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Theorem dimkerim 32322
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 32317 . . . 4 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4 eqid 2736 . . . . 5 (LBasis‘𝐾) = (LBasis‘𝐾)
54lbsex 20626 . . . 4 (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
63, 5syl 17 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7 n0 4306 . . 3 ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
86, 7sylib 217 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
9 simpllr 774 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10 vex 3449 . . . . . . 7 𝑏 ∈ V
1110difexi 5285 . . . . . 6 (𝑏𝑤) ∈ V
1211a1i 11 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ V)
13 disjdif 4431 . . . . . 6 (𝑤 ∩ (𝑏𝑤)) = ∅
1413a1i 11 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∩ (𝑏𝑤)) = ∅)
15 hashunx 14286 . . . . 5 ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏𝑤) ∈ V ∧ (𝑤 ∩ (𝑏𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
169, 12, 14, 15syl3anc 1371 . . . 4 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (♯‘(𝑤 ∪ (𝑏𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
17 simp-4l 781 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑉 ∈ LVec)
18 simpr 485 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤𝑏)
19 undif 4441 . . . . . . 7 (𝑤𝑏 ↔ (𝑤 ∪ (𝑏𝑤)) = 𝑏)
2018, 19sylib 217 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∪ (𝑏𝑤)) = 𝑏)
21 simplr 767 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑏 ∈ (LBasis‘𝑉))
2220, 21eqeltrd 2838 . . . . 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 32300 . . . . 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 728 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐾 ∈ LVec)
274dimval 32300 . . . . . 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 32318 . . . . . . . 8 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
3130ad3antrrr 728 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐼 ∈ LVec)
32 simp-4r 782 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33 lmhmlmod2 20493 . . . . . . . . . . 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 20511 . . . . . . . . . . 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 5646 . . . . . . . . . . 11 (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))
38 imassrn 6024 . . . . . . . . . . . 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 3993 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹)
41 lveclmod 20567 . . . . . . . . . . . . 13 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
4241ad4antr 730 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑉 ∈ LMod)
4323lbslinds 21239 . . . . . . . . . . . . . . 15 (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
4443, 21sselid 3942 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45 difssd 4092 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ 𝑏)
46 lindsss 21230 . . . . . . . . . . . . . 14 ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏𝑤) ⊆ 𝑏) → (𝑏𝑤) ∈ (LIndS‘𝑉))
4742, 44, 45, 46syl3anc 1371 . . . . . . . . . . . . 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 21216 . . . . . . . . . . . . 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 20437 . . . . . . . . . . . 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 20513 . . . . . . . . . . 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 20515 . . . . . . . . . . 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 477 . . . . . . . . . 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 1373 . . . . . . . . 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 20492 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
6362ad4antlr 731 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
6448, 23lbsss 20538 . . . . . . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . . . . . . 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 20418 . . . . . . . . . . . . . . . . 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 19024 . . . . . . . . . . . . . . . 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 20495 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
7473ad4antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
7574ffnd 6669 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 Fn (Base‘𝑉))
7648, 52lspssv 20444 . . . . . . . . . . . . . . . . . . . . . . . . 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))
78 fnssres 6624 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn (Base‘𝑉) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) Fn ((LSpan‘𝑉)‘(𝑏𝑤)))
7975, 77, 78syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) Fn ((LSpan‘𝑉)‘(𝑏𝑤)))
80 fniniseg 7010 . . . . . . . . . . . . . . . . . . . . . . . 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 )))
8180biimpa 477 . . . . . . . . . . . . . . . . . . . . . . 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 ))
8279, 81sylan 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 ))
8382simpld 495 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ ((LSpan‘𝑉)‘(𝑏𝑤)))
8475adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝐹 Fn (Base‘𝑉))
8577adantr 481 . . . . . . . . . . . . . . . . . . . . . . 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))
8685, 83sseldd 3945 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ (Base‘𝑉))
8783fvresd 6862 . . . . . . . . . . . . . . . . . . . . . . 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))
