A1: [.a,b.] = { r where r is Real : ( a <= r & r <= b ) } by RCOMP_1:def_1;
thus ( f is_convex_on [.a,b.] implies ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds
for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) ) ) ::_thesis: ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in [.a,b.] & s in [.a,b.] holds
f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) ) implies f is_convex_on [.a,b.] )
proof
assume A2: f is_convex_on [.a,b.] ; ::_thesis: ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds
for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) )
hence [.a,b.] c= dom f by Def12; ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) )
assume that
A3: 0 <= p and
A4: p <= 1 ; ::_thesis: for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
A5: 0 <= 1 - p by A4, XREAL_1:48;
A6: (p * b) + ((1 - p) * b) = b ;
A7: (p * a) + ((1 - p) * a) = a ;
let x, y be Real; ::_thesis: ( x in [.a,b.] & y in [.a,b.] implies f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) )
assume that
A8: x in [.a,b.] and
A9: y in [.a,b.] ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
A10: ex r2 being Real st
( r2 = y & a <= r2 & r2 <= b ) by A1, A9;
then A11: (1 - p) * y <= (1 - p) * b by A5, XREAL_1:64;
A12: ex r1 being Real st
( r1 = x & a <= r1 & r1 <= b ) by A1, A8;
then p * x <= p * b by A3, XREAL_1:64;
then A13: (p * x) + ((1 - p) * y) <= b by A11, A6, XREAL_1:7;
A14: (1 - p) * a <= (1 - p) * y by A5, A10, XREAL_1:64;
p * a <= p * x by A3, A12, XREAL_1:64;
then a <= (p * x) + ((1 - p) * y) by A14, A7, XREAL_1:7;
then (p * x) + ((1 - p) * y) in { r where r is Real : ( a <= r & r <= b ) } by A13;
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by A1, A2, A3, A4, A8, A9, Def12; ::_thesis: verum
end;
assume that
A15: [.a,b.] c= dom f and
A16: for p being Real st 0 <= p & p <= 1 holds
for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: f is_convex_on [.a,b.]
thus [.a,b.] c= dom f by A15; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in [.a,b.] & s in [.a,b.] & (p * r) + ((1 - p) * s) in [.a,b.] holds
f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in [.a,b.] & s in [.a,b.] & (p * r) + ((1 - p) * s) in [.a,b.] holds
f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) )
assume A17: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in [.a,b.] & s in [.a,b.] & (p * r) + ((1 - p) * s) in [.a,b.] holds
f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s))
let x, y be Real; ::_thesis: ( x in [.a,b.] & y in [.a,b.] & (p * x) + ((1 - p) * y) in [.a,b.] implies f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) )
assume that
A18: ( x in [.a,b.] & y in [.a,b.] ) and
(p * x) + ((1 - p) * y) in [.a,b.] ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
thus f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by A16, A17, A18; ::_thesis: verum
end;
theorem :: RFUNCT_3:53
for a, b being Real
for F being PartFunc of REAL,REAL holds
( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((F . x1) - (F . x2)) / (x1 - x2) <= ((F . x2) - (F . x3)) / (x2 - x3) ) ) )
proof
let a, b be Real; ::_thesis: for F being PartFunc of REAL,REAL holds
( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((F . x1) - (F . x2)) / (x1 - x2) <= ((F . x2) - (F . x3)) / (x2 - x3) ) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convex_on [.a,b.] iff ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) ) )
thus ( f is_convex_on [.a,b.] implies ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) ) ) ::_thesis: ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) implies f is_convex_on [.a,b.] )
proof
assume A1: f is_convex_on [.a,b.] ; ::_thesis: ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) )
hence [.a,b.] c= dom f by Def12; ::_thesis: for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3)
let x1, x2, x3 be Real; ::_thesis: ( x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 implies ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) )
assume that
A2: ( x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] ) and
A3: x1 < x2 and
A4: x2 < x3 ; ::_thesis: ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3)
A5: x2 - x3 < 0 by A4, XREAL_1:49;
set r = (x2 - x3) / (x1 - x3);
A6: ( ((x2 - x3) / (x1 - x3)) * ((f . x2) - (f . x1)) = (((x2 - x3) / (x1 - x3)) * (f . x2)) - (((x2 - x3) / (x1 - x3)) * (f . x1)) & (1 - ((x2 - x3) / (x1 - x3))) * ((f . x3) - (f . x2)) = ((1 - ((x2 - x3) / (x1 - x3))) * (f . x3)) - ((1 - ((x2 - x3) / (x1 - x3))) * (f . x2)) ) ;
A7: (x1 - x2) / (x1 - x3) = ((x1 - x3) ") * (x1 - x2) by XCMPLX_0:def_9;
x1 < x3 by A3, A4, XXREAL_0:2;
then A8: x1 - x3 < 0 by XREAL_1:49;
A9: ((x2 - x3) / (x1 - x3)) + ((x1 - x2) / (x1 - x3)) = ((x1 - x2) + (x2 - x3)) / (x1 - x3) by XCMPLX_1:62
.= 1 by A8, XCMPLX_1:60 ;
then A10: (((x2 - x3) / (x1 - x3)) * x1) + ((1 - ((x2 - x3) / (x1 - x3))) * x3) = ((x1 * (x2 - x3)) / (x1 - x3)) + (x3 * ((x1 - x2) / (x1 - x3))) by XCMPLX_1:74
.= ((x1 * (x2 - x3)) / (x1 - x3)) + ((x3 * (x1 - x2)) / (x1 - x3)) by XCMPLX_1:74
.= (((x2 * x1) - (x3 * x1)) + ((x1 - x2) * x3)) / (x1 - x3) by XCMPLX_1:62
.= (x2 * (x1 - x3)) / (x1 - x3)
.= x2 by A8, XCMPLX_1:89 ;
A11: x1 - x2 < 0 by A3, XREAL_1:49;
then (x2 - x3) / (x1 - x3) <= 1 by A8, A9, XREAL_1:29, XREAL_1:140;
then (((x2 - x3) / (x1 - x3)) * (f . x2)) + ((1 - ((x2 - x3) / (x1 - x3))) * (f . x2)) <= (((x2 - x3) / (x1 - x3)) * (f . x1)) + ((1 - ((x2 - x3) / (x1 - x3))) * (f . x3)) by A1, A2, A5, A8, A10, Def12;
then ((x2 - x3) / (x1 - x3)) * ((f . x2) - (f . x1)) <= (1 - ((x2 - x3) / (x1 - x3))) * ((f . x3) - (f . x2)) by A6, XREAL_1:21;
then - ((1 - ((x2 - x3) / (x1 - x3))) * ((f . x3) - (f . x2))) <= - (((x2 - x3) / (x1 - x3)) * ((f . x2) - (f . x1))) by XREAL_1:24;
then (1 - ((x2 - x3) / (x1 - x3))) * (- ((f . x3) - (f . x2))) <= ((x2 - x3) / (x1 - x3)) * (- ((f . x2) - (f . x1))) ;
then (((x1 - x3) ") * (x1 - x2)) * ((f . x2) - (f . x3)) <= (((x1 - x3) ") * (x2 - x3)) * ((f . x1) - (f . x2)) by A9, A7, XCMPLX_0:def_9;
then A12: (x1 - x3) * ((((x1 - x3) ") * (x2 - x3)) * ((f . x1) - (f . x2))) <= (x1 - x3) * ((((x1 - x3) ") * (x1 - x2)) * ((f . x2) - (f . x3))) by A8, XREAL_1:65;
set v = (x1 - x2) * ((f . x2) - (f . x3));
set u = (x2 - x3) * ((f . x1) - (f . x2));
A13: (x1 - x3) * ((((x1 - x3) ") * (x1 - x2)) * ((f . x2) - (f . x3))) = ((x1 - x3) * ((x1 - x3) ")) * ((x1 - x2) * ((f . x2) - (f . x3)))
.= 1 * ((x1 - x2) * ((f . x2) - (f . x3))) by A8, XCMPLX_0:def_7
.= (x1 - x2) * ((f . x2) - (f . x3)) ;
(x1 - x3) * ((((x1 - x3) ") * (x2 - x3)) * ((f . x1) - (f . x2))) = ((x1 - x3) * ((x1 - x3) ")) * ((x2 - x3) * ((f . x1) - (f . x2)))
.= 1 * ((x2 - x3) * ((f . x1) - (f . x2))) by A8, XCMPLX_0:def_7
.= (x2 - x3) * ((f . x1) - (f . x2)) ;
hence ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) by A5, A11, A12, A13, XREAL_1:103; ::_thesis: verum
end;
assume that
A14: [.a,b.] c= dom f and
A15: for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ; ::_thesis: f is_convex_on [.a,b.]
now__::_thesis:_for_p_being_Real_st_0_<=_p_&_p_<=_1_holds_
for_x,_y_being_Real_st_x_in_[.a,b.]_&_y_in_[.a,b.]_holds_
f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) )
assume that
A16: 0 <= p and
A17: p <= 1 ; ::_thesis: for x, y being Real st x in [.a,b.] & y in [.a,b.] holds
f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
A18: 0 <= 1 - p by A17, XREAL_1:48;
let x, y be Real; ::_thesis: ( x in [.a,b.] & y in [.a,b.] implies f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) )
set r = (p * x) + ((1 - p) * y);
assume that
A19: x in [.a,b.] and
A20: y in [.a,b.] ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
A21: (p * y) + ((1 - p) * y) = y ;
A22: { s where s is Real : ( a <= s & s <= b ) } = [.a,b.] by RCOMP_1:def_1;
then A23: ex t being Real st
( t = y & a <= t & t <= b ) by A20;
then A24: (1 - p) * y <= (1 - p) * b by A18, XREAL_1:64;
A25: ex t being Real st
( t = x & a <= t & t <= b ) by A22, A19;
then ( (p * b) + ((1 - p) * b) = b & p * x <= p * b ) by A16, XREAL_1:64;
then A26: (p * x) + ((1 - p) * y) <= b by A24, XREAL_1:7;
A27: (1 - p) * a <= (1 - p) * y by A18, A23, XREAL_1:64;
( (p * a) + ((1 - p) * a) = a & p * a <= p * x ) by A16, A25, XREAL_1:64;
then a <= (p * x) + ((1 - p) * y) by A27, XREAL_1:7;
then A28: (p * x) + ((1 - p) * y) in [.a,b.] by A22, A26;
A29: (p * x) + ((1 - p) * x) = x ;
now__::_thesis:_(_(_x_=_y_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_x_<>_y_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_)
percases ( x = y or x <> y ) ;
case x = y ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum
end;
caseA30: x <> y ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
now__::_thesis:_(_(_p_=_0_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_p_<>_0_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_)
percases ( p = 0 or p <> 0 ) ;
case p = 0 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum
end;
caseA31: p <> 0 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
now__::_thesis:_(_(_p_=_1_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_p_<>_1_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_)
percases ( p = 1 or p <> 1 ) ;
case p = 1 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum
end;
case p <> 1 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
then p < 1 by A17, XXREAL_0:1;
then A32: 0 < 1 - p by XREAL_1:50;
now__::_thesis:_(_(_x_<_y_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_y_<_x_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_)
percases ( x < y or y < x ) by A30, XXREAL_0:1;
caseA33: x < y ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
then (1 - p) * x < (1 - p) * y by A32, XREAL_1:68;
then A34: x < (p * x) + ((1 - p) * y) by A29, XREAL_1:8;
then A35: x - ((p * x) + ((1 - p) * y)) < 0 by XREAL_1:49;
p * x < p * y by A16, A31, A33, XREAL_1:68;
then A36: (p * x) + ((1 - p) * y) < y by A21, XREAL_1:8;
then A37: ((p * x) + ((1 - p) * y)) - y < 0 by XREAL_1:49;
A38: x - y < 0 by A33, XREAL_1:49;
((f . x) - (f . ((p * x) + ((1 - p) * y)))) / (x - ((p * x) + ((1 - p) * y))) <= ((f . ((p * x) + ((1 - p) * y))) - (f . y)) / (((p * x) + ((1 - p) * y)) - y) by A15, A19, A20, A28, A36, A34;
then ((f . x) - (f . ((p * x) + ((1 - p) * y)))) * (p * (x - y)) <= ((f . ((p * x) + ((1 - p) * y))) - (f . y)) * ((1 - p) * (x - y)) by A37, A35, XREAL_1:107;
then ((((f . x) - (f . ((p * x) + ((1 - p) * y)))) * p) * (x - y)) / (x - y) >= ((((f . ((p * x) + ((1 - p) * y))) - (f . y)) * (1 - p)) * (x - y)) / (x - y) by A38, XREAL_1:73;
then ((((f . ((p * x) + ((1 - p) * y))) - (f . y)) * (1 - p)) * (x - y)) / (x - y) <= ((f . x) - (f . ((p * x) + ((1 - p) * y)))) * p by A38, XCMPLX_1:89;
then ((f . ((p * x) + ((1 - p) * y))) * (1 - p)) - ((f . y) * (1 - p)) <= ((f . x) * p) - ((f . ((p * x) + ((1 - p) * y))) * p) by A38, XCMPLX_1:89;
then ((f . ((p * x) + ((1 - p) * y))) * p) + (((f . ((p * x) + ((1 - p) * y))) * (1 - p)) - ((f . y) * (1 - p))) <= (f . x) * p by XREAL_1:19;
then (((f . ((p * x) + ((1 - p) * y))) * p) + ((f . ((p * x) + ((1 - p) * y))) * (1 - p))) - ((f . y) * (1 - p)) <= (f . x) * p ;
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by XREAL_1:20; ::_thesis: verum
end;
caseA39: y < x ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y))
then (1 - p) * y < (1 - p) * x by A32, XREAL_1:68;
then A40: (p * x) + ((1 - p) * y) < x by A29, XREAL_1:8;
then A41: ((p * x) + ((1 - p) * y)) - x < 0 by XREAL_1:49;
p * y < p * x by A16, A31, A39, XREAL_1:68;
then A42: y < (p * x) + ((1 - p) * y) by A21, XREAL_1:8;
then A43: y - ((p * x) + ((1 - p) * y)) < 0 by XREAL_1:49;
A44: y - x < 0 by A39, XREAL_1:49;
((f . y) - (f . ((p * x) + ((1 - p) * y)))) / (y - ((p * x) + ((1 - p) * y))) <= ((f . ((p * x) + ((1 - p) * y))) - (f . x)) / (((p * x) + ((1 - p) * y)) - x) by A15, A19, A20, A28, A42, A40;
then ((f . y) - (f . ((p * x) + ((1 - p) * y)))) * ((1 - p) * (y - x)) <= ((f . ((p * x) + ((1 - p) * y))) - (f . x)) * (p * (y - x)) by A43, A41, XREAL_1:107;
then ((((f . y) - (f . ((p * x) + ((1 - p) * y)))) * (1 - p)) * (y - x)) / (y - x) >= ((((f . ((p * x) + ((1 - p) * y))) - (f . x)) * p) * (y - x)) / (y - x) by A44, XREAL_1:73;
then ((((f . ((p * x) + ((1 - p) * y))) - (f . x)) * p) * (y - x)) / (y - x) <= ((f . y) - (f . ((p * x) + ((1 - p) * y)))) * (1 - p) by A44, XCMPLX_1:89;
then ((f . ((p * x) + ((1 - p) * y))) * p) - ((f . x) * p) <= ((f . y) * (1 - p)) - ((f . ((p * x) + ((1 - p) * y))) * (1 - p)) by A44, XCMPLX_1:89;
