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Theorem dimkerim 33679
Description: Given a linear map 𝐹 between vector spaces 𝑉 and 𝑈, the dimension of the vector space 𝑉 is the sum of the dimension of 𝐹 's kernel and the dimension of 𝐹's image. Second part of theorem 5.3 in [Lang] p. 141 This can also be described as the Rank-nullity theorem, (dim‘𝐼) being the rank of 𝐹 (the dimension of its image), and (dim‘𝐾) its nullity (the dimension of its kernel). (Contributed by Thierry Arnoux, 17-May-2023.)
Hypotheses
Ref Expression
dimkerim.0 0 = (0g𝑈)
dimkerim.k 𝐾 = (𝑉s (𝐹 “ { 0 }))
dimkerim.i 𝐼 = (𝑈s ran 𝐹)
Assertion
Ref Expression
dimkerim ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))

Proof of Theorem dimkerim
Dummy variables 𝑏 𝑢 𝑣 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dimkerim.0 . . . . 5 0 = (0g𝑈)
2 dimkerim.k . . . . 5 𝐾 = (𝑉s (𝐹 “ { 0 }))
31, 2kerlmhm 33672 . . . 4 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4 eqid 2736 . . . . 5 (LBasis‘𝐾) = (LBasis‘𝐾)
54lbsex 21168 . . . 4 (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
63, 5syl 17 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7 n0 4352 . . 3 ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
86, 7sylib 218 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
9 simpllr 775 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10 vex 3483 . . . . . . 7 𝑏 ∈ V
1110difexi 5329 . . . . . 6 (𝑏𝑤) ∈ V
1211a1i 11 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ V)
13 disjdif 4471 . . . . . 6 (𝑤 ∩ (𝑏𝑤)) = ∅
1413a1i 11 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∩ (𝑏𝑤)) = ∅)
15 hashunx 14426 . . . . 5 ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏𝑤) ∈ V ∧ (𝑤 ∩ (𝑏𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
169, 12, 14, 15syl3anc 1372 . . . 4 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (♯‘(𝑤 ∪ (𝑏𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
17 simp-4l 782 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑉 ∈ LVec)
18 simpr 484 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤𝑏)
19 undif 4481 . . . . . . 7 (𝑤𝑏 ↔ (𝑤 ∪ (𝑏𝑤)) = 𝑏)
2018, 19sylib 218 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∪ (𝑏𝑤)) = 𝑏)
21 simplr 768 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑏 ∈ (LBasis‘𝑉))
2220, 21eqeltrd 2840 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∪ (𝑏𝑤)) ∈ (LBasis‘𝑉))
23 eqid 2736 . . . . . 6 (LBasis‘𝑉) = (LBasis‘𝑉)
2423dimval 33652 . . . . 5 ((𝑉 ∈ LVec ∧ (𝑤 ∪ (𝑏𝑤)) ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏𝑤))))
2517, 22, 24syl2anc 584 . . . 4 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏𝑤))))
263ad3antrrr 730 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐾 ∈ LVec)
274dimval 33652 . . . . . 6 ((𝐾 ∈ LVec ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑤))
2826, 9, 27syl2anc 584 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝐾) = (♯‘𝑤))
29 dimkerim.i . . . . . . . . 9 𝐼 = (𝑈s ran 𝐹)
3029imlmhm 33673 . . . . . . . 8 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
3130ad3antrrr 730 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐼 ∈ LVec)
32 simp-4r 783 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33 lmhmlmod2 21032 . . . . . . . . . . 11 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod)
3432, 33syl 17 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑈 ∈ LMod)
35 lmhmrnlss 21050 . . . . . . . . . . 11 (𝐹 ∈ (𝑉 LMHom 𝑈) → ran 𝐹 ∈ (LSubSp‘𝑈))
3632, 35syl 17 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran 𝐹 ∈ (LSubSp‘𝑈))
37 df-ima 5697 . . . . . . . . . . 11 (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))
38 imassrn 6088 . . . . . . . . . . . 12 (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹
3938a1i 11 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹)
4037, 39eqsstrrid 4022 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹)
41 lveclmod 21106 . . . . . . . . . . . . 13 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
4241ad4antr 732 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑉 ∈ LMod)
4323lbslinds 21854 . . . . . . . . . . . . . . 15 (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
4443, 21sselid 3980 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45 difssd 4136 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ 𝑏)
46 lindsss 21845 . . . . . . . . . . . . . 14 ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏𝑤) ⊆ 𝑏) → (𝑏𝑤) ∈ (LIndS‘𝑉))
4742, 44, 45, 46syl3anc 1372 . . . . . . . . . . . . 13 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LIndS‘𝑉))
48 eqid 2736 . . . . . . . . . . . . . 14 (Base‘𝑉) = (Base‘𝑉)
4948linds1 21831 . . . . . . . . . . . . 13 ((𝑏𝑤) ∈ (LIndS‘𝑉) → (𝑏𝑤) ⊆ (Base‘𝑉))
5047, 49syl 17 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ (Base‘𝑉))
51 eqid 2736 . . . . . . . . . . . . 13 (LSubSp‘𝑉) = (LSubSp‘𝑉)
52 eqid 2736 . . . . . . . . . . . . 13 (LSpan‘𝑉) = (LSpan‘𝑉)
5348, 51, 52lspcl 20975 . . . . . . . . . . . 12 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉))
5442, 50, 53syl2anc 584 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉))