8882simprd 496 . . . . . . . . . . . . . . . . . . . . . . 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 )
8987, 88eqtr3d 2778 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → (𝐹𝑥) = 0 )
90 fniniseg 7010 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹𝑥) = 0 )))
9190biimpar 478 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹𝑥) = 0 )) → 𝑥 ∈ (𝐹 “ { 0 }))
9284, 86, 89, 91syl12anc 835 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ (𝐹 “ { 0 }))
9383, 92elind 4154 . . . . . . . . . . . . . . . . . . . 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 })))
94 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
95 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘𝐾) = (Base‘𝐾)
96 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (LSpan‘𝐾) = (LSpan‘𝐾)
9795, 4, 96lbssp 20540 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (LBasis‘𝐾) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
9894, 97syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
9941ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
100 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 “ { 0 }) = (𝐹 “ { 0 })
101100, 1, 51lmhmkerlss 20512 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
102101ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
10395, 4lbsss 20538 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ (LBasis‘𝐾) → 𝑤 ⊆ (Base‘𝐾))
10494, 103syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (Base‘𝐾))
105 cnvimass 6033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹 “ { 0 }) ⊆ dom 𝐹
106105, 73fssdm 6688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) ⊆ (Base‘𝑉))
1072, 48ressbas2 17120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹 “ { 0 }) ⊆ (Base‘𝑉) → (𝐹 “ { 0 }) = (Base‘𝐾))
108106, 107syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) = (Base‘𝐾))
109108ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝐹 “ { 0 }) = (Base‘𝐾))
110104, 109sseqtrrd 3985 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (𝐹 “ { 0 }))
1112, 52, 96, 51lsslsp 20476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
11299, 102, 110, 111syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
11398, 112, 1093eqtr4d 2786 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (𝐹 “ { 0 }))
114113ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) = (𝐹 “ { 0 }))
115114ineq2d 4172 . . . . . . . . . . . . . . . . . . . . . 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 32321 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g𝑉)})
118117ad5ant145 1369 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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)})
12193, 120eleqtrd 2840 . . . . . . . . . . . . . . . . . . 19 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ {(0g𝑉)})
122121ex 413 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) → 𝑥 ∈ {(0g𝑉)}))
123122ssrdv 3950 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) ⊆ {(0g𝑉)})
124116, 48, 520ellsp 32158 . . . . . . . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V)
127125fvresd 6862 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = (𝐹‘(0g𝑉)))
128116, 1ghmid 19014 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝑉 GrpHom 𝑈) → (𝐹‘(0g𝑉)) = 0 )
12962, 128syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹‘(0g𝑉)) = 0 )
130129ad4antlr 731 . . . . . . . . . . . . . . . . . . . . 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 4600 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V → (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ { 0 } ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = 0 ))
133132biimpar 478 . . . . . . . . . . . . . . . . . . . 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 })
13579, 125, 134elpreimad 7009 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }))
136135snssd 4769 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → {(0g𝑉)} ⊆ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }))
137123, 136eqssd 3961 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) = {(0g𝑉)})
138 lmodgrp 20329 . . . . . . . . . . . . . . . . . . 19 (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139 grpmnd 18755 . . . . . . . . . . . . . . . . . . 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 18584 . . . . . . . . . . . . . . . . . 18 ((𝑉 ∈ Mnd ∧ (0g𝑉) ∈ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉)) → (0g𝑉) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
142140, 125, 77, 141syl3anc 1371 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
143142sneqd 4598 . . . . . . . . . . . . . . . 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 20177 . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . 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 17120 . . . . . . . . . . . . . . . 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 6776 . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . 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 6745 . . . . . . . . . . . . 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 6795 . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . . . . . . 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 7010 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 Fn (Base‘𝑉) → (𝑢 ∈ (𝐹 “ { 0 }) ↔ (𝑢 ∈ (Base‘𝑉) ∧ (𝐹𝑢) = 0 )))