then (((f . ((p * x) + ((1 - p) * y))) * p) - ((f . x) * p)) + ((f . ((p * x) + ((1 - p) * y))) * (1 - p)) <= (f . y) * (1 - p) by XREAL_1:19;
then (((f . ((p * x) + ((1 - p) * y))) * p) + ((f . ((p * x) + ((1 - p) * y))) * (1 - p))) - ((f . x) * p) <= (f . y) * (1 - p) ;
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by XREAL_1:20; ::_thesis: verum
end;
end;
end;
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum
end;
end;
end;
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum
end;
end;
end;
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum
end;
end;
end;
hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum
end;
hence f is_convex_on [.a,b.] by A14, Th52; ::_thesis: verum
end;
theorem :: RFUNCT_3:54
for F being PartFunc of REAL,REAL
for X, Y being set st F is_convex_on X & Y c= X holds
F is_convex_on Y
proof
let F be PartFunc of REAL,REAL; ::_thesis: for X, Y being set st F is_convex_on X & Y c= X holds
F is_convex_on Y
let X, Y be set ; ::_thesis: ( F is_convex_on X & Y c= X implies F is_convex_on Y )
assume that
A1: F is_convex_on X and
A2: Y c= X ; ::_thesis: F is_convex_on Y
X c= dom F by A1, Def12;
hence Y c= dom F by A2, XBOOLE_1:1; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in Y & s in Y & (p * r) + ((1 - p) * s) in Y holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in Y & s in Y & (p * r) + ((1 - p) * s) in Y holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) )
assume A3: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in Y & s in Y & (p * r) + ((1 - p) * s) in Y holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s))
let x, y be Real; ::_thesis: ( x in Y & y in Y & (p * x) + ((1 - p) * y) in Y implies F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) )
assume ( x in Y & y in Y & (p * x) + ((1 - p) * y) in Y ) ; ::_thesis: F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y))
hence F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, A2, A3, Def12; ::_thesis: verum
end;
theorem :: RFUNCT_3:55
for F being PartFunc of REAL,REAL
for X being set
for r being Real holds
( F is_convex_on X iff F - r is_convex_on X )
proof
let F be PartFunc of REAL,REAL; ::_thesis: for X being set
for r being Real holds
( F is_convex_on X iff F - r is_convex_on X )
let X be set ; ::_thesis: for r being Real holds
( F is_convex_on X iff F - r is_convex_on X )
let r be Real; ::_thesis: ( F is_convex_on X iff F - r is_convex_on X )
A1: dom F = dom (F - r) by VALUED_1:3;
thus ( F is_convex_on X implies F - r is_convex_on X ) ::_thesis: ( F - r is_convex_on X implies F is_convex_on X )
proof
assume A2: F is_convex_on X ; ::_thesis: F - r is_convex_on X
hence A3: X c= dom (F - r) by A1, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(F - r) . ((p * r) + ((1 - p) * s)) <= (p * ((F - r) . r)) + ((1 - p) * ((F - r) . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(F - r) . ((p * r) + ((1 - p) * s)) <= (p * ((F - r) . r)) + ((1 - p) * ((F - r) . s)) )
assume A4: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(F - r) . ((p * r) + ((1 - p) * s)) <= (p * ((F - r) . r)) + ((1 - p) * ((F - r) . s))
let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) )
assume that
A5: x in X and
A6: y in X and
A7: (p * x) + ((1 - p) * y) in X ; ::_thesis: (F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y))
F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A2, A4, A5, A6, A7, Def12;
then A8: (F . ((p * x) + ((1 - p) * y))) - r <= ((p * (F . x)) + ((1 - p) * (F . y))) - r by XREAL_1:9;
((p * (F . x)) + ((1 - p) * (F . y))) - r = (p * ((F . x) - r)) + ((1 - p) * ((F . y) - r))
.= (p * ((F - r) . x)) + ((1 - p) * ((F . y) - r)) by A1, A3, A5, VALUED_1:3
.= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A1, A3, A6, VALUED_1:3 ;
hence (F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A1, A3, A7, A8, VALUED_1:3; ::_thesis: verum
end;
assume A9: F - r is_convex_on X ; ::_thesis: F is_convex_on X
hence A10: X c= dom F by A1, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) )
assume A11: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s))
let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) )
assume that
A12: x in X and
A13: y in X and
A14: (p * x) + ((1 - p) * y) in X ; ::_thesis: F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y))
(F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A9, A11, A12, A13, A14, Def12;
then A15: (F . ((p * x) + ((1 - p) * y))) - r <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A10, A14, VALUED_1:3;
(p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) = (p * ((F - r) . x)) + ((1 - p) * ((F . y) - r)) by A10, A13, VALUED_1:3
.= (p * ((F . x) - r)) + (((1 - p) * (F . y)) - ((1 - p) * r)) by A10, A12, VALUED_1:3
.= ((p * (F . x)) + ((1 - p) * (F . y))) - r ;
hence F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A15, XREAL_1:9; ::_thesis: verum
end;
theorem :: RFUNCT_3:56
for F being PartFunc of REAL,REAL
for X being set
for r being Real st 0 < r holds
( F is_convex_on X iff r (#) F is_convex_on X )
proof
let F be PartFunc of REAL,REAL; ::_thesis: for X being set
for r being Real st 0 < r holds
( F is_convex_on X iff r (#) F is_convex_on X )
let X be set ; ::_thesis: for r being Real st 0 < r holds
( F is_convex_on X iff r (#) F is_convex_on X )
let r be Real; ::_thesis: ( 0 < r implies ( F is_convex_on X iff r (#) F is_convex_on X ) )
assume A1: 0 < r ; ::_thesis: ( F is_convex_on X iff r (#) F is_convex_on X )
A2: dom F = dom (r (#) F) by VALUED_1:def_5;
thus ( F is_convex_on X implies r (#) F is_convex_on X ) ::_thesis: ( r (#) F is_convex_on X implies F is_convex_on X )
proof
assume A3: F is_convex_on X ; ::_thesis: r (#) F is_convex_on X
then A4: X c= dom F by Def12;
thus X c= dom (r (#) F) by A2, A3, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s)) )
assume A5: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s))
let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) )
assume that
A6: x in X and
A7: y in X and
A8: (p * x) + ((1 - p) * y) in X ; ::_thesis: (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y))
F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A3, A5, A6, A7, A8, Def12;
then r * (F . ((p * x) + ((1 - p) * y))) <= r * ((p * (F . x)) + ((1 - p) * (F . y))) by A1, XREAL_1:64;
then (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * (r * (F . x))) + (((1 - p) * r) * (F . y)) by A2, A4, A8, VALUED_1:def_5;
then (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * (r * (F . y))) by A2, A4, A6, VALUED_1:def_5;
hence (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A2, A4, A7, VALUED_1:def_5; ::_thesis: verum
end;
assume A9: r (#) F is_convex_on X ; ::_thesis: F is_convex_on X
then A10: X c= dom (r (#) F) by Def12;
hence X c= dom F by VALUED_1:def_5; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) )
assume A11: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s))
let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) )
assume that
A12: x in X and
A13: y in X and
A14: (p * x) + ((1 - p) * y) in X ; ::_thesis: F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y))
(r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A9, A11, A12, A13, A14, Def12;
then r * (F . ((p * x) + ((1 - p) * y))) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A10, A14, VALUED_1:def_5;
then r * (F . ((p * x) + ((1 - p) * y))) <= (p * (r * (F . x))) + ((1 - p) * ((r (#) F) . y)) by A10, A12, VALUED_1:def_5;
then r * (F . ((p * x) + ((1 - p) * y))) <= (p * (r * (F . x))) + ((1 - p) * (r * (F . y))) by A10, A13, VALUED_1:def_5;
then (r * (F . ((p * x) + ((1 - p) * y)))) / r <= (r * ((p * (F . x)) + ((1 - p) * (F . y)))) / r by A1, XREAL_1:72;
then F . ((p * x) + ((1 - p) * y)) <= (r * ((p * (F . x)) + ((1 - p) * (F . y)))) / r by A1, XCMPLX_1:89;
hence F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, XCMPLX_1:89; ::_thesis: verum
end;
theorem :: RFUNCT_3:57
for F being PartFunc of REAL,REAL
for X being set st X c= dom F holds
0 (#) F is_convex_on X
proof
let F be PartFunc of REAL,REAL; ::_thesis: for X being set st X c= dom F holds
0 (#) F is_convex_on X
let X be set ; ::_thesis: ( X c= dom F implies 0 (#) F is_convex_on X )
assume A1: X c= dom F ; ::_thesis: 0 (#) F is_convex_on X
hence X c= dom (0 (#) F) by VALUED_1:def_5; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(0 (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((0 (#) F) . r)) + ((1 - p) * ((0 (#) F) . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(0 (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((0 (#) F) . r)) + ((1 - p) * ((0 (#) F) . s)) )
assume that
0 <= p and
p <= 1 ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(0 (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((0 (#) F) . r)) + ((1 - p) * ((0 (#) F) . s))
let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (0 (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y)) )
assume that
A2: x in X and
A3: y in X and
A4: (p * x) + ((1 - p) * y) in X ; ::_thesis: (0 (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y))
A5: dom F = dom (0 (#) F) by VALUED_1:def_5;
then A6: (0 (#) F) . ((p * x) + ((1 - p) * y)) = 0 * (F . ((p * x) + ((1 - p) * y))) by A1, A4, VALUED_1:def_5
.= 0 ;
(p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y)) = (p * (0 * (F . x))) + ((1 - p) * ((0 (#) F) . y)) by A1, A5, A2, VALUED_1:def_5
.= (p * 0) + ((1 - p) * (0 * (F . y))) by A1, A5, A3, VALUED_1:def_5
.= 0 + 0 ;
hence (0 (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y)) by A6; ::_thesis: verum
end;
theorem :: RFUNCT_3:58
for F, G being PartFunc of REAL,REAL
for X being set st F is_convex_on X & G is_convex_on X holds
F + G is_convex_on X
proof
let F, G be PartFunc of REAL,REAL; ::_thesis: for X being set st F is_convex_on X & G is_convex_on X holds
F + G is_convex_on X
let X be set ; ::_thesis: ( F is_convex_on X & G is_convex_on X implies F + G is_convex_on X )
A1: dom (F + G) = (dom F) /\ (dom G) by VALUED_1:def_1;
assume A2: ( F is_convex_on X & G is_convex_on X ) ; ::_thesis: F + G is_convex_on X
then ( X c= dom F & X c= dom G ) by Def12;
hence A3: X c= dom (F + G) by A1, XBOOLE_1:19; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(F + G) . ((p * r) + ((1 - p) * s)) <= (p * ((F + G) . r)) + ((1 - p) * ((F + G) . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(F + G) . ((p * r) + ((1 - p) * s)) <= (p * ((F + G) . r)) + ((1 - p) * ((F + G) . s)) )
assume A4: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(F + G) . ((p * r) + ((1 - p) * s)) <= (p * ((F + G) . r)) + ((1 - p) * ((F + G) . s))
let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (F + G) . ((p * x) + ((1 - p) * y)) <= (p * ((F + G) . x)) + ((1 - p) * ((F + G) . y)) )
assume that
A5: x in X and
A6: y in X and
A7: (p * x) + ((1 - p) * y) in X ; ::_thesis: (F + G) . ((p * x) + ((1 - p) * y)) <= (p * ((F + G) . x)) + ((1 - p) * ((F + G) . y))
( F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) & G . ((p * x) + ((1 - p) * y)) <= (p * (G . x)) + ((1 - p) * (G . y)) ) by A2, A4, A5, A6, A7, Def12;
then (F . ((p * x) + ((1 - p) * y))) + (G . ((p * x) + ((1 - p) * y))) <= ((p * (F . x)) + ((1 - p) * (F . y))) + ((p * (G . x)) + ((1 - p) * (G . y))) by XREAL_1:7;
then A8: (F + G) . ((p * x) + ((1 - p) * y)) <= ((p * (F . x)) + ((1 - p) * (F . y))) + ((p * (G . x)) + ((1 - p) * (G . y))) by A3, A7, VALUED_1:def_1;
((p * (F . x)) + ((1 - p) * (F . y))) + ((p * (G . x)) + ((1 - p) * (G . y))) = ((p * ((F . x) + (G . x))) + ((1 - p) * (F . y))) + ((1 - p) * (G . y))
.= ((p * ((F + G) . x)) + ((1 - p) * (F . y))) + ((1 - p) * (G . y)) by A3, A5, VALUED_1:def_1
.= (p * ((F + G) . x)) + ((1 - p) * ((F . y) + (G . y))) ;
hence (F + G) . ((p * x) + ((1 - p) * y)) <= (p * ((F + G) . x)) + ((1 - p) * ((F + G) . y)) by A3, A6, A8, VALUED_1:def_1; ::_thesis: verum
end;
theorem Th59: :: RFUNCT_3:59
for F being PartFunc of REAL,REAL
for X being set
for r being Real st F is_convex_on X holds
max+ (F - r) is_convex_on X
proof
let F be PartFunc of REAL,REAL; ::_thesis: for X being set
for r being Real st F is_convex_on X holds
max+ (F - r) is_convex_on X
let X be set ; ::_thesis: for r being Real st F is_convex_on X holds
max+ (F - r) is_convex_on X
let r be Real; ::_thesis: ( F is_convex_on X implies max+ (F - r) is_convex_on X )
assume A1: F is_convex_on X ; ::_thesis: max+ (F - r) is_convex_on X
then A2: X c= dom F by Def12;
A3: ( dom F = dom (F - r) & dom (max+ (F - r)) = dom (F - r) ) by Def10, VALUED_1:3;
hence X c= dom (max+ (F - r)) by A1, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s)) )