55 eqid 2736 . . . . . . . . . . . 12 (𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) = (𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))
5651, 55reslmhm 21052 . . . . . . . . . . 11 ((𝐹 ∈ (𝑉 LMHom 𝑈) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝑈))
5732, 54, 56syl2anc 584 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝑈))
58 eqid 2736 . . . . . . . . . . . 12 (LSubSp‘𝑈) = (LSubSp‘𝑈)
5929, 58reslmhm2b 21054 . . . . . . . . . . 11 ((𝑈 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑈) ∧ ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝑈) ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝐼)))
6059biimpa 476 . . . . . . . . . 10 (((𝑈 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑈) ∧ ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹) ∧ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝑈)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝐼))
6134, 36, 40, 57, 60syl31anc 1374 . . . . . . . . 9 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝐼))
62 lmghm 21031 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
6362ad4antlr 733 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
6448, 23lbsss 21077 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ (LBasis‘𝑉) → 𝑏 ⊆ (Base‘𝑉))
6521, 64syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑏 ⊆ (Base‘𝑉))
6645, 65sstrd 3993 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ (Base‘𝑉))
6742, 66, 53syl2anc 584 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉))
6851lsssubg 20956 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉)) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (SubGrp‘𝑉))
6942, 67, 68syl2anc 584 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (SubGrp‘𝑉))
7055resghm 19251 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ (𝑉 GrpHom 𝑈) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (SubGrp‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) GrpHom 𝑈))
7163, 69, 70syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) GrpHom 𝑈))
72 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘𝑈) = (Base‘𝑈)
7348, 72lmhmf 21034 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
7473ad4antlr 733 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
7574ffnd 6736 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 Fn (Base‘𝑉))
7648, 52lspssv 20982 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
7742, 66, 76syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
7875, 77fnssresd 6691 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) Fn ((LSpan‘𝑉)‘(𝑏𝑤)))
79 fniniseg 7079 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) Fn ((LSpan‘𝑉)‘(𝑏𝑤)) → (𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) ↔ (𝑥 ∈ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘𝑥) = 0 )))
8079biimpa 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) Fn ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → (𝑥 ∈ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘𝑥) = 0 ))
8178, 80sylan 580 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → (𝑥 ∈ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘𝑥) = 0 ))
8281simpld 494 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ ((LSpan‘𝑉)‘(𝑏𝑤)))
8375adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝐹 Fn (Base‘𝑉))
8477adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
8584, 82sseldd 3983 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ (Base‘𝑉))
8682fvresd 6925 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘𝑥) = (𝐹𝑥))
8781simprd 495 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘𝑥) = 0 )
8886, 87eqtr3d 2778 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → (𝐹𝑥) = 0 )
89 fniniseg 7079 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹𝑥) = 0 )))
9089biimpar 477 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹𝑥) = 0 )) → 𝑥 ∈ (𝐹 “ { 0 }))
9183, 85, 88, 90syl12anc 836 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ (𝐹 “ { 0 }))
9282, 91elind 4199 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ (𝐹 “ { 0 })))
93 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
94 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘𝐾) = (Base‘𝐾)
95 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (LSpan‘𝐾) = (LSpan‘𝐾)
9694, 4, 95lbssp 21079 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (LBasis‘𝐾) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
9793, 96syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
9841ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
99 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 “ { 0 }) = (𝐹 “ { 0 })
10099, 1, 51lmhmkerlss 21051 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
101100ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
10294, 4lbsss 21077 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ (LBasis‘𝐾) → 𝑤 ⊆ (Base‘𝐾))
10393, 102syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (Base‘𝐾))
104 cnvimass 6099 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹 “ { 0 }) ⊆ dom 𝐹
105104, 73fssdm 6754 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) ⊆ (Base‘𝑉))
