:: RMOD_4 semantic presentation begin theorem :: RMOD_4:1 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")))) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))))) ")" )) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "G")) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")))))))) ; theorem :: RMOD_4:2 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" ) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))))) ; theorem :: RMOD_4:3 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "v")) "," (Set (Var "u")) ($#k2_finseq_4 :::"*>"::: ) ) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "u")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")) ")" ))))))) ; theorem :: RMOD_4:4 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "v")) "," (Set (Var "u")) "," (Set (Var "w")) ($#k3_finseq_4 :::"*>"::: ) ) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "u")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "w")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")) ")" ))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "T" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"Sum"::: "T" -> ($#m1_subset_1 :::"Vector":::) "of" "V" means :: RMOD_4:def 1 (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" "V" "st" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) "T") & (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "F")))) ")" )); end; :: deftheorem defines :::"Sum"::: RMOD_4:def 1 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "T")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Var "T"))) & (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "F")))) ")" )) ")" ))))); theorem :: RMOD_4:5 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" ($#k1_struct_0 :::"{}"::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: RMOD_4:6 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "v")))))) ; theorem :: RMOD_4:7 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "v1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v2"))))))) ; theorem :: RMOD_4:8 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set (Var "v2")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3"))) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3")))) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) ($#k8_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v2")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v3"))))))) ; theorem :: RMOD_4:9 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "T")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "S")))) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "T")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "S")) ")" )))))) ; theorem :: RMOD_4:10 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "T")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "S")) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "S")) ")" ) ")" )))))) ; theorem :: RMOD_4:11 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "T")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "S")) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "S")) ")" ) ")" )))))) ; theorem :: RMOD_4:12 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k7_subset_1 :::"\"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "S")) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "S")) ")" )))))) ; theorem :: RMOD_4:13 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k7_subset_1 :::"\"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set (Var "T")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "S")) ")" ) ")" )))))) ; theorem :: RMOD_4:14 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k5_subset_1 :::"\+\"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "S")) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "S")) ")" ) ")" )))))) ; theorem :: RMOD_4:15 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k5_subset_1 :::"\+\"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "T")) ($#k7_subset_1 :::"\"::: ) (Set (Var "S")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k1_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "S")) ($#k7_subset_1 :::"\"::: ) (Set (Var "T")) ")" ) ")" )))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); mode :::"Linear_Combination"::: "of" "V" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") ")" ) means :: RMOD_4:def 2 (Bool "ex" (Set (Var "T")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" "V" "st" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" "V" "st" (Bool (Bool (Bool "not" (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "T"))))) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "R")))); end; :: deftheorem defines :::"Linear_Combination"::: RMOD_4:def 2 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b3")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V"))) "iff" (Bool "ex" (Set (Var "T")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Bool "not" (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "T"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R")))))) ")" )))); definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "L" be ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")); func :::"Carrier"::: "L" -> ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: RMOD_4:def 3 "{" (Set (Var "v")) where v "is" ($#m1_subset_1 :::"Vector":::) "of" "V" : (Bool (Set "L" ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "R")) "}" ; end; :: deftheorem defines :::"Carrier"::: RMOD_4:def 3 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "v")) where v "is" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) : (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R")))) "}" )))); theorem :: RMOD_4:16 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L")))) "iff" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R")))) ")" )) ")" ))))) ; theorem :: RMOD_4:17 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R")))) "iff" (Bool (Bool "not" (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L"))))) ")" ))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); func :::"ZeroLC"::: "V" -> ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" "V" means :: RMOD_4:def 4 (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"ZeroLC"::: RMOD_4:def 4 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b3")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V")))) "iff" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )))); theorem :: RMOD_4:18 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); mode :::"Linear_Combination"::: "of" "A" -> ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" "V" means :: RMOD_4:def 5 (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) it) ($#r1_tarski :::"c="::: ) "A"); end; :: deftheorem defines :::"Linear_Combination"::: RMOD_4:def 5 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b4")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A"))) "iff" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "b4"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ))))); theorem :: RMOD_4:19 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "l")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "B"))))))) ; theorem :: RMOD_4:20 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V"))) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")))))) ; theorem :: RMOD_4:21 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) "holds" (Bool (Set (Var "l")) ($#r2_funct_2 :::"="::: ) (Set ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V"))))))) ; theorem :: RMOD_4:22 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Var "L")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L"))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "F" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Const "V")); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "V")) "," (Set (Const "R")); func "f" :::"(#)"::: "F" -> ($#m2_finseq_1 :::"FinSequence":::) "of" "V" means :: RMOD_4:def 6 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "F")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "F" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set "(" "F" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"(#)"::: RMOD_4:def 6 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "R")) (Bool "for" (Set (Var "b5")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set (Var "F")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b5"))))) "holds" (Bool (Set (Set (Var "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ))) ")" ) ")" ) ")" )))))); theorem :: RMOD_4:23 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "R")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")))) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set (Var "F")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ))))))))) ; theorem :: RMOD_4:24 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "R")) "holds" (Bool (Set (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))))))) ; theorem :: RMOD_4:25 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "R")) "holds" (Bool (Set (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "v")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: RMOD_4:26 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "R")) "holds" (Bool (Set (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k2_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set "(" (Set (Var "v1")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ")" ) "," (Set "(" (Set (Var "v2")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ")" ) ($#k2_finseq_4 :::"*>"::: ) )))))) ; theorem :: RMOD_4:27 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "R")) "holds" (Bool (Set (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set "(" (Set (Var "v1")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ")" ) "," (Set "(" (Set (Var "v2")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ")" ) "," (Set "(" (Set (Var "v3")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v3")) ")" ) ")" ) ($#k3_finseq_4 :::"*>"::: ) )))))) ; theorem :: RMOD_4:28 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "R")) "holds" (Bool (Set (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set "(" (Set (Var "F")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "G")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set (Var "F")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f")) ($#k4_rmod_4 :::"(#)"::: ) (Set (Var "G")) ")" ))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "L" be ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")); func :::"Sum"::: "L" -> ($#m1_subset_1 :::"Vector":::) "of" "V" means :: RMOD_4:def 7 (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" "V" "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rmod_4 :::"Carrier"::: ) "L")) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" "L" ($#k4_rmod_4 :::"(#)"::: ) (Set (Var "F")) ")" ))) ")" )); end; :: deftheorem defines :::"Sum"::: RMOD_4:def 7 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L")))) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k4_rmod_4 :::"(#)"::: ) (Set (Var "F")) ")" ))) ")" )) ")" ))))); theorem :: RMOD_4:29 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set (Var "R"))))) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "A")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) ) ")" ) "iff" (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l"))) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) ")" )))) ; theorem :: RMOD_4:30 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set "(" ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: RMOD_4:31 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))))) ; theorem :: RMOD_4:32 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "l")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ))))))) ; theorem :: RMOD_4:33 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v1")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "l")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "v2")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "l")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ")" ))))))) ; theorem :: RMOD_4:34 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))))) ; theorem :: RMOD_4:35 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ))))))) ; theorem :: RMOD_4:36 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v1")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "v2")) ($#k5_vectsp_2 :::"*"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ")" ))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "L1", "L2" be ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")); :: original: :::"="::: redefine pred "L1" :::"="::: "L2" means :: RMOD_4:def 8 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" "V" "holds" (Bool (Set "L1" ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set "L2" ($#k3_funct_2 :::"."::: ) (Set (Var "v"))))); end; :: deftheorem defines :::"="::: RMOD_4:def 8 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "L1")) ($#r1_rmod_4 :::"="::: ) (Set (Var "L2"))) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "L2")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))))) ")" )))); definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "L1", "L2" be ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")); func "L1" :::"+"::: "L2" -> ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" "V" means :: RMOD_4:def 9 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" "V" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "L1" ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" "L2" ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))); end; :: deftheorem defines :::"+"::: RMOD_4:def 9 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "b5")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2")))) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "b5")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L1")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "L2")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))) ")" )))); theorem :: RMOD_4:37 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set "(" (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L1")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L2")) ")" )))))) ; theorem :: RMOD_4:38 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L1")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A"))) & (Bool (Set (Var "L2")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")))) "holds" (Bool (Set (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2"))) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A"))))))) ; theorem :: RMOD_4:39 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"comRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2"))) ($#r1_rmod_4 :::"="::: ) (Set (Set (Var "L2")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L1"))))))) ; theorem :: RMOD_4:40 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set "(" (Set (Var "L2")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L3")) ")" )) ($#r1_rmod_4 :::"="::: ) (Set (Set "(" (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2")) ")" ) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L3"))))))) ; theorem :: RMOD_4:41 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"comRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k6_rmod_4 :::"+"::: ) (Set "(" ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V")) ")" )) ($#r1_rmod_4 :::"="::: ) (Set (Var "L"))) & (Bool (Set (Set "(" ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V")) ")" ) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L"))) ($#r1_rmod_4 :::"="::: ) (Set (Var "L"))) ")" )))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "a" be ($#m1_subset_1 :::"Scalar":::) "of" (Set (Const "R")); let "L" be ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")); func "L" :::"*"::: "a" -> ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" "V" means :: RMOD_4:def 10 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" "V" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "L" ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k6_algstr_0 :::"*"::: ) "a"))); end; :: deftheorem defines :::"*"::: RMOD_4:def 10 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "," (Set (Var "b5")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a")))) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "b5")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a"))))) ")" ))))); theorem :: RMOD_4:42 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set "(" (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L")))))))) ; theorem :: RMOD_4:43 (Bool "for" (Set (Var "RR")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "VV")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "RR")) (Bool "for" (Set (Var "LL")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "VV")) (Bool "for" (Set (Var "aa")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "RR")) "st" (Bool (Bool (Set (Var "aa")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "RR"))))) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set "(" (Set (Var "LL")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "aa")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "LL")))))))) ; theorem :: RMOD_4:44 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" )) ($#r1_rmod_4 :::"="::: ) (Set ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V"))))))) ; theorem :: RMOD_4:45 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")))) "holds" (Bool (Set (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a"))) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")))))))) ; theorem :: RMOD_4:46 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "b")) ")" )) ($#r1_rmod_4 :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a")) ")" ) ($#k6_rmod_4 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "b")) ")" ))))))) ; theorem :: RMOD_4:47 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2")) ")" ) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a"))) ($#r1_rmod_4 :::"="::: ) (Set (Set "(" (Set (Var "L1")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a")) ")" ) ($#k6_rmod_4 :::"+"::: ) (Set "(" (Set (Var "L2")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a")) ")" ))))))) ; theorem :: RMOD_4:48 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "b")) ")" ) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a"))) ($#r1_rmod_4 :::"="::: ) (Set (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a")) ")" ))))))) ; theorem :: RMOD_4:49 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "R")) ")" )) ($#r1_rmod_4 :::"="::: ) (Set (Var "L")))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "L" be ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")); func :::"-"::: "L" -> ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" "V" equals :: RMOD_4:def 11 (Set "L" ($#k7_rmod_4 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) "R" ")" ) ")" )); involutiveness (Bool "for" (Set (Var "b1")) "," (Set (Var "b2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")) "st" (Bool (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b2")) ($#k7_rmod_4 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Const "R")) ")" ) ")" )))) "holds" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b1")) ($#k7_rmod_4 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Const "R")) ")" ) ")" )))) ; end; :: deftheorem defines :::"-"::: RMOD_4:def 11 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k8_rmod_4 :::"-"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "R")) ")" ) ")" )))))); theorem :: RMOD_4:50 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k8_rmod_4 :::"-"::: ) (Set (Var "L")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ))))))) ; theorem :: RMOD_4:51 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2"))) ($#r1_rmod_4 :::"="::: ) (Set ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V"))))) "holds" (Bool (Set (Var "L2")) ($#r1_rmod_4 :::"="::: ) (Set ($#k8_rmod_4 :::"-"::: ) (Set (Var "L1"))))))) ; theorem :: RMOD_4:52 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set "(" ($#k8_rmod_4 :::"-"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L"))))))) ; theorem :: RMOD_4:53 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")))) "holds" (Bool (Set ($#k8_rmod_4 :::"-"::: ) (Set (Var "L"))) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A"))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "L1", "L2" be ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Const "V")); func "L1" :::"-"::: "L2" -> ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" "V" equals :: RMOD_4:def 12 (Set "L1" ($#k6_rmod_4 :::"+"::: ) (Set "(" ($#k8_rmod_4 :::"-"::: ) "L2" ")" )); end; :: deftheorem defines :::"-"::: RMOD_4:def 12 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k9_rmod_4 :::"-"::: ) (Set (Var "L2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set "(" ($#k8_rmod_4 :::"-"::: ) (Set (Var "L2")) ")" )))))); theorem :: RMOD_4:54 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "L1")) ($#k9_rmod_4 :::"-"::: ) (Set (Var "L2")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L1")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "L2")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ))))))) ; theorem :: RMOD_4:55 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set "(" (Set (Var "L1")) ($#k9_rmod_4 :::"-"::: ) (Set (Var "L2")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L1")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "L2")) ")" )))))) ; theorem :: RMOD_4:56 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L1")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A"))) & (Bool (Set (Var "L2")) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")))) "holds" (Bool (Set (Set (Var "L1")) ($#k9_rmod_4 :::"-"::: ) (Set (Var "L2"))) "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A"))))))) ; theorem :: RMOD_4:57 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L")) ($#k9_rmod_4 :::"-"::: ) (Set (Var "L"))) ($#r1_rmod_4 :::"="::: ) (Set ($#k3_rmod_4 :::"ZeroLC"::: ) (Set (Var "V"))))))) ; theorem :: RMOD_4:58 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "L1")) ($#k6_rmod_4 :::"+"::: ) (Set (Var "L2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L2")) ")" )))))) ; theorem :: RMOD_4:59 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k7_rmod_4 :::"*"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L")) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")))))))) ; theorem :: RMOD_4:60 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set "(" ($#k8_rmod_4 :::"-"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L")) ")" )))))) ; theorem :: RMOD_4:61 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set "(" (Set (Var "L1")) ($#k9_rmod_4 :::"-"::: ) (Set (Var "L2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "L2")) ")" )))))) ; begin definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "IT" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "IT" is :::"linearly-independent"::: means :: RMOD_4:def 13 (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" "IT" "st" (Bool (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "V"))) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))); end; :: deftheorem defines :::"linearly-independent"::: RMOD_4:def 13 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_rmod_4 :::"linearly-independent"::: ) ) "iff" (Bool "for" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "IT")) "st" (Bool (Bool (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))) "holds" (Bool (Set ($#k2_rmod_4 :::"Carrier"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" )))); notationlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "IT" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); antonym :::"linearly-dependent"::: "IT" for :::"linearly-independent"::: ; end; theorem :: RMOD_4:62 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Var "B")) "is" ($#v1_rmod_4 :::"linearly-independent"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v1_rmod_4 :::"linearly-independent"::: ) )))) ; theorem :: RMOD_4:63 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set (Var "R")))) & (Bool (Set (Var "A")) "is" ($#v1_rmod_4 :::"linearly-independent"::: ) )) "holds" (Bool "not" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))))) ; theorem :: RMOD_4:64 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) "is" ($#v1_rmod_4 :::"linearly-independent"::: ) ))) ; theorem :: RMOD_4:65 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set (Var "R")))) & (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "is" ($#v1_rmod_4 :::"linearly-independent"::: ) )) "holds" (Bool "(" (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set (Var "v2")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) ")" )))) ; theorem :: RMOD_4:66 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set (Var "R"))))) "holds" (Bool "(" (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k7_domain_1 :::"}"::: ) ) "is" ($#v1_rmod_4 :::"linearly-dependent"::: ) ) & (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) "," (Set (Var "v")) ($#k7_domain_1 :::"}"::: ) ) "is" ($#v1_rmod_4 :::"linearly-dependent"::: ) ) ")" )))) ; definitionlet "R" be ($#l6_algstr_0 :::"domRing":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"Lin"::: "A" -> ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" "V" means :: RMOD_4:def 14 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l")) ")" ) where l "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" "A" : (Bool verum) "}" ); end; :: deftheorem defines :::"Lin"::: RMOD_4:def 14 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b4")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A")))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b4"))) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l")) ")" ) where l "is" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")) : (Bool verum) "}" ) ")" ))))); theorem :: RMOD_4:67 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A")))) "iff" (Bool "ex" (Set (Var "l")) "being" ($#m2_rmod_4 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k5_rmod_4 :::"Sum"::: ) (Set (Var "l"))))) ")" ))))) ; theorem :: RMOD_4:68 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A")))))))) ; theorem :: RMOD_4:69 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set "(" ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V")))))) ; theorem :: RMOD_4:70 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" "not" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V")))) "or" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ")" )))) ; theorem :: RMOD_4:71 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set (Var "R")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "W"))))))) ; theorem :: RMOD_4:72 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set (Var "R")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))))) "holds" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "V")))))) ; theorem :: RMOD_4:73 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A"))) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "B"))))))) ; theorem :: RMOD_4:74 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "V"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Var "V")))))) ; theorem :: RMOD_4:75 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A")) ")" ) ($#k1_rmod_3 :::"+"::: ) (Set "(" ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "B")) ")" )))))) ; theorem :: RMOD_4:76 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"domRing":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k10_rmod_4 :::"Lin"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" )) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Set "(" ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "A")) ")" ) ($#k2_rmod_3 :::"/\"::: ) (Set "(" ($#k10_rmod_4 :::"Lin"::: ) (Set (Var "B")) ")" )))))) ;