assume that
A4: 0 <= p and
A5: p <= 1 ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s))
let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y)) )
assume that
A6: x in X and
A7: y in X and
A8: (p * x) + ((1 - p) * y) in X ; ::_thesis: (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y))
F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, A4, A5, A6, A7, A8, Def12;
then (F . ((p * x) + ((1 - p) * y))) - r <= ((p * (F . x)) + ((1 - p) * (F . y))) - r by XREAL_1:9;
then A9: max+ ((F . ((p * x) + ((1 - p) * y))) - r) <= max ((((p * (F . x)) + ((1 - p) * (F . y))) - r),0) by XXREAL_0:26;
0 + p <= 1 by A5;
then 0 <= 1 - p by XREAL_1:19;
then A10: max+ ((1 - p) * ((F - r) . y)) = (1 - p) * (max+ ((F - r) . y)) by Th4;
A11: max+ ((p * ((F - r) . x)) + ((1 - p) * ((F - r) . y))) <= (max+ (p * ((F - r) . x))) + (max+ ((1 - p) * ((F - r) . y))) by Th5;
A12: max+ (p * ((F - r) . x)) = p * (max+ ((F - r) . x)) by A4, Th4;
((p * (F . x)) + ((1 - p) * (F . y))) - r = (p * ((F . x) - r)) + ((1 - p) * ((F . y) - r))
.= (p * ((F - r) . x)) + ((1 - p) * ((F . y) - r)) by A6, A2, VALUED_1:3
.= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A7, A2, VALUED_1:3 ;
then max+ ((F . ((p * x) + ((1 - p) * y))) - r) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A9, A11, A12, A10, XXREAL_0:2;
then max+ ((F - r) . ((p * x) + ((1 - p) * y))) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A8, A2, VALUED_1:3;
then (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A3, A8, A2, Def10;
then (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * (max+ ((F - r) . y))) by A3, A6, A2, Def10;
hence (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y)) by A3, A7, A2, Def10; ::_thesis: verum
end;
theorem :: RFUNCT_3:60
for F being PartFunc of REAL,REAL
for X being set st F is_convex_on X holds
max+ F is_convex_on X
proof
let F be PartFunc of REAL,REAL; ::_thesis: for X being set st F is_convex_on X holds
max+ F is_convex_on X
let X be set ; ::_thesis: ( F is_convex_on X implies max+ F is_convex_on X )
assume F is_convex_on X ; ::_thesis: max+ F is_convex_on X
then max+ (F - 0) is_convex_on X by Th59;
hence max+ F is_convex_on X by Th48; ::_thesis: verum
end;
theorem Th61: :: RFUNCT_3:61
id ([#] REAL) is_convex_on REAL
proof
set i = id ([#] REAL);
thus REAL c= dom (id ([#] REAL)) by FUNCT_1:17; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in REAL & s in REAL & (p * r) + ((1 - p) * s) in REAL holds
(id ([#] REAL)) . ((p * r) + ((1 - p) * s)) <= (p * ((id ([#] REAL)) . r)) + ((1 - p) * ((id ([#] REAL)) . s))
let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in REAL & s in REAL & (p * r) + ((1 - p) * s) in REAL holds
(id ([#] REAL)) . ((p * r) + ((1 - p) * s)) <= (p * ((id ([#] REAL)) . r)) + ((1 - p) * ((id ([#] REAL)) . s)) )
assume that
0 <= p and
p <= 1 ; ::_thesis: for r, s being Real st r in REAL & s in REAL & (p * r) + ((1 - p) * s) in REAL holds
(id ([#] REAL)) . ((p * r) + ((1 - p) * s)) <= (p * ((id ([#] REAL)) . r)) + ((1 - p) * ((id ([#] REAL)) . s))
let x, y be Real; ::_thesis: ( x in REAL & y in REAL & (p * x) + ((1 - p) * y) in REAL implies (id ([#] REAL)) . ((p * x) + ((1 - p) * y)) <= (p * ((id ([#] REAL)) . x)) + ((1 - p) * ((id ([#] REAL)) . y)) )
assume that
x in REAL and
y in REAL and
(p * x) + ((1 - p) * y) in REAL ; ::_thesis: (id ([#] REAL)) . ((p * x) + ((1 - p) * y)) <= (p * ((id ([#] REAL)) . x)) + ((1 - p) * ((id ([#] REAL)) . y))
( (id ([#] REAL)) . x = x & (id ([#] REAL)) . y = y ) by FUNCT_1:17;
hence (id ([#] REAL)) . ((p * x) + ((1 - p) * y)) <= (p * ((id ([#] REAL)) . x)) + ((1 - p) * ((id ([#] REAL)) . y)) by FUNCT_1:17; ::_thesis: verum
end;
theorem :: RFUNCT_3:62
for r being Real holds max+ ((id ([#] REAL)) - r) is_convex_on REAL by Th59, Th61;
definition
let D be non empty set ;
let F be PartFunc of D,REAL;
let X be set ;
assume A1: dom (F | X) is finite ;
func FinS (F,X) -> non-increasing FinSequence of REAL means :Def13: :: RFUNCT_3:def 13
F | X,it are_fiberwise_equipotent ;
existence
ex b1 being non-increasing FinSequence of REAL st F | X,b1 are_fiberwise_equipotent
proof
set x = dom (F | X);
consider b being FinSequence such that
A2: F | (dom (F | X)),b are_fiberwise_equipotent by A1, RFINSEQ:5;
rng (F | (dom (F | X))) = rng b by A2, CLASSES1:75;
then reconsider b = b as FinSequence of REAL by FINSEQ_1:def_4;
consider a being non-increasing FinSequence of REAL such that
A3: b,a are_fiberwise_equipotent by RFINSEQ:22;
take a ; ::_thesis: F | X,a are_fiberwise_equipotent
dom (F | X) = (dom F) /\ X by RELAT_1:61;
then F | (dom (F | X)) = (F | (dom F)) | X by RELAT_1:71
.= F | X by RELAT_1:68 ;
hence F | X,a are_fiberwise_equipotent by A2, A3, CLASSES1:76; ::_thesis: verum
end;
uniqueness
for b1, b2 being non-increasing FinSequence of REAL st F | X,b1 are_fiberwise_equipotent & F | X,b2 are_fiberwise_equipotent holds
b1 = b2 by CLASSES1:76, RFINSEQ:23;
end;
:: deftheorem Def13 defines FinS RFUNCT_3:def_13_:_
for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
for b4 being non-increasing FinSequence of REAL holds
( b4 = FinS (F,X) iff F | X,b4 are_fiberwise_equipotent );
theorem Th63: :: RFUNCT_3:63
for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
FinS (F,(dom (F | X))) = FinS (F,X)
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
FinS (F,(dom (F | X))) = FinS (F,X)
let F be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite holds
FinS (F,(dom (F | X))) = FinS (F,X)
let X be set ; ::_thesis: ( dom (F | X) is finite implies FinS (F,(dom (F | X))) = FinS (F,X) )
A1: F | (dom (F | X)) = F | ((dom F) /\ X) by RELAT_1:61
.= (F | (dom F)) | X by RELAT_1:71
.= F | X by RELAT_1:68 ;
assume A2: dom (F | X) is finite ; ::_thesis: FinS (F,(dom (F | X))) = FinS (F,X)
then FinS (F,X),F | X are_fiberwise_equipotent by Def13;
hence FinS (F,(dom (F | X))) = FinS (F,X) by A2, A1, Def13; ::_thesis: verum
end;
theorem Th64: :: RFUNCT_3:64
for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
FinS ((F | X),X) = FinS (F,X)
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
FinS ((F | X),X) = FinS (F,X)
let F be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite holds
FinS ((F | X),X) = FinS (F,X)
let X be set ; ::_thesis: ( dom (F | X) is finite implies FinS ((F | X),X) = FinS (F,X) )
A1: (F | X) | X = F | X by RELAT_1:72;
assume A2: dom (F | X) is finite ; ::_thesis: FinS ((F | X),X) = FinS (F,X)
then FinS (F,X),F | X are_fiberwise_equipotent by Def13;
hence FinS ((F | X),X) = FinS (F,X) by A2, A1, Def13; ::_thesis: verum
end;
theorem Th65: :: RFUNCT_3:65
for D being non empty set
for d being Element of D
for X being set
for F being PartFunc of D,REAL st X is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent
proof
for D being non empty set
for X being finite set
for F being PartFunc of D,REAL
for x being set st x in dom (F | X) holds
(FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent
proof
let D be non empty set ; ::_thesis: for X being finite set
for F being PartFunc of D,REAL
for x being set st x in dom (F | X) holds
(FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent
let X be finite set ; ::_thesis: for F being PartFunc of D,REAL
for x being set st x in dom (F | X) holds
(FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent
let F be PartFunc of D,REAL; ::_thesis: for x being set st x in dom (F | X) holds
(FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent
let x be set ; ::_thesis: ( x in dom (F | X) implies (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent )
set Y = X \ {x};
set A = (FinS (F,(X \ {x}))) ^ <*(F . x)*>;
assume x in dom (F | X) ; ::_thesis: (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent
then A1: x in (dom F) /\ X by RELAT_1:61;
then x in X by XBOOLE_0:def_4;
then A2: {x} c= X by ZFMISC_1:31;
x in dom F by A1, XBOOLE_0:def_4;
then A3: {x} c= dom F by ZFMISC_1:31;
dom (F | (X \ {x})) is finite ;
then A4: F | (X \ {x}), FinS (F,(X \ {x})) are_fiberwise_equipotent by Def13;
now__::_thesis:_for_y_being_set_holds_card_(Coim_((F_|_X),y))_=_card_(Coim_(((FinS_(F,(X_\_{x})))_^_<*(F_._x)*>),y))
let y be set ; ::_thesis: card (Coim ((F | X),y)) = card (Coim (((FinS (F,(X \ {x}))) ^ <*(F . x)*>),y))
A5: X \ {x} misses {x} by XBOOLE_1:79;
A6: card (Coim ((F | (X \ {x})),y)) = card (Coim ((FinS (F,(X \ {x}))),y)) by A4, CLASSES1:def_9;
A7: dom (F | {x}) = {x} by A3, RELAT_1:62;
A8: dom <*(F . x)*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
A9: now__::_thesis:_(_(_y_=_F_._x_&_card_((F_|_{x})_"_{y})_=_card_(<*(F_._x)*>_"_{y})_)_or_(_y_<>_F_._x_&_card_((F_|_{x})_"_{y})_=_card_(<*(F_._x)*>_"_{y})_)_)
percases ( y = F . x or y <> F . x ) ;
caseA10: y = F . x ; ::_thesis: card ((F | {x}) " {y}) = card (<*(F . x)*> " {y})
A11: {x} c= (F | {x}) " {y}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {x} or z in (F | {x}) " {y} )
A12: y in {y} by TARSKI:def_1;
assume A13: z in {x} ; ::_thesis: z in (F | {x}) " {y}
then z = x by TARSKI:def_1;
then y = (F | {x}) . z by A7, A10, A13, FUNCT_1:47;
hence z in (F | {x}) " {y} by A7, A13, A12, FUNCT_1:def_7; ::_thesis: verum
end;
(F | {x}) " {y} c= {x} by A7, RELAT_1:132;
then (F | {x}) " {y} = {x} by A11, XBOOLE_0:def_10;
then A14: card ((F | {x}) " {y}) = 1 by CARD_1:30;
A15: {1} c= <*(F . x)*> " {y}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {1} or z in <*(F . x)*> " {y} )
A16: y in {y} by TARSKI:def_1;
assume A17: z in {1} ; ::_thesis: z in <*(F . x)*> " {y}
then z = 1 by TARSKI:def_1;
then y = <*(F . x)*> . z by A10, FINSEQ_1:40;
hence z in <*(F . x)*> " {y} by A8, A17, A16, FUNCT_1:def_7; ::_thesis: verum
end;
<*(F . x)*> " {y} c= {1} by A8, RELAT_1:132;
then <*(F . x)*> " {y} = {1} by A15, XBOOLE_0:def_10;
hence card ((F | {x}) " {y}) = card (<*(F . x)*> " {y}) by A14, CARD_1:30; ::_thesis: verum
end;
caseA18: y <> F . x ; ::_thesis: card ((F | {x}) " {y}) = card (<*(F . x)*> " {y})
A19: now__::_thesis:_not_<*(F_._x)*>_"_{y}_<>_{}
set z = the Element of <*(F . x)*> " {y};
assume A20: <*(F . x)*> " {y} <> {} ; ::_thesis: contradiction
then <*(F . x)*> . the Element of <*(F . x)*> " {y} in {y} by FUNCT_1:def_7;
then A21: <*(F . x)*> . the Element of <*(F . x)*> " {y} = y by TARSKI:def_1;
the Element of <*(F . x)*> " {y} in {1} by A8, A20, FUNCT_1:def_7;
then the Element of <*(F . x)*> " {y} = 1 by TARSKI:def_1;
hence contradiction by A18, A21, FINSEQ_1:40; ::_thesis: verum
end;
now__::_thesis:_not_(F_|_{x})_"_{y}_<>_{}
set z = the Element of (F | {x}) " {y};
assume A22: (F | {x}) " {y} <> {} ; ::_thesis: contradiction
then (F | {x}) . the Element of (F | {x}) " {y} in {y} by FUNCT_1:def_7;
then A23: (F | {x}) . the Element of (F | {x}) " {y} = y by TARSKI:def_1;
A24: the Element of (F | {x}) " {y} in {x} by A7, A22, FUNCT_1:def_7;
then the Element of (F | {x}) " {y} = x by TARSKI:def_1;
hence contradiction by A7, A18, A24, A23, FUNCT_1:47; ::_thesis: verum
end;
hence card ((F | {x}) " {y}) = card (<*(F . x)*> " {y}) by A19; ::_thesis: verum
end;
end;
end;
A25: ((F | (X \ {x})) " {y}) \/ ((F | {x}) " {y}) = ((X \ {x}) /\ (F " {y})) \/ ((F | {x}) " {y}) by FUNCT_1:70
.= ((X \ {x}) /\ (F " {y})) \/ ({x} /\ (F " {y})) by FUNCT_1:70
.= ((X \ {x}) \/ {x}) /\ (F " {y}) by XBOOLE_1:23
.= (X \/ {x}) /\ (F " {y}) by XBOOLE_1:39
.= X /\ (F " {y}) by A2, XBOOLE_1:12
.= (F | X) " {y} by FUNCT_1:70 ;
((F | (X \ {x})) " {y}) /\ ((F | {x}) " {y}) = ((X \ {x}) /\ (F " {y})) /\ ((F | {x}) " {y}) by FUNCT_1:70
.= ((X \ {x}) /\ (F " {y})) /\ ({x} /\ (F " {y})) by FUNCT_1:70
.= (((F " {y}) /\ (X \ {x})) /\ {x}) /\ (F " {y}) by XBOOLE_1:16
.= ((F " {y}) /\ ((X \ {x}) /\ {x})) /\ (F " {y}) by XBOOLE_1:16
.= {} /\ (F " {y}) by A5, XBOOLE_0:def_7
.= {} ;
hence card (Coim ((F | X),y)) = ((card ((F | (X \ {x})) " {y})) + (card (<*(F . x)*> " {y}))) - (card {}) by A25, A9, CARD_2:45
.= card (Coim (((FinS (F,(X \ {x}))) ^ <*(F . x)*>),y)) by A6, FINSEQ_3:57 ;
::_thesis: verum
end;
hence (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent by CLASSES1:def_9; ::_thesis: verum
end;
hence for D being non empty set
for d being Element of D
for X being set
for F being PartFunc of D,REAL st X is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent ; ::_thesis: verum
end;
theorem Th66: :: RFUNCT_3:66
for D being non empty set
for d being Element of D
for X being set
for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent
proof
let D be non empty set ; ::_thesis: for d being Element of D
for X being set
for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent
let d be Element of D; ::_thesis: for X being set
for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent
let X be set ; ::_thesis: for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent
let F be PartFunc of D,REAL; ::_thesis: ( dom (F | X) is finite & d in dom (F | X) implies (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent )
set dx = dom (F | X);
assume that
A1: dom (F | X) is finite and
A2: d in dom (F | X) ; ::_thesis: (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent
set Y = X \ {d};
set dy = dom (F | (X \ {d}));
A3: dom (F | (X \ {d})) = (dom F) /\ (X \ {d}) by RELAT_1:61;
A4: dom (F | X) = (dom F) /\ X by RELAT_1:61;
A5: dom (F | (X \ {d})) = (dom (F | X)) \ {d}
proof
thus dom (F | (X \ {d})) c= (dom (F | X)) \ {d} :: according to XBOOLE_0:def_10 ::_thesis: (dom (F | X)) \ {d} c= dom (F | (X \ {d}))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (X \ {d})) or y in (dom (F | X)) \ {d} )
assume A6: y in dom (F | (X \ {d})) ; ::_thesis: y in (dom (F | X)) \ {d}
then y in X \ {d} by A3, XBOOLE_0:def_4;
then A7: not y in {d} by XBOOLE_0:def_5;
y in dom F by A3, A6, XBOOLE_0:def_4;
then y in dom (F | X) by A3, A4, A6, XBOOLE_0:def_4;
hence y in (dom (F | X)) \ {d} by A7, XBOOLE_0:def_5; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (dom (F | X)) \ {d} or y in dom (F | (X \ {d})) )
assume A8: y in (dom (F | X)) \ {d} ; ::_thesis: y in dom (F | (X \ {d}))
then A9: not y in {d} by XBOOLE_0:def_5;
A10: y in dom (F | X) by A8, XBOOLE_0:def_5;
then y in X by A4, XBOOLE_0:def_4;
then A11: y in X \ {d} by A9, XBOOLE_0:def_5;
y in dom F by A4, A10, XBOOLE_0:def_4;
hence y in dom (F | (X \ {d})) by A3, A11, XBOOLE_0:def_4; ::_thesis: verum
end;
F | (dom (F | X)) = F | ((dom F) /\ X) by RELAT_1:61
.= (F | (dom F)) | X by RELAT_1:71
.= F | X by RELAT_1:68 ;
then (FinS (F,((dom (F | X)) \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent by A1, A2, Th65;
hence (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent by A1, A5, Th63; ::_thesis: verum
end;
theorem Th67: :: RFUNCT_3:67
for D being non empty set
for F being PartFunc of D,REAL
for X being set
for Y being finite set st Y = dom (F | X) holds
len (FinS (F,X)) = card Y
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set
for Y being finite set st Y = dom (F | X) holds
len (FinS (F,X)) = card Y
let F be PartFunc of D,REAL; ::_thesis: for X being set
for Y being finite set st Y = dom (F | X) holds
len (FinS (F,X)) = card Y
let X be set ; ::_thesis: for Y being finite set st Y = dom (F | X) holds
len (FinS (F,X)) = card Y
let Y be finite set ; ::_thesis: ( Y = dom (F | X) implies len (FinS (F,X)) = card Y )
reconsider fs = dom (FinS (F,X)) as finite set ;
A1: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3;
assume A2: Y = dom (F | X) ; ::_thesis: len (FinS (F,X)) = card Y
FinS (F,X),F | X are_fiberwise_equipotent by A2, Def13;
hence card Y = card fs by A2, CLASSES1:81
.= len (FinS (F,X)) by A1, FINSEQ_1:57 ;
::_thesis: verum
end;
theorem Th68: :: RFUNCT_3:68
for D being non empty set
for F being PartFunc of D,REAL holds FinS (F,{}) = <*> REAL
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds FinS (F,{}) = <*> REAL
let F be PartFunc of D,REAL; ::_thesis: FinS (F,{}) = <*> REAL
dom (F | {}) = (dom F) /\ {} by RELAT_1:61
.= {} ;
then len (FinS (F,{})) = 0 by Th67, CARD_1:27;
hence FinS (F,{}) = <*> REAL ; ::_thesis: verum
end;
theorem Th69: :: RFUNCT_3:69
for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D st d in dom F holds
FinS (F,{d}) = <*(F . d)*>
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for d being Element of D st d in dom F holds
FinS (F,{d}) = <*(F . d)*>
let F be PartFunc of D,REAL; ::_thesis: for d being Element of D st d in dom F holds
FinS (F,{d}) = <*(F . d)*>
let d be Element of D; ::_thesis: ( d in dom F implies FinS (F,{d}) = <*(F . d)*> )
assume d in dom F ; ::_thesis: FinS (F,{d}) = <*(F . d)*>
then {d} c= dom F by ZFMISC_1:31;
then A1: {d} = (dom F) /\ {d} by XBOOLE_1:28
.= dom (F | {d}) by RELAT_1:61 ;
then FinS (F,{d}),F | {d} are_fiberwise_equipotent by Def13;
then A2: rng (FinS (F,{d})) = rng (F | {d}) by CLASSES1:75;
A3: rng (F | {d}) = {(F . d)}
proof
thus rng (F | {d}) c= {(F . d)} :: according to XBOOLE_0:def_10 ::_thesis: {(F . d)} c= rng (F | {d})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (F | {d}) or x in {(F . d)} )
assume x in rng (F | {d}) ; ::_thesis: x in {(F . d)}
then consider e being Element of D such that
A4: e in dom (F | {d}) and
A5: (F | {d}) . e = x by PARTFUN1:3;
e = d by A1, A4, TARSKI:def_1;
then x = F . d by A4, A5, FUNCT_1:47;
hence x in {(F . d)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(F . d)} or x in rng (F | {d}) )
A6: d in dom (F | {d}) by A1, TARSKI:def_1;
assume x in {(F . d)} ; ::_thesis: x in rng (F | {d})
then x = F . d by TARSKI:def_1;
then x = (F | {d}) . d by A6, FUNCT_1:47;
hence x in rng (F | {d}) by A6, FUNCT_1:def_3; ::_thesis: verum
end;
len (FinS (F,{d})) = card {d} by A1, Th67
.= 1 by CARD_1:30 ;
hence FinS (F,{d}) = <*(F . d)*> by A2, A3, FINSEQ_1:39; ::_thesis: verum
end;
theorem Th70: :: RFUNCT_3:70
for D being non empty set
for F being PartFunc of D,REAL
for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds
FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*>
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds
FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*>
let F be PartFunc of D,REAL; ::_thesis: for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds
FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*>
let X be set ; ::_thesis: for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds
FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*>
let d be Element of D; ::_thesis: ( dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d implies FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> )
set dx = dom (F | X);
set fx = FinS (F,X);
set fy = FinS (F,(X \ {d}));
assume that
A1: dom (F | X) is finite and
A2: d in dom (F | X) and
A3: (FinS (F,X)) . (len (FinS (F,X))) = F . d ; ::_thesis: FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*>
A4: FinS (F,X),F | X are_fiberwise_equipotent by A1, Def13;
then rng (FinS (F,X)) = rng (F | X) by CLASSES1:75;
then FinS (F,X) <> {} by A2, FUNCT_1:3, RELAT_1:38;
then 0 + 1 <= len (FinS (F,X)) by NAT_1:13;
then max (0,((len (FinS (F,X))) - 1)) = (len (FinS (F,X))) - 1 by FINSEQ_2:4;
then reconsider n = (len (FinS (F,X))) - 1 as Element of NAT by FINSEQ_2:5;
len (FinS (F,X)) = n + 1 ;
then A5: FinS (F,X) = ((FinS (F,X)) | n) ^ <*(F . d)*> by A3, RFINSEQ:7;
A6: (FinS (F,X)) | n is non-increasing by RFINSEQ:20;
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent by A1, A2, Th66;
then FinS (F,X),(FinS (F,(X \ {d}))) ^ <*(F . d)*> are_fiberwise_equipotent by A4, CLASSES1:76;
then FinS (F,(X \ {d})),(FinS (F,X)) | n are_fiberwise_equipotent by A5, RFINSEQ:1;
hence FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> by A5, A6, RFINSEQ:23; ::_thesis: verum
end;
defpred S1[ Element of NAT ] means for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set
for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & $1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)));
Lm3: S1[ 0 ]
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X, Y being set
for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
let F be PartFunc of D,REAL; ::_thesis: for X, Y being set
for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
let X, Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
let Z be finite set ; ::_thesis: ( Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) implies FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) )
assume A1: Z = dom (F | Y) ; ::_thesis: ( not dom (F | X) is finite or not Y c= X or not 0 = card Z or ex d1, d2 being Element of D st
( d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) & not F . d1 >= F . d2 ) or FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) )
assume that
A2: dom (F | X) is finite and
Y c= X and
A3: 0 = card Z and
for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ; ::_thesis: FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
A4: dom (F | Y) = {} by A1, A3;
A5: dom (F | (X \ Y)) = (dom F) /\ (X \ Y) by RELAT_1:61;
( dom (F | X) = (dom F) /\ X & dom (F | Y) = (dom F) /\ Y ) by RELAT_1:61;
then dom (F | (X \ Y)) = (dom (F | X)) \ {} by A5, A4, XBOOLE_1:50
.= dom (F | X) ;
then A6: FinS (F,(X \ Y)) = FinS (F,(dom (F | X))) by A2, Th63
.= FinS (F,X) by A2, Th63 ;
FinS (F,Y) = FinS (F,{}) by A4, Th63
.= {} by Th68 ;
hence FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) by A6, FINSEQ_1:34; ::_thesis: verum
end;
Lm4: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A1: S1[n] ; ::_thesis: S1[n + 1]
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X, Y being set
for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
let F be PartFunc of D,REAL; ::_thesis: for X, Y being set
for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
let X, Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
set dx = dom (F | X);
set dxy = dom (F | (X \ Y));
set fy = FinS (F,Y);
set fxy = FinS (F,(X \ Y));
let dy be finite set ; ::_thesis: ( dy = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card dy & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) implies FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) )
assume that
A2: dy = dom (F | Y) and
A3: dom (F | X) is finite and
A4: Y c= X and
A5: n + 1 = card dy and
A6: for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ; ::_thesis: FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
A7: len (FinS (F,Y)) = n + 1 by A2, A5, Th67;
A8: F | Y, FinS (F,Y) are_fiberwise_equipotent by A2, Def13;
then A9: rng (FinS (F,Y)) = rng (F | Y) by CLASSES1:75;
0 + 1 <= n + 1 by NAT_1:13;
then A10: len (FinS (F,Y)) in dom (FinS (F,Y)) by A7, FINSEQ_3:25;
then (FinS (F,Y)) . (len (FinS (F,Y))) in rng (FinS (F,Y)) by FUNCT_1:def_3;
then consider d being Element of D such that
A11: d in dy and
A12: (F | Y) . d = (FinS (F,Y)) . (len (FinS (F,Y))) by A2, A9, PARTFUN1:3;
A13: dom (F | (X \ Y)) = (dom F) /\ (X \ Y) by RELAT_1:61;
A14: dy = (dom F) /\ Y by A2, RELAT_1:61;
then A15: d in Y by A11, XBOOLE_0:def_4;
then A16: {d} c= X by A4, ZFMISC_1:31;
A17: d in dom F by A14, A11, XBOOLE_0:def_4;
then A18: {d} c= dom F by ZFMISC_1:31;
A19: {d} c= Y by A15, ZFMISC_1:31;
A20: (FinS (F,(X \ Y))) ^ <*(F . d)*>,<*(F . d)*> ^ (FinS (F,(X \ Y))) are_fiberwise_equipotent by RFINSEQ:2;
set Yd = Y \ {d};
set dyd = dom (F | (Y \ {d}));
set xyd = dom (F | (X \ (Y \ {d})));
A21: dom (F | (X \ (Y \ {d}))) = (dom F) /\ (X \ (Y \ {d})) by RELAT_1:61;
A22: dom (F | (Y \ {d})) = (dom F) /\ (Y \ {d}) by RELAT_1:61;
A23: dom (F | (Y \ {d})) = dy \ {d}
proof
thus dom (F | (Y \ {d})) c= dy \ {d} :: according to XBOOLE_0:def_10 ::_thesis: dy \ {d} c= dom (F | (Y \ {d}))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (Y \ {d})) or y in dy \ {d} )
assume A24: y in dom (F | (Y \ {d})) ; ::_thesis: y in dy \ {d}
then y in Y \ {d} by A22, XBOOLE_0:def_4;
then A25: not y in {d} by XBOOLE_0:def_5;
y in dom F by A22, A24, XBOOLE_0:def_4;
then y in dy by A14, A22, A24, XBOOLE_0:def_4;
hence y in dy \ {d} by A25, XBOOLE_0:def_5; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dy \ {d} or y in dom (F | (Y \ {d})) )
assume A26: y in dy \ {d} ; ::_thesis: y in dom (F | (Y \ {d}))
then ( not y in {d} & y in Y ) by A14, XBOOLE_0:def_4, XBOOLE_0:def_5;
then A27: y in Y \ {d} by XBOOLE_0:def_5;
y in dom F by A14, A26, XBOOLE_0:def_4;
hence y in dom (F | (Y \ {d})) by A22, A27, XBOOLE_0:def_4; ::_thesis: verum
end;
A28: F . d = (FinS (F,Y)) . (len (FinS (F,Y))) by A2, A11, A12, FUNCT_1:47;
then A29: FinS (F,Y) = ((FinS (F,Y)) | n) ^ <*(F . d)*> by A7, RFINSEQ:7;
reconsider dyd = dom (F | (Y \ {d})) as finite set by A23;
A30: X \ (Y \ {d}) = (X \ Y) \/ (X /\ {d}) by XBOOLE_1:52
.= (X \ Y) \/ {d} by A16, XBOOLE_1:28 ;
then A31: dom (F | (X \ (Y \ {d}))) = (dom (F | (X \ Y))) \/ ((dom F) /\ {d}) by A13, A21, XBOOLE_1:23
.= (dom (F | (X \ Y))) \/ {d} by A18, XBOOLE_1:28 ;
A32: now__::_thesis:_for_d1,_d2_being_Element_of_D_st_d1_in_dyd_&_d2_in_dom_(F_|_(X_\_(Y_\_{d})))_holds_
F_._d1_>=_F_._d2
let d1, d2 be Element of D; ::_thesis: ( d1 in dyd & d2 in dom (F | (X \ (Y \ {d}))) implies F . d1 >= F . d2 )
assume that
A33: d1 in dyd and
A34: d2 in dom (F | (X \ (Y \ {d}))) ; ::_thesis: F . d1 >= F . d2
now__::_thesis:_(_(_d2_in_dom_(F_|_(X_\_Y))_&_F_._d1_>=_F_._d2_)_or_(_d2_in_{d}_&_F_._d1_>=_F_._d2_)_)
