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Theorem dimkerim 30652
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 30647 . . . 4 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4 eqid 2772 . . . . 5 (LBasis‘𝐾) = (LBasis‘𝐾)
54lbsex 19653 . . . 4 (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
63, 5syl 17 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7 n0 4190 . . 3 ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
86, 7sylib 210 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
9 simpllr 763 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10 vex 3412 . . . . . . 7 𝑏 ∈ V
1110difexi 5082 . . . . . 6 (𝑏𝑤) ∈ V
1211a1i 11 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ V)
13 disjdif 4298 . . . . . 6 (𝑤 ∩ (𝑏𝑤)) = ∅
1413a1i 11 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∩ (𝑏𝑤)) = ∅)
15 hashunx 13554 . . . . 5 ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏𝑤) ∈ V ∧ (𝑤 ∩ (𝑏𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
169, 12, 14, 15syl3anc 1351 . . . 4 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (♯‘(𝑤 ∪ (𝑏𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏𝑤))))
17 simp-4l 770 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑉 ∈ LVec)
18 simpr 477 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤𝑏)
19 undif 4307 . . . . . . 7 (𝑤𝑏 ↔ (𝑤 ∪ (𝑏𝑤)) = 𝑏)
2018, 19sylib 210 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∪ (𝑏𝑤)) = 𝑏)
21 simplr 756 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑏 ∈ (LBasis‘𝑉))
2220, 21eqeltrd 2860 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑤 ∪ (𝑏𝑤)) ∈ (LBasis‘𝑉))
23 eqid 2772 . . . . . 6 (LBasis‘𝑉) = (LBasis‘𝑉)
2423dimval 30630 . . . . 5 ((𝑉 ∈ LVec ∧ (𝑤 ∪ (𝑏𝑤)) ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏𝑤))))
2517, 22, 24syl2anc 576 . . . 4 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏𝑤))))
263ad3antrrr 717 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐾 ∈ LVec)
274dimval 30630 . . . . . 6 ((𝐾 ∈ LVec ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑤))
2826, 9, 27syl2anc 576 . . . . 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 30648 . . . . . . . 8 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
3130ad3antrrr 717 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐼 ∈ LVec)
32 simp-4r 771 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33 lmhmlmod2 19520 . . . . . . . . . . 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 19538 . . . . . . . . . . 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 5414 . . . . . . . . . . 11 (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))
38 imassrn 5775 . . . . . . . . . . . 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, 39syl5eqssr 3900 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ⊆ ran 𝐹)
41 lveclmod 19594 . . . . . . . . . . . . 13 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
4241ad4antr 719 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑉 ∈ LMod)
4323lbslinds 20673 . . . . . . . . . . . . . . 15 (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
4443, 21sseldi 3850 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45 difssd 3993 . . . . . . . . . . . . . 14 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ 𝑏)
46 lindsss 20664 . . . . . . . . . . . . . 14 ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏𝑤) ⊆ 𝑏) → (𝑏𝑤) ∈ (LIndS‘𝑉))
4742, 44, 45, 46syl3anc 1351 . . . . . . . . . . . . 13 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LIndS‘𝑉))
48 eqid 2772 . . . . . . . . . . . . . 14 (Base‘𝑉) = (Base‘𝑉)
4948linds1 20650 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . 13 (LSubSp‘𝑉) = (LSubSp‘𝑉)
52 eqid 2772 . . . . . . . . . . . . 13 (LSpan‘𝑉) = (LSpan‘𝑉)
5348, 51, 52lspcl 19464 . . . . . . . . . . . 12 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉))
5442, 50, 53syl2anc 576 . . . . . . . . . . 11 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉))
55 eqid 2772 . . . . . . . . . . . 12 (𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) = (𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))
5651, 55reslmhm 19540 . . . . . . . . . . 11 ((𝐹 ∈ (𝑉 LMHom 𝑈) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) LMHom 𝑈))
5732, 54, 56syl2anc 576 . . . . . . . . . 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 2772 . . . . . . . . . . . 12 (LSubSp‘𝑈) = (LSubSp‘𝑈)
5929, 58reslmhm2b 19542 . . . . . . . . . . 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 469 . . . . . . . . . 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 1353 . . . . . . . . 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 19519 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
6362ad4antlr 720 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
6448, 23lbsss 19565 . . . . . . . . . . . . . . . . . . . 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 3862 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ (Base‘𝑉))
6742, 66, 53syl2anc 576 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉))
6851lsssubg 19445 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉)) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (SubGrp‘𝑉))