166165simplbda 500 . . . . . . . . . . . . . . . . . . . . 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 7372 . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . . 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 20445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
17342, 65, 18, 172syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
17448, 23, 52lbssp 20540 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177176ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . . . . . . . . . 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 19013 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (𝑉 GrpHom 𝑈) ∧ 𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑉)) → (𝐹‘(𝑢(+g𝑉)𝑣)) = ((𝐹𝑢)(+g𝑈)(𝐹𝑣)))
187171, 179, 183, 186syl3anc 1371 . . . . . . . . . . . . . . . . . . . 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 32316 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190 lveclmod 20567 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈 ∈ LVec → 𝑈 ∈ LMod)
191 lmodgrp 20329 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈 ∈ LMod → 𝑈 ∈ Grp)
192189, 190, 1913syl 18 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
193192ad9antr 740 . . . . . . . . . . . . . . . . . . . 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)
19474ad6antr 734 . . . . . . . . . . . . . . . . . . . . 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))
195194, 183ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . 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))
19672, 185, 1grplid 18780 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 ∈ Grp ∧ (𝐹𝑣) ∈ (Base‘𝑈)) → ( 0 (+g𝑈)(𝐹𝑣)) = (𝐹𝑣))
197193, 195, 196syl2anc 584 . . . . . . . . . . . . . . . . . . 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))
198168, 188, 1973eqtr3d 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))
199160, 198eqtr3d 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))
200161, 183, 182fnfvimad 7184 . . . . . . . . . . . . . . . . 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))))
201199, 200eqeltrd 2838 . . . . . . . . . . . . . . . 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))))
202 simp-7l 787 . . . . . . . . . . . . . . . . 17 ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹𝑥) = 𝑦) → 𝑉 ∈ LVec)
203 simplr 767 . . . . . . . . . . . . . . . . . 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))
204110ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ⊆ (𝐹 “ { 0 }))
205106ad4antlr 731 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 “ { 0 }) ⊆ (Base‘𝑉))
206204, 205sstrd 3954 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ⊆ (Base‘𝑉))
207 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (LSSum‘𝑉) = (LSSum‘𝑉)
20848, 52, 207lsmsp2 20548 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 ∈ LMod ∧ 𝑤 ⊆ (Base‘𝑉) ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏𝑤))) = ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏𝑤))))
20942, 206, 66, 208syl3anc 1371 . . . . . . . . . . . . . . . . . . . 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))))
21020fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏𝑤))) = ((LSpan‘𝑉)‘𝑏))
211209, 210, 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))))
212211ad3antrrr 728 . . . . . . . . . . . . . . . . . 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))))
213203, 212eleqtrd 2840 . . . . . . . . . . . . . . . . 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))))
21448, 184, 207lsmelvalx 19422 . . . . . . . . . . . . . . . . . 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)))
215214biimpa 477 . . . . . . . . . . . . . . . . 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))
216202, 177, 180, 213, 215syl31anc 1373 . . . . . . . . . . . . . . . 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))
217201, 216r19.29vva 3207 . . . . . . . . . . . . . . 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))))
218 fvelrnb 6903 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦))
219218biimpa 477 . . . . . . . . . . . . . . . 16 ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦)
22075, 219sylan 580 . . . . . . . . . . . . . . 15 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦)
221217, 220r19.29a 3159 . . . . . . . . . . . . . 14 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))))
22239, 221eqelssd 3965 . . . . . . . . . . . . 13 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran 𝐹)
22337, 222eqtr3id 2790 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran 𝐹)
224223f1oeq3d 6781 . . . . . . . . . . 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))
225159, 224mpbid 231 . . . . . . . . . 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)
22642, 50, 76syl2anc 584 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
227226, 151syl 17 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
228 frn 6675 . . . . . . . . . . . . 13 (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
22929, 72ressbas2 17120 . . . . . . . . . . . . 13 (ran 𝐹 ⊆ (Base‘𝑈) → ran 𝐹 = (Base‘𝐼))
23073, 228, 2293syl 18 . . . . . . . . . . . 12 (𝐹 ∈ (𝑉 LMHom 𝑈) → ran 𝐹 = (Base‘𝐼))
23132, 230syl 17 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran 𝐹 = (Base‘𝐼))
232150, 227, 231f1oeq123d 6778 . . . . . . . . . 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)))
233225, 232mpbid 231 . . . . . . . . 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))