percases ( d2 in dom (F | (X \ Y)) or d2 in {d} ) by A31, A34, XBOOLE_0:def_3;
case d2 in dom (F | (X \ Y)) ; ::_thesis: F . d1 >= F . d2
hence F . d1 >= F . d2 by A2, A6, A23, A33; ::_thesis: verum
end;
case d2 in {d} ; ::_thesis: F . d1 >= F . d2
then A35: d2 = d by TARSKI:def_1;
(F | Y) . d1 in rng (F | Y) by A2, A23, A33, FUNCT_1:def_3;
then F . d1 in rng (F | Y) by A2, A23, A33, FUNCT_1:47;
then consider m being Nat such that
A36: m in dom (FinS (F,Y)) and
A37: (FinS (F,Y)) . m = F . d1 by A9, FINSEQ_2:10;
A38: m <= len (FinS (F,Y)) by A36, FINSEQ_3:25;
now__::_thesis:_(_(_m_=_len_(FinS_(F,Y))_&_F_._d1_>=_F_._d2_)_or_(_m_<>_len_(FinS_(F,Y))_&_F_._d1_>=_F_._d2_)_)
percases ( m = len (FinS (F,Y)) or m <> len (FinS (F,Y)) ) ;
case m = len (FinS (F,Y)) ; ::_thesis: F . d1 >= F . d2
hence F . d1 >= F . d2 by A2, A11, A12, A35, A37, FUNCT_1:47; ::_thesis: verum
end;
case m <> len (FinS (F,Y)) ; ::_thesis: F . d1 >= F . d2
then m < len (FinS (F,Y)) by A38, XXREAL_0:1;
hence F . d1 >= F . d2 by A10, A28, A35, A36, A37, RFINSEQ:19; ::_thesis: verum
end;
end;
end;
hence F . d1 >= F . d2 ; ::_thesis: verum
end;
end;
end;
hence F . d1 >= F . d2 ; ::_thesis: verum
end;
dom (F | X) = (dom F) /\ X by RELAT_1:61;
then A39: dom (F | (X \ Y)) is finite by A3, A13, FINSET_1:1, XBOOLE_1:26;
then F | (X \ Y), FinS (F,(X \ Y)) are_fiberwise_equipotent by Def13;
then A40: rng (FinS (F,(X \ Y))) = rng (F | (X \ Y)) by CLASSES1:75;
A41: <*(F . d)*> ^ (FinS (F,(X \ Y))) is non-increasing
proof
set xfy = <*(F . d)*> ^ (FinS (F,(X \ Y)));
let n be Element of NAT ; :: according to RFINSEQ:def_3 ::_thesis: ( not n in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) or not n + 1 in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) or (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) <= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n )
assume that
A42: n in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) and
A43: n + 1 in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) ; ::_thesis: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) <= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n
A44: 1 <= n by A42, FINSEQ_3:25;
then max (0,(n - 1)) = n - 1 by FINSEQ_2:4;
then reconsider n1 = n - 1 as Element of NAT by FINSEQ_2:5;
set r = (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n;
set s = (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1);
A45: len <*(F . d)*> = 1 by FINSEQ_1:40;
then len (<*(F . d)*> ^ (FinS (F,(X \ Y)))) = 1 + (len (FinS (F,(X \ Y)))) by FINSEQ_1:22;
then A46: len (FinS (F,(X \ Y))) = (len (<*(F . d)*> ^ (FinS (F,(X \ Y))))) - 1 ;
A47: n + 1 <= len (<*(F . d)*> ^ (FinS (F,(X \ Y)))) by A43, FINSEQ_3:25;
then n1 + 1 <= len (FinS (F,(X \ Y))) by A46, XREAL_1:19;
then A48: n1 + 1 in dom (FinS (F,(X \ Y))) by A44, FINSEQ_3:25;
then (FinS (F,(X \ Y))) . (n1 + 1) in rng (FinS (F,(X \ Y))) by FUNCT_1:def_3;
then consider d1 being Element of D such that
A49: ( d1 in dom (F | (X \ Y)) & (F | (X \ Y)) . d1 = (FinS (F,(X \ Y))) . (n1 + 1) ) by A40, PARTFUN1:3;
1 < n + 1 by A44, NAT_1:13;
then A50: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) = (FinS (F,(X \ Y))) . ((n + 1) - 1) by A45, A47, FINSEQ_1:24
.= (FinS (F,(X \ Y))) . (n1 + 1) ;
A51: n <= len (<*(F . d)*> ^ (FinS (F,(X \ Y)))) by A42, FINSEQ_3:25;
then A52: n1 <= len (FinS (F,(X \ Y))) by A46, XREAL_1:9;
A53: ( F . d1 = (FinS (F,(X \ Y))) . (n1 + 1) & F . d >= F . d1 ) by A2, A6, A11, A49, FUNCT_1:47;
now__::_thesis:_(<*(F_._d)*>_^_(FinS_(F,(X_\_Y))))_._n_>=_(<*(F_._d)*>_^_(FinS_(F,(X_\_Y))))_._(n_+_1)
percases ( n = 1 or n <> 1 ) ;
suppose n = 1 ; ::_thesis: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1)
hence (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) by A50, A53, FINSEQ_1:41; ::_thesis: verum
end;
suppose n <> 1 ; ::_thesis: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1)
then A54: 1 < n by A44, XXREAL_0:1;
then 1 + 1 <= n by NAT_1:13;
then 1 <= n1 by XREAL_1:19;
then A55: n1 in dom (FinS (F,(X \ Y))) by A52, FINSEQ_3:25;
(<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n = (FinS (F,(X \ Y))) . n1 by A45, A51, A54, FINSEQ_1:24;
hence (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) by A48, A50, A55, RFINSEQ:def_3; ::_thesis: verum
end;
end;
end;
hence (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) ; ::_thesis: verum
end;
d in {d} by TARSKI:def_1;
then d in X \ (Y \ {d}) by A30, XBOOLE_0:def_3;
then A56: d in dom (F | (X \ (Y \ {d}))) by A21, A17, XBOOLE_0:def_4;
(X \ (Y \ {d})) \ {d} = X \ ((Y \ {d}) \/ {d}) by XBOOLE_1:41
.= X \ (Y \/ {d}) by XBOOLE_1:39
.= X \ Y by A19, XBOOLE_1:12 ;
then (FinS (F,(X \ Y))) ^ <*(F . d)*>,F | (X \ (Y \ {d})) are_fiberwise_equipotent by A39, A31, A56, Th66;
then <*(F . d)*> ^ (FinS (F,(X \ Y))),F | (X \ (Y \ {d})) are_fiberwise_equipotent by A20, CLASSES1:76;
then A57: <*(F . d)*> ^ (FinS (F,(X \ Y))) = FinS (F,(X \ (Y \ {d}))) by A39, A31, A41, Def13;
{d} c= dy by A11, ZFMISC_1:31;
then card dyd = (card dy) - (card {d}) by A23, CARD_2:44
.= (n + 1) - 1 by A5, CARD_1:30
.= n ;
then FinS (F,X) = (FinS (F,(Y \ {d}))) ^ (FinS (F,(X \ (Y \ {d})))) by A1, A3, A4, A32, XBOOLE_1:1;
then A58: FinS (F,X) = ((FinS (F,(Y \ {d}))) ^ <*(F . d)*>) ^ (FinS (F,(X \ Y))) by A57, FINSEQ_1:32;
A59: (FinS (F,Y)) | n is non-increasing by RFINSEQ:20;
F | Y,(FinS (F,(Y \ {d}))) ^ <*(F . d)*> are_fiberwise_equipotent by A2, A11, Th66;
then (FinS (F,(Y \ {d}))) ^ <*(F . d)*>, FinS (F,Y) are_fiberwise_equipotent by A8, CLASSES1:76;
then FinS (F,(Y \ {d})),(FinS (F,Y)) | n are_fiberwise_equipotent by A29, RFINSEQ:1;
hence FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) by A58, A29, A59, RFINSEQ:23; ::_thesis: verum
end;
theorem :: RFUNCT_3:71
for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
proof
A1: for n being Element of NAT holds S1[n] from NAT_1:sch_1(Lm3, Lm4);
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X, Y being set st dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
let X, Y be set ; ::_thesis: ( dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) implies FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) )
assume that
A2: dom (F | X) is finite and
A3: Y c= X and
A4: for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ; ::_thesis: FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))
F | Y c= F | X by A3, RELAT_1:75;
then reconsider dFY = dom (F | Y) as finite set by A2, FINSET_1:1, RELAT_1:11;
card dFY = card dFY ;
hence FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) by A1, A2, A3, A4; ::_thesis: verum
end;
theorem Th72: :: RFUNCT_3:72
for D being non empty set
for F being PartFunc of D,REAL
for r being Real
for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds
( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d )
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r being Real
for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds
( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d )
let F be PartFunc of D,REAL; ::_thesis: for r being Real
for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds
( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d )
let r be Real; ::_thesis: for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds
( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d )
let X be set ; ::_thesis: for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds
( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d )
let d be Element of D; ::_thesis: ( dom (F | X) is finite & d in dom (F | X) implies ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) )
set dx = dom (F | X);
set drx = dom ((F - r) | X);
set frx = FinS ((F - r),X);
set fx = FinS (F,X);
assume that
A1: dom (F | X) is finite and
A2: d in dom (F | X) ; ::_thesis: ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d )
reconsider dx = dom (F | X) as finite set by A1;
A3: dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61
.= (dom F) /\ X by VALUED_1:3
.= dx by RELAT_1:61 ;
then FinS (F,X),F | X are_fiberwise_equipotent by Def13;
then A4: rng (FinS (F,X)) = rng (F | X) by CLASSES1:75;
then FinS (F,X) <> {} by A2, FUNCT_1:3, RELAT_1:38;
then 0 + 1 <= len (FinS (F,X)) by NAT_1:13;
then A5: len (FinS (F,X)) in dom (FinS (F,X)) by FINSEQ_3:25;
(F | X) . d in rng (F | X) by A2, FUNCT_1:def_3;
then F . d in rng (F | X) by A2, FUNCT_1:47;
then consider n being Nat such that
A6: n in dom (FinS (F,X)) and
A7: (FinS (F,X)) . n = F . d by A4, FINSEQ_2:10;
A8: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3;
FinS ((F - r),X),(F - r) | X are_fiberwise_equipotent by A3, Def13;
then A9: rng (FinS ((F - r),X)) = rng ((F - r) | X) by CLASSES1:75;
A10: ( len (FinS (F,X)) = card dx & dom (FinS ((F - r),X)) = Seg (len (FinS ((F - r),X))) ) by Th67, FINSEQ_1:def_3;
A11: len (FinS ((F - r),X)) = card dx by A3, Th67;
then (FinS ((F - r),X)) . (len (FinS ((F - r),X))) in rng (FinS ((F - r),X)) by A10, A8, A5, FUNCT_1:def_3;
then consider d1 being Element of D such that
A12: d1 in dom ((F - r) | X) and
A13: ((F - r) | X) . d1 = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A9, PARTFUN1:3;
(F | X) . d1 = F . d1 by A3, A12, FUNCT_1:47;
then F . d1 in rng (F | X) by A3, A12, FUNCT_1:def_3;
then consider m being Nat such that
A14: m in dom (FinS (F,X)) and
A15: (FinS (F,X)) . m = F . d1 by A4, FINSEQ_2:10;
A16: dom (F - r) = dom F by VALUED_1:3;
A17: dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61;
then A18: d1 in dom (F - r) by A12, XBOOLE_0:def_4;
A19: (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d1 by A12, A13, FUNCT_1:47
.= (F . d1) - r by A16, A18, VALUED_1:3 ;
A20: d in dom (F - r) by A2, A3, A17, XBOOLE_0:def_4;
then A21: (F - r) . d = (F . d) - r by A16, VALUED_1:3;
A22: n <= len (FinS (F,X)) by A6, FINSEQ_3:25;
thus ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d implies (FinS (F,X)) . (len (FinS (F,X))) = F . d ) ::_thesis: ( (FinS (F,X)) . (len (FinS (F,X))) = F . d implies (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d )
proof
(FinS (F,X)) . (len (FinS (F,X))) in rng (FinS (F,X)) by A5, FUNCT_1:def_3;
then consider d1 being Element of D such that
A23: d1 in dx and
A24: (F | X) . d1 = (FinS (F,X)) . (len (FinS (F,X))) by A4, PARTFUN1:3;
A25: d1 in dom (F - r) by A3, A17, A23, XBOOLE_0:def_4;
A26: F . d1 = (FinS (F,X)) . (len (FinS (F,X))) by A23, A24, FUNCT_1:47;
((F - r) | X) . d1 = (F - r) . d1 by A3, A23, FUNCT_1:47
.= (F . d1) - r by A16, A25, VALUED_1:3 ;
then (F . d1) - r in rng ((F - r) | X) by A3, A23, FUNCT_1:def_3;
then consider m being Nat such that
A27: m in dom (FinS ((F - r),X)) and
A28: (FinS ((F - r),X)) . m = (F . d1) - r by A9, FINSEQ_2:10;
A29: m <= len (FinS ((F - r),X)) by A27, FINSEQ_3:25;
assume that
A30: (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d and
A31: (FinS (F,X)) . (len (FinS (F,X))) <> F . d ; ::_thesis: contradiction
n < len (FinS (F,X)) by A7, A22, A31, XXREAL_0:1;
then A32: F . d >= F . d1 by A5, A6, A7, A26, RFINSEQ:19;
now__::_thesis:_(_(_len_(FinS_((F_-_r),X))_=_m_&_contradiction_)_or_(_len_(FinS_((F_-_r),X))_<>_m_&_contradiction_)_)
percases ( len (FinS ((F - r),X)) = m or len (FinS ((F - r),X)) <> m ) ;
case len (FinS ((F - r),X)) = m ; ::_thesis: contradiction
then (F . d1) + (- r) = (F . d) - r by A16, A20, A30, A28, VALUED_1:3;
hence contradiction by A31, A23, A24, FUNCT_1:47; ::_thesis: verum
end;
case len (FinS ((F - r),X)) <> m ; ::_thesis: contradiction
then m < len (FinS ((F - r),X)) by A29, XXREAL_0:1;
then (F . d1) - r >= (F . d) - r by A11, A10, A8, A21, A5, A30, A27, A28, RFINSEQ:19;
then F . d1 >= F . d by XREAL_1:9;
hence contradiction by A31, A26, A32, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
assume that
A33: (FinS (F,X)) . (len (FinS (F,X))) = F . d and
A34: (FinS ((F - r),X)) . (len (FinS ((F - r),X))) <> (F - r) . d ; ::_thesis: contradiction
((F - r) | X) . d in rng ((F - r) | X) by A2, A3, FUNCT_1:def_3;
then (F - r) . d in rng ((F - r) | X) by A2, A3, FUNCT_1:47;
then consider n1 being Nat such that
A35: n1 in dom (FinS ((F - r),X)) and
A36: (FinS ((F - r),X)) . n1 = (F . d) - r by A9, A21, FINSEQ_2:10;
n1 <= len (FinS ((F - r),X)) by A35, FINSEQ_3:25;
then n1 < len (FinS ((F - r),X)) by A21, A34, A36, XXREAL_0:1;
then (F . d) - r >= (F . d1) - r by A11, A10, A8, A5, A19, A35, A36, RFINSEQ:19;
then A37: F . d >= F . d1 by XREAL_1:9;
A38: m <= len (FinS (F,X)) by A14, FINSEQ_3:25;
now__::_thesis:_(_(_len_(FinS_(F,X))_=_m_&_contradiction_)_or_(_len_(FinS_(F,X))_<>_m_&_contradiction_)_)