1062, 48ressbas2 17284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹 “ { 0 }) ⊆ (Base‘𝑉) → (𝐹 “ { 0 }) = (Base‘𝐾))
107105, 106syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) = (Base‘𝐾))
108107ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝐹 “ { 0 }) = (Base‘𝐾))
109103, 108sseqtrrd 4020 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (𝐹 “ { 0 }))
1102, 52, 95, 51lsslsp 21014 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) → ((LSpan‘𝐾)‘𝑤) = ((LSpan‘𝑉)‘𝑤))
111110eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
11298, 101, 109, 111syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
11397, 112, 1083eqtr4d 2786 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (𝐹 “ { 0 }))
114113ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) = (𝐹 “ { 0 }))
115114ineq2d 4219 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ (𝐹 “ { 0 })))
116 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . 24 (0g𝑉) = (0g𝑉)
11723, 52, 116lbsdiflsp0 33678 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g𝑉)})
118117ad5ant145 1370 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g𝑉)})
119115, 118eqtr3d 2778 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ (𝐹 “ { 0 })) = {(0g𝑉)})
120119adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ (𝐹 “ { 0 })) = {(0g𝑉)})
12192, 120eleqtrd 2842 . . . . . . . . . . . . . . . . . . 19 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ {(0g𝑉)})
122121ex 412 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) → 𝑥 ∈ {(0g𝑉)}))
123122ssrdv 3988 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) ⊆ {(0g𝑉)})
124116, 48, 520ellsp 33398 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → (0g𝑉) ∈ ((LSpan‘𝑉)‘(𝑏𝑤)))
12542, 66, 124syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) ∈ ((LSpan‘𝑉)‘(𝑏𝑤)))
126 fvexd 6920 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V)
127125fvresd 6925 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = (𝐹‘(0g𝑉)))
128116, 1ghmid 19241 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝑉 GrpHom 𝑈) → (𝐹‘(0g𝑉)) = 0 )
12962, 128syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹‘(0g𝑉)) = 0 )
130129ad4antlr 733 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹‘(0g𝑉)) = 0 )
131127, 130eqtrd 2776 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = 0 )
132 elsng 4639 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V → (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ { 0 } ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = 0 ))
133132biimpar 477 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = 0 ) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ { 0 })
134126, 131, 133syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ { 0 })
13578, 125, 134elpreimad 7078 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }))
136135snssd 4808 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → {(0g𝑉)} ⊆ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }))
137123, 136eqssd 4000 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) = {(0g𝑉)})
138 lmodgrp 20866 . . . . . . . . . . . . . . . . . . 19 (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139 grpmnd 18959 . . . . . . . . . . . . . . . . . . 19 (𝑉 ∈ Grp → 𝑉 ∈ Mnd)
14042, 138, 1393syl 18 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑉 ∈ Mnd)
14155, 48, 116ress0g 18776 . . . . . . . . . . . . . . . . . 18 ((𝑉 ∈ Mnd ∧ (0g𝑉) ∈ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉)) → (0g𝑉) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
142140, 125, 77, 141syl3anc 1372 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
143142sneqd 4637 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → {(0g𝑉)} = {(0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))})
144137, 143eqtrd 2776 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) = {(0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))})
145 eqid 2736 . . . . . . . . . . . . . . . . 17 (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
146 eqid 2736 . . . . . . . . . . . . . . . . 17 (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
147145, 72, 146, 1kerf1ghm 19266 . . . . . . . . . . . . . . . 16 ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) GrpHom 𝑈) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):(Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))–1-1→(Base‘𝑈) ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) = {(0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))}))
148147biimpar 477 . . . . . . . . . . . . . . 15 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) GrpHom 𝑈) ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) = {(0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))}) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):(Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))–1-1→(Base‘𝑈))
14971, 144, 148syl2anc 584 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):(Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))–1-1→(Base‘𝑈))
150 eqidd 2737 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) = (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))))