6942, 67, 68syl2anc 576 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (SubGrp‘𝑉))
7055resghm 18139 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ (𝑉 GrpHom 𝑈) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (SubGrp‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) ∈ ((𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))) GrpHom 𝑈))
7163, 69, 70syl2anc 576 . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘𝑈) = (Base‘𝑈)
7348, 72lmhmf 19522 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
7473ad4antlr 720 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
7574ffnd 6339 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝐹 Fn (Base‘𝑉))
7648, 52lspssv 19471 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
7742, 66, 76syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉))
78 fnssres 6297 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn (Base‘𝑉) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) Fn ((LSpan‘𝑉)‘(𝑏𝑤)))
7975, 77, 78syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) Fn ((LSpan‘𝑉)‘(𝑏𝑤)))
80 fniniseg 6649 . . . . . . . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . 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 3853 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ (Base‘𝑉))
8783fvresd 6513 . . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . . 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 2810 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → (𝐹𝑥) = 0 )
90 fniniseg 6649 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹𝑥) = 0 )))
9190biimpar 470 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹𝑥) = 0 )) → 𝑥 ∈ (𝐹 “ { 0 }))
9284, 86, 89, 91syl12anc 824 . . . . . . . . . . . . . . . . . . . . 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 4053 . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
95 eqid 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘𝐾) = (Base‘𝐾)
96 eqid 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (LSpan‘𝐾) = (LSpan‘𝐾)
9795, 4, 96lbssp 19567 . . . . . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
100 eqid 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 “ { 0 }) = (𝐹 “ { 0 })
101100, 1, 51lmhmkerlss 19539 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
102101ad2antlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
10395, 4lbsss 19565 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹 “ { 0 }) ⊆ dom 𝐹
106105, 73fssdm 6354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) ⊆ (Base‘𝑉))
1072, 48ressbas2 16405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹 “ { 0 }) ⊆ (Base‘𝑉) → (𝐹 “ { 0 }) = (Base‘𝐾))
108106, 107syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹 “ { 0 }) = (Base‘𝐾))
109108ad2antlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝐹 “ { 0 }) = (Base‘𝐾))
110104, 109sseqtr4d 3892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (𝐹 “ { 0 }))
1112, 52, 96, 51lsslsp 19503 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
11299, 102, 110, 111syl3anc 1351 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
11398, 112, 1093eqtr4d 2818 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (𝐹 “ { 0 }))
114113ad2antrr 713 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) = (𝐹 “ { 0 }))
115114ineq2d 4070 . . . . . . . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . . . . . . . 24 (0g𝑉) = (0g𝑉)
11723, 52, 116lbsdiflsp0 30651 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g𝑉)})
118117ad5ant145 1349 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g𝑉)})
119115, 118eqtr3d 2810 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (((LSpan‘𝑉)‘(𝑏𝑤)) ∩ (𝐹 “ { 0 })) = {(0g𝑉)})
120119adantr 473 . . . . . . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . . . . . . . 19 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 })) → 𝑥 ∈ {(0g𝑉)})
122121ex 405 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑥 ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) → 𝑥 ∈ {(0g𝑉)}))
123122ssrdv 3858 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) ⊆ {(0g𝑉)})
124116, 48, 520ellsp 30607 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → (0g𝑉) ∈ ((LSpan‘𝑉)‘(𝑏𝑤)))
12542, 66, 124syl2anc 576 . . . . . . . . . . . . . . . . . . 19 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) ∈ ((LSpan‘𝑉)‘(𝑏𝑤)))
126 fvexd 6508 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V)
127125fvresd 6513 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = (𝐹‘(0g𝑉)))
128116, 1ghmid 18129 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝑉 GrpHom 𝑈) → (𝐹‘(0g𝑉)) = 0 )
12962, 128syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹‘(0g𝑉)) = 0 )
130129ad4antlr 720 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹‘(0g𝑉)) = 0 )
131127, 130eqtrd 2808 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = 0 )
132 elsng 4449 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V → (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ { 0 } ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = 0 ))
133132biimpar 470 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ V ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) = 0 ) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ { 0 })