percases ( len (FinS (F,X)) = m or len (FinS (F,X)) <> m ) ;
case len (FinS (F,X)) = m ; ::_thesis: contradiction
hence contradiction by A16, A20, A33, A34, A19, A15, VALUED_1:3; ::_thesis: verum
end;
case len (FinS (F,X)) <> m ; ::_thesis: contradiction
then m < len (FinS (F,X)) by A38, XXREAL_0:1;
then F . d1 >= F . d by A5, A33, A14, A15, RFINSEQ:19;
hence contradiction by A21, A34, A19, A37, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem Th73: :: RFUNCT_3:73
for D being non empty set
for F being PartFunc of D,REAL
for r being Real
for X being set
for Z being finite set st Z = dom (F | X) holds
FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r)
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for r being Real
for X being set
for Z being finite set st Z = dom (F | X) holds
FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r)
let F be PartFunc of D,REAL; ::_thesis: for r being Real
for X being set
for Z being finite set st Z = dom (F | X) holds
FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r)
let r be Real; ::_thesis: for X being set
for Z being finite set st Z = dom (F | X) holds
FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r)
let X be set ; ::_thesis: for Z being finite set st Z = dom (F | X) holds
FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r)
defpred S2[ Element of NAT ] means for X being set
for G being finite set st G = dom (F | X) & $1 = card G holds
FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r);
A1: for n being Element of NAT st S2[n] holds
S2[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] )
assume A2: S2[n] ; ::_thesis: S2[n + 1]
let X be set ; ::_thesis: for G being finite set st G = dom (F | X) & n + 1 = card G holds
FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r)
let G be finite set ; ::_thesis: ( G = dom (F | X) & n + 1 = card G implies FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) )
assume A3: G = dom (F | X) ; ::_thesis: ( not n + 1 = card G or FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) )
set frx = FinS ((F - r),X);
set fx = FinS (F,X);
A4: dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61
.= (dom F) /\ X by VALUED_1:3
.= dom (F | X) by RELAT_1:61 ;
then A5: len (FinS ((F - r),X)) = card G by A3, Th67;
A6: FinS ((F - r),X),(F - r) | X are_fiberwise_equipotent by A3, A4, Def13;
then A7: rng (FinS ((F - r),X)) = rng ((F - r) | X) by CLASSES1:75;
assume A8: n + 1 = card G ; ::_thesis: FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r)
then A9: len (FinS (F,X)) = n + 1 by A3, Th67;
0 + 1 <= n + 1 by NAT_1:13;
then len (FinS ((F - r),X)) in dom (FinS ((F - r),X)) by A8, A5, FINSEQ_3:25;
then (FinS ((F - r),X)) . (len (FinS ((F - r),X))) in rng (FinS ((F - r),X)) by FUNCT_1:def_3;
then consider d being Element of D such that
A10: d in dom ((F - r) | X) and
A11: ((F - r) | X) . d = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A7, PARTFUN1:3;
set Y = X \ {d};
set dx = dom (F | X);
set dy = dom (F | (X \ {d}));
set fry = FinS ((F - r),(X \ {d}));
set fy = FinS (F,(X \ {d}));
set n1r = (n + 1) |-> r;
set nr = n |-> r;
A12: {d} c= dom (F | X) by A4, A10, ZFMISC_1:31;
(F - r) . d = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A10, A11, FUNCT_1:47;
then A13: (FinS (F,X)) . (len (FinS (F,X))) = F . d by A3, A4, A10, Th72;
len (FinS (F,X)) = card G by A3, Th67;
then A14: FinS (F,X) = ((FinS (F,X)) | n) ^ <*(F . d)*> by A8, A13, RFINSEQ:7;
FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> by A3, A4, A10, A13, Th70;
then A15: FinS (F,(X \ {d})) = (FinS (F,X)) | n by A14, FINSEQ_1:33;
A16: dom ((FinS (F,(X \ {d}))) - (n |-> r)) = Seg (len ((FinS (F,(X \ {d}))) - (n |-> r))) by FINSEQ_1:def_3;
A17: dom (F | (X \ {d})) = (dom F) /\ (X \ {d}) by RELAT_1:61;
A18: dom (F | X) = (dom F) /\ X by RELAT_1:61;
A19: dom (F | (X \ {d})) = (dom (F | X)) \ {d}
proof
thus dom (F | (X \ {d})) c= (dom (F | X)) \ {d} :: according to XBOOLE_0:def_10 ::_thesis: (dom (F | X)) \ {d} c= dom (F | (X \ {d}))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (X \ {d})) or y in (dom (F | X)) \ {d} )
assume A20: y in dom (F | (X \ {d})) ; ::_thesis: y in (dom (F | X)) \ {d}
then y in X \ {d} by A17, XBOOLE_0:def_4;
then A21: not y in {d} by XBOOLE_0:def_5;
y in dom F by A17, A20, XBOOLE_0:def_4;
then y in dom (F | X) by A17, A18, A20, XBOOLE_0:def_4;
hence y in (dom (F | X)) \ {d} by A21, XBOOLE_0:def_5; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (dom (F | X)) \ {d} or y in dom (F | (X \ {d})) )
assume A22: y in (dom (F | X)) \ {d} ; ::_thesis: y in dom (F | (X \ {d}))
then A23: not y in {d} by XBOOLE_0:def_5;
A24: y in dom (F | X) by A22, XBOOLE_0:def_5;
then y in X by A18, XBOOLE_0:def_4;
then A25: y in X \ {d} by A23, XBOOLE_0:def_5;
y in dom F by A18, A24, XBOOLE_0:def_4;
hence y in dom (F | (X \ {d})) by A17, A25, XBOOLE_0:def_4; ::_thesis: verum
end;
then reconsider dx = dom (F | X), dy = dom (F | (X \ {d})) as finite set by A3;
A26: card dy = (card dx) - (card {d}) by A12, A19, CARD_2:44
.= (n + 1) - 1 by A3, A8, CARD_1:30
.= n ;
then ( len (n |-> r) = n & len (FinS (F,(X \ {d}))) = n ) by Th67, CARD_1:def_7;
then A27: len ((FinS (F,(X \ {d}))) - (n |-> r)) = n by FINSEQ_2:72;
(F - r) . d = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A10, A11, FUNCT_1:47;
then A28: FinS ((F - r),X) = ((FinS ((F - r),X)) | n) ^ <*((F - r) . d)*> by A8, A5, RFINSEQ:7;
(FinS ((F - r),(X \ {d}))) ^ <*((F - r) . d)*>,(F - r) | X are_fiberwise_equipotent by A3, A4, A10, Th66;
then (FinS ((F - r),(X \ {d}))) ^ <*((F - r) . d)*>, FinS ((F - r),X) are_fiberwise_equipotent by A6, CLASSES1:76;
then ( (FinS ((F - r),X)) | n is non-increasing & FinS ((F - r),(X \ {d})),(FinS ((F - r),X)) | n are_fiberwise_equipotent ) by A28, RFINSEQ:1, RFINSEQ:20;
then A29: FinS ((F - r),(X \ {d})) = (FinS ((F - r),X)) | n by RFINSEQ:23;
len ((n + 1) |-> r) = n + 1 by CARD_1:def_7;
then A30: len ((FinS (F,X)) - ((n + 1) |-> r)) = n + 1 by A9, FINSEQ_2:72;
then A31: dom ((FinS (F,X)) - ((n + 1) |-> r)) = Seg (n + 1) by FINSEQ_1:def_3;
dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61;
then d in dom (F - r) by A10, XBOOLE_0:def_4;
then d in dom F by VALUED_1:3;
then (F - r) . d = (F . d) - r by VALUED_1:3;
then A32: <*((F - r) . d)*> = <*(F . d)*> - <*r*> by RVSUM_1:29;
A33: n < n + 1 by NAT_1:13;
A34: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3;
( len <*(F . d)*> = 1 & len <*r*> = 1 ) by FINSEQ_1:40;
then A35: len (<*(F . d)*> - <*r*>) = 1 by FINSEQ_2:72;
then A36: len (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) = n + 1 by A27, FINSEQ_1:22;
1 in Seg 1 by FINSEQ_1:1;
then A37: 1 in dom (<*(F . d)*> - <*r*>) by A35, FINSEQ_1:def_3;
A38: ( <*(F . d)*> . 1 = F . d & <*r*> . 1 = r ) by FINSEQ_1:40;
A39: now__::_thesis:_for_m_being_Nat_st_m_in_dom_((FinS_(F,X))_-_((n_+_1)_|->_r))_holds_
((FinS_(F,X))_-_((n_+_1)_|->_r))_._m_=_(((FinS_(F,(X_\_{d})))_-_(n_|->_r))_^_(<*(F_._d)*>_-_<*r*>))_._m
let m be Nat; ::_thesis: ( m in dom ((FinS (F,X)) - ((n + 1) |-> r)) implies ((FinS (F,X)) - ((n + 1) |-> r)) . b1 = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1 )
assume A40: m in dom ((FinS (F,X)) - ((n + 1) |-> r)) ; ::_thesis: ((FinS (F,X)) - ((n + 1) |-> r)) . b1 = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1
percases ( m = n + 1 or m <> n + 1 ) ;
supposeA41: m = n + 1 ; ::_thesis: ((FinS (F,X)) - ((n + 1) |-> r)) . b1 = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1
then A42: ((n + 1) |-> r) . m = r by FINSEQ_1:4, FUNCOP_1:7;
(((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m = (<*(F . d)*> - <*r*>) . ((n + 1) - n) by A27, A36, A33, A41, FINSEQ_1:24
.= (F . d) - r by A38, A37, VALUED_1:13 ;
hence ((FinS (F,X)) - ((n + 1) |-> r)) . m = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m by A13, A9, A40, A41, A42, VALUED_1:13; ::_thesis: verum
end;
supposeA43: m <> n + 1 ; ::_thesis: (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1 = ((FinS (F,X)) - ((n + 1) |-> r)) . b1
m <= n + 1 by A31, A40, FINSEQ_1:1;
then m < n + 1 by A43, XXREAL_0:1;
then A44: m <= n by NAT_1:13;
reconsider s = (FinS (F,X)) . m as Real ;
A45: n <= n + 1 by NAT_1:11;
A46: ((n + 1) |-> r) . m = r by A31, A40, FUNCOP_1:7;
A47: 1 <= m by A31, A40, FINSEQ_1:1;
then A48: m in Seg n by A44, FINSEQ_1:1;
then A49: m in dom ((FinS (F,(X \ {d}))) - (n |-> r)) by A27, FINSEQ_1:def_3;
1 <= n by A47, A44, XXREAL_0:2;
then n in Seg (n + 1) by A45, FINSEQ_1:1;
then A50: ((FinS (F,X)) | n) . m = (FinS (F,X)) . m by A9, A34, A48, RFINSEQ:6;
( (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m = ((FinS (F,(X \ {d}))) - (n |-> r)) . m & (n |-> r) . m = r ) by A27, A16, A48, FINSEQ_1:def_7, FUNCOP_1:7;
hence (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m = s - r by A15, A50, A49, VALUED_1:13
.= ((FinS (F,X)) - ((n + 1) |-> r)) . m by A40, A46, VALUED_1:13 ;
::_thesis: verum
end;
end;
end;
FinS ((F - r),(X \ {d})) = (FinS (F,(X \ {d}))) - (n |-> r) by A2, A26;
hence FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) by A8, A28, A29, A32, A30, A36, A39, FINSEQ_2:9; ::_thesis: verum
end;
A51: S2[ 0 ]
proof
let X be set ; ::_thesis: for G being finite set st G = dom (F | X) & 0 = card G holds
FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r)
let G be finite set ; ::_thesis: ( G = dom (F | X) & 0 = card G implies FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) )
assume A52: G = dom (F | X) ; ::_thesis: ( not 0 = card G or FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) )
assume 0 = card G ; ::_thesis: FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r)
then A53: dom (F | X) = {} by A52;
then FinS (F,X) = FinS (F,{}) by Th63
.= <*> REAL by Th68 ;
then A54: (FinS (F,X)) - ((card G) |-> r) = <*> REAL by FINSEQ_2:73;
dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61
.= (dom F) /\ X by VALUED_1:3
.= dom (F | X) by RELAT_1:61 ;
hence FinS ((F - r),X) = FinS ((F - r),{}) by A53, Th63
.= (FinS (F,X)) - ((card G) |-> r) by A54, Th68 ;
::_thesis: verum
end;
A55: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A51, A1);
let G be finite set ; ::_thesis: ( G = dom (F | X) implies FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) )
assume G = dom (F | X) ; ::_thesis: FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r)
hence FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) by A55; ::_thesis: verum
end;
theorem :: RFUNCT_3:74
for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds
F . d >= 0 ) holds
FinS ((max+ F),X) = FinS (F,X)
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds
F . d >= 0 ) holds
FinS ((max+ F),X) = FinS (F,X)
let F be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds
F . d >= 0 ) holds
FinS ((max+ F),X) = FinS (F,X)
let X be set ; ::_thesis: ( dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds
F . d >= 0 ) implies FinS ((max+ F),X) = FinS (F,X) )
assume that
A1: dom (F | X) is finite and
A2: for d being Element of D st d in dom (F | X) holds
F . d >= 0 ; ::_thesis: FinS ((max+ F),X) = FinS (F,X)
now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_(F_|_X)_holds_
(F_|_X)_._d_>=_0
let d be Element of D; ::_thesis: ( d in dom (F | X) implies (F | X) . d >= 0 )
assume A3: d in dom (F | X) ; ::_thesis: (F | X) . d >= 0
then F . d >= 0 by A2;
hence (F | X) . d >= 0 by A3, FUNCT_1:47; ::_thesis: verum
end;
then A4: F | X = max+ (F | X) by Th46
.= (max+ F) | X by Th44 ;
hence FinS (F,X) = FinS (((max+ F) | X),X) by A1, Th64
.= FinS ((max+ F),X) by A1, A4, Th64 ;
::_thesis: verum
end;
theorem :: RFUNCT_3:75
for D being non empty set
for F being PartFunc of D,REAL
for X being set
for r being Real
for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds
FinS (F,X) = (card Z) |-> r
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set
for r being Real
for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds
FinS (F,X) = (card Z) |-> r
let F be PartFunc of D,REAL; ::_thesis: for X being set
for r being Real
for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds
FinS (F,X) = (card Z) |-> r
let X be set ; ::_thesis: for r being Real
for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds
FinS (F,X) = (card Z) |-> r
let r be Real; ::_thesis: for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds
FinS (F,X) = (card Z) |-> r
let dx be finite set ; ::_thesis: ( dx = dom (F | X) & rng (F | X) = {r} implies FinS (F,X) = (card dx) |-> r )