15155, 48ressbas2 17284 . . . . . . . . . . . . . . . 16 (((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉) → ((LSpan‘𝑉)‘(𝑏𝑤)) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
15277, 151syl 17 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
153 eqidd 2737 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (Base‘𝑈) = (Base‘𝑈))
154150, 152, 153f1eq123d 6839 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→(Base‘𝑈) ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):(Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))–1-1→(Base‘𝑈)))
155149, 154mpbird 257 . . . . . . . . . . . . 13 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→(Base‘𝑈))
156 f1ssr 6809 . . . . . . . . . . . . 13 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→(Base‘𝑈) ∧ ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→ran 𝐹)
157155, 40, 156syl2anc 584 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→ran 𝐹)
158 f1f1orn 6858 . . . . . . . . . . . 12 ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→ran 𝐹 → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1-onto→ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))))
159157, 158syl 17 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1-onto→ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))))
160 simp-4r 783 . . . . . . . . . . . . . . . . . 18 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → (𝐹𝑥) = 𝑦)
16175ad6antr 736 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝐹 Fn (Base‘𝑉))
162 simpllr 775 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑢 ∈ ((LSpan‘𝑉)‘𝑤))
163113ad8antr 740 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → ((LSpan‘𝑉)‘𝑤) = (𝐹 “ { 0 }))
164162, 163eleqtrd 2842 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑢 ∈ (𝐹 “ { 0 }))
165 fniniseg 7079 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 Fn (Base‘𝑉) → (𝑢 ∈ (𝐹 “ { 0 }) ↔ (𝑢 ∈ (Base‘𝑉) ∧ (𝐹𝑢) = 0 )))
166165simplbda 499 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn (Base‘𝑉) ∧ 𝑢 ∈ (𝐹 “ { 0 })) → (𝐹𝑢) = 0 )
167161, 164, 166syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → (𝐹𝑢) = 0 )
168167oveq1d 7447 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → ((𝐹𝑢)(+g𝑈)(𝐹𝑣)) = ( 0 (+g𝑈)(𝐹𝑣)))
169 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑥 = (𝑢(+g𝑉)𝑣))
170169fveq2d 6909 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → (𝐹𝑥) = (𝐹‘(𝑢(+g𝑉)𝑣)))
17163ad6antr 736 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
17248, 52lspss 20983 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
17342, 65, 18, 172syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
17448, 23, 52lbssp 21079 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (LBasis‘𝑉) → ((LSpan‘𝑉)‘𝑏) = (Base‘𝑉))
17521, 174syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑏) = (Base‘𝑉))
176173, 175sseqtrd 4019 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177176ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
178177ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
179178, 162sseldd 3983 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑢 ∈ (Base‘𝑉))
18077ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
181180ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
182 simplr 768 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤)))
183181, 182sseldd 3983 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑣 ∈ (Base‘𝑉))
184 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (+g𝑉) = (+g𝑉)
185 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (+g𝑈) = (+g𝑈)
18648, 184, 185ghmlin 19240 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (𝑉 GrpHom 𝑈) ∧ 𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑉)) → (𝐹‘(𝑢(+g𝑉)𝑣)) = ((𝐹𝑢)(+g𝑈)(𝐹𝑣)))
187171, 179, 183, 186syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → (𝐹‘(𝑢(+g𝑉)𝑣)) = ((𝐹𝑢)(+g𝑈)(𝐹𝑣)))
188170, 187eqtr2d 2777 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → ((𝐹𝑢)(+g𝑈)(𝐹𝑣)) = (𝐹𝑥))
189 lmhmlvec2 33671 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190189lvecgrpd 21108 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
191190ad9antr 742 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑈 ∈ Grp)
19274ad6antr 736 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
193192, 183ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → (𝐹𝑣) ∈ (Base‘𝑈))
19472, 185, 1, 191, 193grplidd 18988 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → ( 0 (+g𝑈)(𝐹𝑣)) = (𝐹𝑣))
195168, 188, 1943eqtr3d 2784 . . . . . . . . . . . . . . . . . 18 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → (𝐹𝑥) = (𝐹𝑣))
196160, 195eqtr3d 2778 . . . . . . . . . . . . . . . . 17 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑦 = (𝐹𝑣))
197161, 183, 182fnfvimad 7255 . . . . . . . . . . . . . . . . 17 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → (𝐹𝑣) ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))))
198196, 197eqeltrd 2840 . . . . . . . . . . . . . . . 16 (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ 𝑥 = (𝑢(+g𝑉)𝑣)) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))))