234 eqid 2736 . . . . . . . . . 10 (Base‘𝐼) = (Base‘𝐼)
235145, 234islmim 20523 . . . . . . . . 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)))
23661, 233, 235sylanbrc 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))
23748, 52lspssid 20446 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)))
23842, 50, 237syl2anc 584 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)))
23951, 55lsslinds 21237 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) → ((𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) ↔ (𝑏𝑤) ∈ (LIndS‘𝑉)))
240239biimpar 478 . . . . . . . . . 10 (((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) ∧ (𝑏𝑤) ∈ (LIndS‘𝑉)) → (𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
24142, 67, 238, 47, 240syl31anc 1373 . . . . . . . . 9 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
242 eqid 2736 . . . . . . . . . . . 12 (LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
24355, 52, 242, 51lsslsp 20476 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) → ((LSpan‘𝑉)‘(𝑏𝑤)) = ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)))
24442, 54, 238, 243syl3anc 1371 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) = ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)))
245244, 227eqtr3d 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)))))
246 eqid 2736 . . . . . . . . . 10 (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
247145, 246, 242islbs4 21238 . . . . . . . . 9 ((𝑏𝑤) ∈ (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) ↔ ((𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) ∧ ((LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))‘(𝑏𝑤)) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))))
248241, 245, 247sylanbrc 583 . . . . . . . 8 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
249 eqid 2736 . . . . . . . . 9 (LBasis‘𝐼) = (LBasis‘𝐼)
250246, 249lmimlbs 21242 . . . . . . . 8 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMIso 𝐼) ∧ (𝑏𝑤) ∈ (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼))
251236, 248, 250syl2anc 584 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼))
252249dimval 32300 . . . . . . 7 ((𝐼 ∈ LVec ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼)) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))))
25331, 251, 252syl2anc 584 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))))
254 f1imaeng 8954 . . . . . . . 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))
255 hasheni 14248 . . . . . . . 8 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ≈ (𝑏𝑤) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))) = (♯‘(𝑏𝑤)))
256254, 255syl 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)))
257157, 238, 47, 256syl3anc 1371 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))) = (♯‘(𝑏𝑤)))
258253, 257eqtrd 2776 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝐼) = (♯‘(𝑏𝑤)))
25928, 258oveq12d 7375 . . . 4 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((dim‘𝐾) +𝑒 (dim‘𝐼)) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
26016, 25, 2593eqtr4d 2786 . . 3 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
2614lbslinds 21239 . . . . . 6 (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
262261, 94sselid 3942 . . . . 5 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
26351, 2lsslinds 21237 . . . . . 6 ((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
264263biimpa 477 . . . . 5 (((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
26599, 102, 110, 262, 264syl31anc 1373 . . . 4 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
26623islinds4 21241 . . . . 5 (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏))
267266ad2antrr 724 . . . 4 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏))
268265, 267mpbid 231 . . 3 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏)
269260, 268r19.29a 3159 . 2 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
2708, 269exlimddv 1938 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wrex 3073  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586   class class class wbr 5105  ccnv 5632  ran crn 5634  cres 5635  cima 5636   Fn wfn 6491  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cen 8880   +𝑒 cxad 13031  chash 14230  Basecbs 17083  s cress 17112  +gcplusg 17133  0gc0g 17321  Mndcmnd 18556  Grpcgrp 18748  SubGrpcsubg 18922   GrpHom cghm 19005  LSSumclsm 19416  LModclmod 20322  LSubSpclss 20392  LSpanclspn 20432   LMHom clmhm 20480   LMIso clmim 20481  LBasisclbs 20535  LVecclvec 20563  LIndSclinds 21211  dimcldim 32298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-rpss 7660  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-r1 9700  df-rank 9701  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-xadd 13034  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ocomp 17154  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-mri 17468  df-acs 17469  df-proset 18184  df-drs 18185  df-poset 18202  df-ipo 18417  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-drng 20187  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lmhm 20483  df-lmim 20484  df-lbs 20536  df-lvec 20564  df-sra 20633  df-rgmod 20634  df-nzr 20728  df-dsmm 21138  df-frlm 21153  df-uvc 21189  df-lindf 21212  df-linds 21213  df-dim 32299
This theorem is referenced by:  qusdimsum  32323
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