assume A1: dx = dom (F | X) ; ::_thesis: ( not rng (F | X) = {r} or FinS (F,X) = (card dx) |-> r )
set fx = FinS (F,X);
assume A2: rng (F | X) = {r} ; ::_thesis: FinS (F,X) = (card dx) |-> r
F | X, FinS (F,X) are_fiberwise_equipotent by A1, Def13;
then A3: rng (FinS (F,X)) = {r} by A2, CLASSES1:75;
A4: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3;
len (FinS (F,X)) = card dx by A1, Th67;
hence FinS (F,X) = (card dx) |-> r by A3, A4, FUNCOP_1:9; ::_thesis: verum
end;
theorem Th76: :: RFUNCT_3:76
for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
let X, Y be set ; ::_thesis: ( dom (F | (X \/ Y)) is finite & X misses Y implies FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent )
assume A1: dom (F | (X \/ Y)) is finite ; ::_thesis: ( not X misses Y or FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent )
F | Y c= F | (X \/ Y) by RELAT_1:75, XBOOLE_1:7;
then reconsider dfy = dom (F | Y) as finite set by A1, FINSET_1:1, RELAT_1:11;
defpred S2[ Element of NAT ] means for Y being set
for Z being finite set st Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & $1 = card Z holds
FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ;
A2: card dfy = card dfy ;
A3: for n being Element of NAT st S2[n] holds
S2[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] )
assume A4: S2[n] ; ::_thesis: S2[n + 1]
let Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & n + 1 = card Z holds
FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
let Z be finite set ; ::_thesis: ( Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & n + 1 = card Z implies FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent )
assume that
A5: Z = dom (F | Y) and
A6: dom (F | (X \/ Y)) is finite and
A7: X /\ Y = {} and
A8: n + 1 = card Z ; ::_thesis: FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
set x = the Element of dom (F | Y);
reconsider x = the Element of dom (F | Y) as Element of D by A5, A8, CARD_1:27, TARSKI:def_3;
set y1 = Y \ {x};
A9: dom (F | Y) = (dom F) /\ Y by RELAT_1:61;
now__::_thesis:_not_x_in_X
assume A10: x in X ; ::_thesis: contradiction
x in Y by A5, A8, A9, CARD_1:27, XBOOLE_0:def_4;
hence contradiction by A7, A10, XBOOLE_0:def_4; ::_thesis: verum
end;
then X \ {x} = X by ZFMISC_1:57;
then A11: (X \/ Y) \ {x} = X \/ (Y \ {x}) by XBOOLE_1:42;
A12: dom (F | (Y \ {x})) = (dom F) /\ (Y \ {x}) by RELAT_1:61;
A13: dom (F | (Y \ {x})) = (dom (F | Y)) \ {x}
proof
thus dom (F | (Y \ {x})) c= (dom (F | Y)) \ {x} :: according to XBOOLE_0:def_10 ::_thesis: (dom (F | Y)) \ {x} c= dom (F | (Y \ {x}))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (Y \ {x})) or y in (dom (F | Y)) \ {x} )
assume A14: y in dom (F | (Y \ {x})) ; ::_thesis: y in (dom (F | Y)) \ {x}
then y in Y \ {x} by A12, XBOOLE_0:def_4;
then A15: not y in {x} by XBOOLE_0:def_5;
y in dom F by A12, A14, XBOOLE_0:def_4;
then y in dom (F | Y) by A12, A9, A14, XBOOLE_0:def_4;
hence y in (dom (F | Y)) \ {x} by A15, XBOOLE_0:def_5; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (dom (F | Y)) \ {x} or y in dom (F | (Y \ {x})) )
assume A16: y in (dom (F | Y)) \ {x} ; ::_thesis: y in dom (F | (Y \ {x}))
then A17: not y in {x} by XBOOLE_0:def_5;
A18: y in dom (F | Y) by A16, XBOOLE_0:def_5;
then y in Y by A9, XBOOLE_0:def_4;
then A19: y in Y \ {x} by A17, XBOOLE_0:def_5;
y in dom F by A9, A18, XBOOLE_0:def_4;
hence y in dom (F | (Y \ {x})) by A12, A19, XBOOLE_0:def_4; ::_thesis: verum
end;
then reconsider dFy = dom (F | (Y \ {x})) as finite set by A5;
{x} c= dom (F | Y) by A5, A8, CARD_1:27, ZFMISC_1:31;
then A20: card dFy = (n + 1) - (card {x}) by A5, A8, A13, CARD_2:44
.= (n + 1) - 1 by CARD_1:30
.= n ;
X \/ (Y \ {x}) c= X \/ Y by XBOOLE_1:13;
then (dom F) /\ (X \/ (Y \ {x})) c= (dom F) /\ (X \/ Y) by XBOOLE_1:27;
then dom (F | (X \/ (Y \ {x}))) c= (dom F) /\ (X \/ Y) by RELAT_1:61;
then A21: dom (F | (X \/ (Y \ {x}))) c= dom (F | (X \/ Y)) by RELAT_1:61;
A22: FinS (F,(X \/ Y)),F | (X \/ Y) are_fiberwise_equipotent by A6, Def13;
dom (F | (X \/ Y)) = (dom F) /\ (X \/ Y) by RELAT_1:61
.= ((dom F) /\ X) \/ ((dom F) /\ Y) by XBOOLE_1:23
.= (dom (F | X)) \/ ((dom F) /\ Y) by RELAT_1:61
.= (dom (F | X)) \/ (dom (F | Y)) by RELAT_1:61 ;
then x in dom (F | (X \/ Y)) by A5, A8, CARD_1:27, XBOOLE_0:def_3;
then (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,F | (X \/ Y) are_fiberwise_equipotent by A6, A11, Th66;
then A23: (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>, FinS (F,(X \/ Y)) are_fiberwise_equipotent by A22, CLASSES1:76;
X /\ (Y \ {x}) c= X /\ Y by XBOOLE_1:27;
then FinS (F,(X \/ (Y \ {x}))),(FinS (F,X)) ^ (FinS (F,(Y \ {x}))) are_fiberwise_equipotent by A4, A6, A7, A21, A20, XBOOLE_1:3;
then (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,((FinS (F,X)) ^ (FinS (F,(Y \ {x})))) ^ <*(F . x)*> are_fiberwise_equipotent by RFINSEQ:1;
then A24: (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,(FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>) are_fiberwise_equipotent by FINSEQ_1:32;
( (FinS (F,(Y \ {x}))) ^ <*(F . x)*>,F | Y are_fiberwise_equipotent & FinS (F,Y),F | Y are_fiberwise_equipotent ) by A5, A8, Def13, Th66, CARD_1:27;
then (FinS (F,(Y \ {x}))) ^ <*(F . x)*>, FinS (F,Y) are_fiberwise_equipotent by CLASSES1:76;
then A25: ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>) ^ (FinS (F,X)),(FinS (F,Y)) ^ (FinS (F,X)) are_fiberwise_equipotent by RFINSEQ:1;
(FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>),((FinS (F,(Y \ {x}))) ^ <*(F . x)*>) ^ (FinS (F,X)) are_fiberwise_equipotent by RFINSEQ:2;
then ( (FinS (F,Y)) ^ (FinS (F,X)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent & (FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>),(FinS (F,Y)) ^ (FinS (F,X)) are_fiberwise_equipotent ) by A25, CLASSES1:76, RFINSEQ:2;
then (FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by CLASSES1:76;
then (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by A24, CLASSES1:76;
hence FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by A23, CLASSES1:76; ::_thesis: verum
end;
A26: S2[ 0 ]
proof
let Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & 0 = card Z holds
FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
let Z be finite set ; ::_thesis: ( Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & 0 = card Z implies FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent )
assume that
A27: Z = dom (F | Y) and
A28: dom (F | (X \/ Y)) is finite and
X /\ Y = {} and
A29: 0 = card Z ; ::_thesis: FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
A30: dom (F | (X \/ Y)) = (dom F) /\ (X \/ Y) by RELAT_1:61
.= ((dom F) /\ X) \/ ((dom F) /\ Y) by XBOOLE_1:23
.= (dom (F | X)) \/ ((dom F) /\ Y) by RELAT_1:61
.= (dom (F | X)) \/ (dom (F | Y)) by RELAT_1:61 ;
then A31: dom (F | X) is finite by A28, FINSET_1:1, XBOOLE_1:7;
A32: dom (F | Y) = {} by A27, A29;
then FinS (F,(X \/ Y)) = FinS (F,(dom (F | X))) by A28, A30, Th63
.= FinS (F,X) by A31, Th63
.= (FinS (F,X)) ^ (<*> REAL) by FINSEQ_1:34
.= (FinS (F,X)) ^ (FinS (F,(dom (F | Y)))) by A32, Th68
.= (FinS (F,X)) ^ (FinS (F,Y)) by A27, Th63 ;
hence FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ; ::_thesis: verum
end;
A33: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A26, A3);
assume X /\ Y = {} ; :: according to XBOOLE_0:def_7 ::_thesis: FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent
hence FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by A1, A33, A2; ::_thesis: verum
end;
definition
let D be non empty set ;
let F be PartFunc of D,REAL;
let X be set ;
func Sum (F,X) -> Real equals :: RFUNCT_3:def 14
Sum (FinS (F,X));
correctness
coherence
Sum (FinS (F,X)) is Real;
;
end;
:: deftheorem defines Sum RFUNCT_3:def_14_:_
for D being non empty set
for F being PartFunc of D,REAL
for X being set holds Sum (F,X) = Sum (FinS (F,X));
theorem Th77: :: RFUNCT_3:77
for D being non empty set
for F being PartFunc of D,REAL
for X being set
for r being Real st dom (F | X) is finite holds
Sum ((r (#) F),X) = r * (Sum (F,X))
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set
for r being Real st dom (F | X) is finite holds
Sum ((r (#) F),X) = r * (Sum (F,X))
let F be PartFunc of D,REAL; ::_thesis: for X being set
for r being Real st dom (F | X) is finite holds
Sum ((r (#) F),X) = r * (Sum (F,X))
let X be set ; ::_thesis: for r being Real st dom (F | X) is finite holds
Sum ((r (#) F),X) = r * (Sum (F,X))
let r be Real; ::_thesis: ( dom (F | X) is finite implies Sum ((r (#) F),X) = r * (Sum (F,X)) )
set x = dom (F | X);
assume A1: dom (F | X) is finite ; ::_thesis: Sum ((r (#) F),X) = r * (Sum (F,X))
then reconsider FX = F | X as finite Function by FINSET_1:10;
dom ((r (#) F) | X) = dom (r (#) (F | X)) by RFUNCT_1:49
.= dom (F | X) by VALUED_1:def_5 ;
then reconsider rFX = (r (#) F) | X as finite Function by A1, FINSET_1:10;
consider b being FinSequence such that
A2: F | (dom (F | X)),b are_fiberwise_equipotent by A1, RFINSEQ:5;
rng (F | (dom (F | X))) = rng b by A2, CLASSES1:75;
then reconsider b = b as FinSequence of REAL by FINSEQ_1:def_4;
dom (F | X) = (dom F) /\ X by RELAT_1:61;
then A3: F | (dom (F | X)) = (F | (dom F)) | X by RELAT_1:71
.= F | X by RELAT_1:68 ;
then A4: rng b = rng (F | X) by A2, CLASSES1:75;
A5: now__::_thesis:_for_x_being_Real_holds_card_(Coim_((r_*_b),x))_=_card_(Coim_(rFX,x))
let x be Real; ::_thesis: card (Coim ((r * b),x)) = card (Coim (rFX,x))
A6: len (r * b) = len b by FINSEQ_2:33;
now__::_thesis:_(_(_not_x_in_rng_(r_*_b)_&_card_((r_*_b)_"_{x})_=_card_(rFX_"_{x})_)_or_(_x_in_rng_(r_*_b)_&_card_((r_*_b)_"_{x})_=_card_(rFX_"_{x})_)_)
percases ( not x in rng (r * b) or x in rng (r * b) ) ;
caseA7: not x in rng (r * b) ; ::_thesis: card ((r * b) " {x}) = card (rFX " {x})
A8: now__::_thesis:_not_x_in_rng_((r_(#)_F)_|_X)
assume x in rng ((r (#) F) | X) ; ::_thesis: contradiction
then x in rng (r (#) (F | X)) by RFUNCT_1:49;
then consider d being Element of D such that
A9: d in dom (r (#) (F | X)) and
A10: (r (#) (F | X)) . d = x by PARTFUN1:3;
d in dom (F | X) by A9, VALUED_1:def_5;
then (F | X) . d in rng (F | X) by FUNCT_1:def_3;
then consider n being Nat such that
A11: n in dom b and
A12: b . n = (F | X) . d by A4, FINSEQ_2:10;
x = r * ((F | X) . d) by A9, A10, VALUED_1:def_5;
then A13: x = (r * b) . n by A12, RVSUM_1:44;
n in dom (r * b) by A6, A11, FINSEQ_3:29;
hence contradiction by A7, A13, FUNCT_1:def_3; ::_thesis: verum
end;
(r * b) " {x} = {} by A7, Lm2;
hence card ((r * b) " {x}) = card (rFX " {x}) by A8, Lm2; ::_thesis: verum
end;
case x in rng (r * b) ; ::_thesis: card ((r * b) " {x}) = card (rFX " {x})
then consider n being Nat such that
n in dom (r * b) and
A14: (r * b) . n = x by FINSEQ_2:10;
reconsider p = b . n as Real ;
A15: x = r * p by A14, RVSUM_1:44;
now__::_thesis:_(_(_r_=_0_&_card_((r_*_b)_"_{x})_=_card_(rFX_"_{x})_)_or_(_r_<>_0_&_card_(Coim_((r_*_b),x))_=_card_(Coim_(rFX,x))_)_)
percases ( r = 0 or r <> 0 ) ;
caseA16: r = 0 ; ::_thesis: card ((r * b) " {x}) = card (rFX " {x})
then A17: (r * b) " {x} = dom b by A15, RFINSEQ:25;
dom FX = (r (#) (F | X)) " {x} by A15, A16, Th7
.= ((r (#) F) | X) " {x} by RFUNCT_1:49 ;
hence card ((r * b) " {x}) = card (rFX " {x}) by A2, A3, A17, CLASSES1:81; ::_thesis: verum
end;
caseA18: r <> 0 ; ::_thesis: card (Coim ((r * b),x)) = card (Coim (rFX,x))
then A19: Coim ((r * b),x) = Coim (b,(x / r)) by RFINSEQ:24;
Coim (((r (#) F) | X),x) = (r (#) (F | X)) " {x} by RFUNCT_1:49
.= Coim (FX,(x / r)) by A18, Th6 ;
hence card (Coim ((r * b),x)) = card (Coim (rFX,x)) by A2, A3, A19, CLASSES1:def_9; ::_thesis: verum
end;
end;
end;
hence card ((r * b) " {x}) = card (rFX " {x}) ; ::_thesis: verum
end;
end;
end;
hence card (Coim ((r * b),x)) = card (Coim (rFX,x)) ; ::_thesis: verum
end;
( rng (r * b) c= REAL & rng ((r (#) F) | X) c= REAL ) ;
then A20: r * b,(r (#) F) | X are_fiberwise_equipotent by A5, CLASSES1:79;
F | X, FinS (F,X) are_fiberwise_equipotent by A1, Def13;
then A21: Sum b = Sum (F,X) by A2, A3, CLASSES1:76, RFINSEQ:9;
dom ((r (#) F) | X) = (dom (r (#) F)) /\ X by RELAT_1:61
.= (dom F) /\ X by VALUED_1:def_5
.= dom (F | X) by RELAT_1:61 ;
then (r (#) F) | X, FinS ((r (#) F),X) are_fiberwise_equipotent by A1, Def13;
hence Sum ((r (#) F),X) = Sum (r * b) by A20, CLASSES1:76, RFINSEQ:9
.= r * (Sum (F,X)) by A21, RVSUM_1:87 ;
::_thesis: verum
end;
theorem Th78: :: RFUNCT_3:78
for D being non empty set
for F, G being PartFunc of D,REAL
for X being set
for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
proof
let D be non empty set ; ::_thesis: for F, G being PartFunc of D,REAL