199 simp-7l 788 . . . . . . . . . . . . . . . . 17 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → 𝑉 ∈ LVec)
200 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → 𝑥 ∈ (Base‘𝑉))
201109ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ⊆ (𝐹 “ { 0 }))
202105ad4antlr 733 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 “ { 0 }) ⊆ (Base‘𝑉))
203201, 202sstrd 3993 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ⊆ (Base‘𝑉))
204 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (LSSum‘𝑉) = (LSSum‘𝑉)
20548, 52, 204lsmsp2 21087 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 ∈ LMod ∧ 𝑤 ⊆ (Base‘𝑉) ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤))) = ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏𝑤))))
20642, 203, 66, 205syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤))) = ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏𝑤))))
20720fveq2d 6909 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏𝑤))) = ((LSpan‘𝑉)‘𝑏))
208206, 207, 1753eqtrrd 2781 . . . . . . . . . . . . . . . . . . 19 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (Base‘𝑉) = (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤))))
209208ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → (Base‘𝑉) = (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤))))
210200, 209eleqtrd 2842 . . . . . . . . . . . . . . . . 17 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → 𝑥 ∈ (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤))))
21148, 184, 204lsmelvalx 19659 . . . . . . . . . . . . . . . . . 18 ((𝑉 ∈ LVec ∧ ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉)) → (𝑥 ∈ (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤))) ↔ ∃𝑢 ∈ ((LSpan‘𝑉)‘𝑤)∃𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))𝑥 = (𝑢(+g𝑉)𝑣)))
212211biimpa 476 . . . . . . . . . . . . . . . . 17 (((𝑉 ∈ LVec ∧ ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉)) ∧ 𝑥 ∈ (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤)))) → ∃𝑢 ∈ ((LSpan‘𝑉)‘𝑤)∃𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))𝑥 = (𝑢(+g𝑉)𝑣))
213199, 177, 180, 210, 212syl31anc 1374 . . . . . . . . . . . . . . . 16 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → ∃𝑢 ∈ ((LSpan‘𝑉)‘𝑤)∃𝑣 ∈ ((LSpan‘𝑉)‘(𝑏𝑤))𝑥 = (𝑢(+g𝑉)𝑣))
214198, 213r19.29vva 3215 . . . . . . . . . . . . . . 15 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))))
215 fvelrnb 6968 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦))
216215biimpa 476 . . . . . . . . . . . . . . . 16 ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦)
21775, 216sylan 580 . . . . . . . . . . . . . . 15 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦)
218214, 217r19.29a 3161 . . . . . . . . . . . . . 14 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))))
21939, 218eqelssd 4004 . . . . . . . . . . . . 13 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran 𝐹)
22037, 219eqtr3id 2790 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran 𝐹)
221220f1oeq3d 6844 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1-onto→ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1-onto→ran 𝐹))
222159, 221mpbid 232 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1-onto→ran 𝐹)
22342, 50, 76syl2anc 584 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
224223, 151syl 17 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
225 frn 6742 . . . . . . . . . . . 12 (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
22629, 72ressbas2 17284 . . . . . . . . . . . 12 (ran 𝐹 ⊆ (Base‘𝑈) → ran 𝐹 = (Base‘𝐼))
22732, 73, 225, 2264syl 19 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran 𝐹 = (Base‘𝐼))
228150, 224, 227f1oeq123d 6841 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1-onto→ran 𝐹 ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):(Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))–1-1-onto→(Base‘𝐼)))
229222, 228mpbid 232 . . . . . . . . 9 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):(Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))–1-1-onto→(Base‘𝐼))
230 eqid 2736 . . . . . . . . . 10 (Base‘𝐼) = (Base‘𝐼)
231145, 230islmim 21062 . . . . . . . . 9 ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMIso 𝐼) ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝐼) ∧ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):(Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))–1-1-onto→(Base‘𝐼)))
23261, 229, 231sylanbrc 583 . . . . . . . 8 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMIso 𝐼))
23348, 52lspssid 20984 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)))
23442, 50, 233syl2anc 584 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)))
23551, 55lsslinds 21852 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) → ((𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) ↔ (𝑏𝑤) ∈ (LIndS‘𝑉)))
236235biimpar 477 . . . . . . . . . 10 (((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ (𝑏𝑤) ∈ (LIndS‘𝑉)) → (𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
23742, 67, 234, 47, 236syl31anc 1374 . . . . . . . . 9 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
238 eqid 2736 . . . . . . . . . . . . 13 (LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
23955, 52, 238, 51lsslsp 21014 . . . . . . . . . . . 12 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) → ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)) = ((LSpan‘𝑉)‘(𝑏𝑤)))
240239eqcomd 2742 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) → ((LSpan‘𝑉)‘(𝑏𝑤)) = ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)))
24142, 54, 234, 240syl3anc 1372 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) = ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)))
242241, 224eqtr3d 2778 . . . . . . . . 9 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
243 eqid 2736 . . . . . . . . . 10 (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
244145, 243, 238islbs4 21853 . . . . . . . . 9 ((𝑏𝑤) ∈ (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) ↔ ((𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) ∧ ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))))
245237, 242, 244sylanbrc 583 . . . . . . . 8 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
246 eqid 2736 . . . . . . . . 9 (LBasis‘𝐼) = (LBasis‘𝐼)
247243, 246lmimlbs 21857 . . . . . . . 8 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMIso 𝐼) ∧ (𝑏𝑤) ∈ (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼))
248232, 245, 247syl2anc 584 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼))
249246dimval 33652 . . . . . . 7 ((𝐼 ∈ LVec ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼)) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))))
25031, 248, 249syl2anc 584 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))))
251 f1imaeng 9055 . . . . . . . 8 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→ran 𝐹 ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ (𝑏𝑤) ∈ (LIndS‘𝑉)) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ≈ (𝑏𝑤))
252 hasheni 14388 . . . . . . . 8 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ≈ (𝑏𝑤) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))) = (♯‘(𝑏𝑤)))
253251, 252syl 17 . . . . . . 7 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))):((LSpan‘𝑉)‘(𝑏𝑤))–1-1→ran 𝐹 ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ (𝑏𝑤) ∈ (LIndS‘𝑉)) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))) = (♯‘(𝑏𝑤)))
254157, 234, 47, 253syl3anc 1372 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))) = (♯‘(𝑏𝑤)))
255250, 254eqtrd 2776 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝐼) = (♯‘(𝑏𝑤)))
25628, 255oveq12d 7450 . . . 4 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((dim‘𝐾) +𝑒 (dim‘𝐼)) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
25716, 25, 2563eqtr4d 2786 . . 3 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
2584lbslinds 21854 . . . . . 6 (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
259258, 93sselid 3980 . . . . 5 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
26051, 2lsslinds 21852 . . . . . 6 ((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
261260biimpa 476 . . . . 5 (((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
26298, 101, 109, 259, 261syl31anc 1374 . . . 4 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
26323islinds4 21856 . . . . 5 (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏))
264263ad2antrr 726 . . . 4 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏))
265262, 264mpbid 232 . . 3 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏)
266257, 265r19.29a 3161 . 2 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
2678, 266exlimddv 1934 1 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wne 2939  wrex 3069  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  {csn 4625   class class class wbr 5142  ccnv 5683  ran crn 5685  cres 5686  cima 5687   Fn wfn 6555  wf 6556  1-1wf1 6557  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cen 8983   +𝑒 cxad 13153  chash 14370  Basecbs 17248  s cress 17275  +gcplusg 17298  0gc0g 17485  Mndcmnd 18748  Grpcgrp 18952  SubGrpcsubg 19139   GrpHom cghm 19231  LSSumclsm 19653  LModclmod 20859  LSubSpclss 20930  LSpanclspn 20970   LMHom clmhm 21019   LMIso clmim 21020  LBasisclbs 21074  LVecclvec 21102  LIndSclinds 21826  dimcldim 33650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682  ax-ac2 10504  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-rpss 7744  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-r1 9805  df-rank 9806  df-dju 9942  df-card 9980  df-acn 9983  df-ac 10157  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12736  df-uz 12880  df-xadd 13156  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-mri 17632  df-acs 17633  df-proset 18341  df-drs 18342  df-poset 18360  df-ipo 18574  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-lsm 19655  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-nzr 20514  df-subrg 20571  df-drng 20732  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lmim 21023  df-lbs 21075  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-lindf 21827  df-linds 21828  df-dim 33651
This theorem is referenced by:  qusdimsum  33680  lvecendof1f1o  33685
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