for X being set
for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
let F, G be PartFunc of D,REAL; ::_thesis: for X being set
for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
let X be set ; ::_thesis: for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
let Y be finite set ; ::_thesis: ( Y = dom (F | X) & dom (F | X) = dom (G | X) implies Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) )
assume A1: Y = dom (F | X) ; ::_thesis: ( not dom (F | X) = dom (G | X) or Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) )
defpred S2[ Element of NAT ] means for F, G being PartFunc of D,REAL
for X being set
for Y being finite set st card Y = $1 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X));
A2: card Y = card Y ;
A3: for n being Element of NAT st S2[n] holds
S2[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] )
assume A4: S2[n] ; ::_thesis: S2[n + 1]
let F, G be PartFunc of D,REAL; ::_thesis: for X being set
for Y being finite set st card Y = n + 1 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
let X be set ; ::_thesis: for Y being finite set st card Y = n + 1 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
let dx be finite set ; ::_thesis: ( card dx = n + 1 & dx = dom (F | X) & dom (F | X) = dom (G | X) implies Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) )
set gx = dom (G | X);
assume that
A5: card dx = n + 1 and
A6: dx = dom (F | X) and
A7: dom (F | X) = dom (G | X) ; ::_thesis: Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
set x = the Element of dx;
reconsider x = the Element of dx as Element of D by A5, A6, CARD_1:27, TARSKI:def_3;
A8: dx = (dom F) /\ X by A6, RELAT_1:61;
then A9: x in dom F by A5, CARD_1:27, XBOOLE_0:def_4;
set Y = X \ {x};
set dy = dom (F | (X \ {x}));
set gy = dom (G | (X \ {x}));
A10: dom (G | X) = (dom G) /\ X by RELAT_1:61;
then x in dom G by A5, A6, A7, CARD_1:27, XBOOLE_0:def_4;
then x in (dom F) /\ (dom G) by A9, XBOOLE_0:def_4;
then A11: x in dom (F + G) by VALUED_1:def_1;
A12: dom (F | (X \ {x})) = (dom F) /\ (X \ {x}) by RELAT_1:61;
A13: dom (F | (X \ {x})) = dx \ {x}
proof
thus dom (F | (X \ {x})) c= dx \ {x} :: according to XBOOLE_0:def_10 ::_thesis: dx \ {x} c= dom (F | (X \ {x}))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (X \ {x})) or y in dx \ {x} )
assume A14: y in dom (F | (X \ {x})) ; ::_thesis: y in dx \ {x}
then y in X \ {x} by A12, XBOOLE_0:def_4;
then A15: not y in {x} by XBOOLE_0:def_5;
y in dom F by A12, A14, XBOOLE_0:def_4;
then y in dx by A12, A8, A14, XBOOLE_0:def_4;
hence y in dx \ {x} by A15, XBOOLE_0:def_5; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dx \ {x} or y in dom (F | (X \ {x})) )
assume A16: y in dx \ {x} ; ::_thesis: y in dom (F | (X \ {x}))
then ( not y in {x} & y in X ) by A8, XBOOLE_0:def_4, XBOOLE_0:def_5;
then A17: y in X \ {x} by XBOOLE_0:def_5;
y in dom F by A8, A16, XBOOLE_0:def_4;
hence y in dom (F | (X \ {x})) by A12, A17, XBOOLE_0:def_4; ::_thesis: verum
end;
then reconsider dy = dom (F | (X \ {x})) as finite set ;
A18: dom (G | (X \ {x})) = (dom G) /\ (X \ {x}) by RELAT_1:61;
A19: dy = dom (G | (X \ {x}))
proof
thus dy c= dom (G | (X \ {x})) :: according to XBOOLE_0:def_10 ::_thesis: dom (G | (X \ {x})) c= dy
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dy or y in dom (G | (X \ {x})) )
assume A20: y in dy ; ::_thesis: y in dom (G | (X \ {x}))
then y in dom F by A12, XBOOLE_0:def_4;
then y in dom (G | X) by A6, A7, A12, A8, A20, XBOOLE_0:def_4;
then A21: y in dom G by A10, XBOOLE_0:def_4;
y in X \ {x} by A12, A20, XBOOLE_0:def_4;
hence y in dom (G | (X \ {x})) by A18, A21, XBOOLE_0:def_4; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (G | (X \ {x})) or y in dy )
assume A22: y in dom (G | (X \ {x})) ; ::_thesis: y in dy
then y in dom G by A18, XBOOLE_0:def_4;
then y in dx by A6, A7, A18, A10, A22, XBOOLE_0:def_4;
then A23: y in dom F by A8, XBOOLE_0:def_4;
y in X \ {x} by A18, A22, XBOOLE_0:def_4;
hence y in dy by A12, A23, XBOOLE_0:def_4; ::_thesis: verum
end;
{x} c= dx by A5, CARD_1:27, ZFMISC_1:31;
then card dy = (card dx) - (card {x}) by A13, CARD_2:44
.= (n + 1) - 1 by A5, CARD_1:30
.= n ;
then A24: Sum ((F + G),(X \ {x})) = (Sum (F,(X \ {x}))) + (Sum (G,(X \ {x}))) by A4, A19;
A25: dom ((F + G) | X) = dom ((F | X) + (G | X)) by RFUNCT_1:44
.= dx /\ (dom (G | X)) by A6, VALUED_1:def_1 ;
then A26: FinS ((F + G),X),(F + G) | X are_fiberwise_equipotent by Def13;
x in X by A5, A8, CARD_1:27, XBOOLE_0:def_4;
then x in (dom (F + G)) /\ X by A11, XBOOLE_0:def_4;
then x in dom ((F + G) | X) by RELAT_1:61;
then A27: (FinS ((F + G),(X \ {x}))) ^ <*((F + G) . x)*>,(F + G) | X are_fiberwise_equipotent by A25, Th66;
( (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent & FinS (F,X),F | X are_fiberwise_equipotent ) by A5, A6, Def13, Th66, CARD_1:27;
then A28: Sum (F,X) = Sum ((FinS (F,(X \ {x}))) ^ <*(F . x)*>) by CLASSES1:76, RFINSEQ:9
.= (Sum (F,(X \ {x}))) + (F . x) by RVSUM_1:74 ;
( (FinS (G,(X \ {x}))) ^ <*(G . x)*>,G | X are_fiberwise_equipotent & FinS (G,X),G | X are_fiberwise_equipotent ) by A5, A6, A7, Def13, Th66, CARD_1:27;
then Sum (G,X) = Sum ((FinS (G,(X \ {x}))) ^ <*(G . x)*>) by CLASSES1:76, RFINSEQ:9
.= (Sum (G,(X \ {x}))) + (G . x) by RVSUM_1:74 ;
hence (Sum (F,X)) + (Sum (G,X)) = (Sum (FinS ((F + G),(X \ {x})))) + ((F . x) + (G . x)) by A24, A28
.= (Sum (FinS ((F + G),(X \ {x})))) + ((F + G) . x) by A11, VALUED_1:def_1
.= Sum ((FinS ((F + G),(X \ {x}))) ^ <*((F + G) . x)*>) by RVSUM_1:74
.= Sum ((F + G),X) by A27, A26, CLASSES1:76, RFINSEQ:9 ;
::_thesis: verum
end;
A29: S2[ 0 ]
proof
let F, G be PartFunc of D,REAL; ::_thesis: for X being set
for Y being finite set st card Y = 0 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
let X be set ; ::_thesis: for Y being finite set st card Y = 0 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
let Y be finite set ; ::_thesis: ( card Y = 0 & Y = dom (F | X) & dom (F | X) = dom (G | X) implies Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) )
assume that
A30: card Y = 0 and
A31: Y = dom (F | X) and
A32: dom (F | X) = dom (G | X) ; ::_thesis: Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
dom (F | X) = {} by A30, A31;
then A33: rng (F | X) = {} by RELAT_1:42;
(F + G) | X = (F | X) + (G | X) by RFUNCT_1:44;
then dom ((F + G) | X) = (dom (F | X)) /\ (dom (G | X)) by VALUED_1:def_1
.= {} by A30, A31, A32 ;
then ( rng ((F + G) | X) = {} & FinS ((F + G),X),(F + G) | X are_fiberwise_equipotent ) by Def13, RELAT_1:42;
then A34: rng (FinS ((F + G),X)) = {} by CLASSES1:75;
FinS (F,X),F | X are_fiberwise_equipotent by A31, Def13;
then rng (FinS (F,X)) = {} by A33, CLASSES1:75;
then A35: Sum (F,X) = 0 by RELAT_1:41, RVSUM_1:72;
dom (G | X) = {} by A30, A31, A32;
then A36: rng (G | X) = {} by RELAT_1:42;
FinS (G,X),G | X are_fiberwise_equipotent by A31, A32, Def13;
then rng (FinS (G,X)) = {} by A36, CLASSES1:75;
then (Sum (F,X)) + (Sum (G,X)) = 0 + 0 by A35, RELAT_1:41, RVSUM_1:72;
hence Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) by A34, RELAT_1:41, RVSUM_1:72; ::_thesis: verum
end;
A37: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A29, A3);
assume dom (F | X) = dom (G | X) ; ::_thesis: Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))
hence Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) by A1, A37, A2; ::_thesis: verum
end;
theorem :: RFUNCT_3:79
for D being non empty set
for F, G being PartFunc of D,REAL
for X being set st dom (F | X) is finite & dom (F | X) = dom (G | X) holds
Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X))
proof
let D be non empty set ; ::_thesis: for F, G being PartFunc of D,REAL
for X being set st dom (F | X) is finite & dom (F | X) = dom (G | X) holds
Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X))
let F, G be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite & dom (F | X) = dom (G | X) holds
Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X))
let X be set ; ::_thesis: ( dom (F | X) is finite & dom (F | X) = dom (G | X) implies Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X)) )
assume A1: ( dom (F | X) is finite & dom (F | X) = dom (G | X) ) ; ::_thesis: Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X))
dom (((- 1) (#) G) | X) = (dom ((- 1) (#) G)) /\ X by RELAT_1:61
.= (dom G) /\ X by VALUED_1:def_5
.= dom (G | X) by RELAT_1:61 ;
hence Sum ((F - G),X) = (Sum (F,X)) + (Sum (((- 1) (#) G),X)) by A1, Th78
.= (Sum (F,X)) + ((- 1) * (Sum (G,X))) by A1, Th77
.= (Sum (F,X)) - (Sum (G,X)) ;
::_thesis: verum
end;
theorem :: RFUNCT_3:80
for D being non empty set
for F being PartFunc of D,REAL
for X being set
for r being Real
for Y being finite set st Y = dom (F | X) holds
Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y))
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X being set
for r being Real
for Y being finite set st Y = dom (F | X) holds
Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y))
let F be PartFunc of D,REAL; ::_thesis: for X being set
for r being Real
for Y being finite set st Y = dom (F | X) holds
Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y))
let X be set ; ::_thesis: for r being Real
for Y being finite set st Y = dom (F | X) holds
Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y))
let r be Real; ::_thesis: for Y being finite set st Y = dom (F | X) holds
Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y))
set fx = FinS (F,X);
let Y be finite set ; ::_thesis: ( Y = dom (F | X) implies Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) )
set dr = (card Y) |-> r;
assume A1: Y = dom (F | X) ; ::_thesis: Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y))
then len (FinS (F,X)) = card Y by Th67;
then reconsider xf = FinS (F,X), rd = (card Y) |-> r as Element of (card Y) -tuples_on REAL by FINSEQ_2:92;
FinS ((F - r),X) = (FinS (F,X)) - ((card Y) |-> r) by A1, Th73;
hence Sum ((F - r),X) = (Sum xf) - (Sum rd) by RVSUM_1:90
.= (Sum (F,X)) - (r * (card Y)) by RVSUM_1:80 ;
::_thesis: verum
end;
theorem :: RFUNCT_3:81
for D being non empty set
for F being PartFunc of D,REAL holds Sum (F,{}) = 0 by Th68, RVSUM_1:72;
theorem :: RFUNCT_3:82
for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D st d in dom F holds
Sum (F,{d}) = F . d
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for d being Element of D st d in dom F holds
Sum (F,{d}) = F . d
let F be PartFunc of D,REAL; ::_thesis: for d being Element of D st d in dom F holds
Sum (F,{d}) = F . d
let d be Element of D; ::_thesis: ( d in dom F implies Sum (F,{d}) = F . d )
assume d in dom F ; ::_thesis: Sum (F,{d}) = F . d
hence Sum (F,{d}) = Sum <*(F . d)*> by Th69
.= F . d by FINSOP_1:11 ;
::_thesis: verum
end;
theorem :: RFUNCT_3:83
for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
let X, Y be set ; ::_thesis: ( dom (F | (X \/ Y)) is finite & X misses Y implies Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) )
assume ( dom (F | (X \/ Y)) is finite & X misses Y ) ; ::_thesis: Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
hence Sum (F,(X \/ Y)) = Sum ((FinS (F,X)) ^ (FinS (F,Y))) by Th76, RFINSEQ:9
.= (Sum (F,X)) + (Sum (F,Y)) by RVSUM_1:75 ;
::_thesis: verum
end;
theorem :: RFUNCT_3:84
for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
proof
let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
let X, Y be set ; ::_thesis: ( dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) implies Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) )
assume that
A1: dom (F | (X \/ Y)) is finite and
A2: dom (F | X) misses dom (F | Y) ; ::_thesis: Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))
A3: dom (F | (X \/ Y)) = (dom F) /\ (X \/ Y) by RELAT_1:61
.= ((dom F) /\ X) \/ ((dom F) /\ Y) by XBOOLE_1:23
.= (dom (F | X)) \/ ((dom F) /\ Y) by RELAT_1:61
.= (dom (F | X)) \/ (dom (F | Y)) by RELAT_1:61 ;
then dom (F | X) is finite by A1, FINSET_1:1, XBOOLE_1:7;
then A4: FinS (F,X) = FinS (F,(dom (F | X))) by Th63;
dom (F | Y) is finite by A1, A3, FINSET_1:1, XBOOLE_1:7;
then A5: FinS (F,Y) = FinS (F,(dom (F | Y))) by Th63;
A6: dom (F | (dom (F | (X \/ Y)))) = (dom F) /\ (dom (F | (X \/ Y))) by RELAT_1:61
.= (dom F) /\ ((dom F) /\ (X \/ Y)) by RELAT_1:61
.= ((dom F) /\ (dom F)) /\ (X \/ Y) by XBOOLE_1:16
.= dom (F | (X \/ Y)) by RELAT_1:61 ;
FinS (F,(X \/ Y)) = FinS (F,(dom (F | (X \/ Y)))) by A1, Th63;
hence Sum (F,(X \/ Y)) = Sum ((FinS (F,X)) ^ (FinS (F,Y))) by A1, A2, A3, A4, A5, A6, Th76, RFINSEQ:9
.= (Sum (F,X)) + (Sum (F,Y)) by RVSUM_1:75 ;
::_thesis: verum
end;