134126, 131, 133syl2anc 576 . . . . . . . . . . . . . . . . . . 19 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤)))‘(0g𝑉)) ∈ { 0 })
13579, 125, 134elpreimad 6648 . . . . . . . . . . . . . . . . . 18 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) ∈ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }))
136135snssd 4610 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → {(0g𝑉)} ⊆ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }))
137123, 136eqssd 3869 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ { 0 }) = {(0g𝑉)})
138 lmodgrp 19357 . . . . . . . . . . . . . . . . . . 19 (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139 grpmnd 17892 . . . . . . . . . . . . . . . . . . 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 17781 . . . . . . . . . . . . . . . . . 18 ((𝑉 ∈ Mnd ∧ (0g𝑉) ∈ ((LSpan‘𝑉)‘(𝑏𝑤)) ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ⊆ (Base‘𝑉)) → (0g𝑉) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
142140, 125, 77, 141syl3anc 1351 . . . . . . . . . . . . . . . . 17 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (0g𝑉) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
143142sneqd 4447 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → {(0g𝑉)} = {(0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))})
144137, 143eqtrd 2808 . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . 17 (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (Base‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
146 eqid 2772 . . . . . . . . . . . . . . . . 17 (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (0g‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
147145, 72, 146, 1kerf1ghm 19214 . . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . . 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 576 . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) = (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))))
15155, 48ressbas2 16405 . . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (Base‘𝑈) = (Base‘𝑈))
154150, 152, 153f1eq123d 6431 . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . 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 6404 . . . . . . . . . . . . 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 576 . . . . . . . . . . . 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 6449 . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . . . . . . . . . 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 6649 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 Fn (Base‘𝑉) → (𝑢 ∈ (𝐹 “ { 0 }) ↔ (𝑢 ∈ (Base‘𝑉) ∧ (𝐹𝑢) = 0 )))
166165simplbda 492 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn (Base‘𝑉) ∧ 𝑢 ∈ (𝐹 “ { 0 })) → (𝐹𝑢) = 0 )
167161, 164, 166syl2anc 576 . . . . . . . . . . . . . . . . . . . 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 6985 . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . 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 6497 . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . 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 19472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
17342, 65, 18, 172syl3anc 1351 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
17448, 23, 52lbssp 19567 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3891 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177176ad3antrrr 717 . . . . . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . . . . . 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 3853 . . . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . . . 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 3853 . . . . . . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . . . . . 22 (+g𝑉) = (+g𝑉)
185 eqid 2772 . . . . . . . . . . . . . . . . . . . . . 22 (+g𝑈) = (+g𝑈)
18648, 184, 185ghmlin 18128 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (𝑉 GrpHom 𝑈) ∧ 𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑉)) → (𝐹‘(𝑢(+g𝑉)𝑣)) = ((𝐹𝑢)(+g𝑈)(𝐹𝑣)))
187171, 179, 183, 186syl3anc 1351 . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . 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 30646 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190 lveclmod 19594 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈 ∈ LVec → 𝑈 ∈ LMod)
191 lmodgrp 19357 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈 ∈ LMod → 𝑈 ∈ Grp)
192189, 190, 1913syl 18 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
193192ad9antr 729 . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . 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, 183ffvelrnd 6671 . . . . . . . . . . . . . . . . . . . 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 17915 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 ∈ Grp ∧ (𝐹𝑣) ∈ (Base‘𝑈)) → ( 0 (+g𝑈)(𝐹𝑣)) = (𝐹𝑣))
197193, 195, 196syl2anc 576 . . . . . . . . . . . . . . . . . . 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 2816 . . . . . . . . . . . . . . . . . 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 2810 . . . . . . . . . . . . . . . . 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 6815 . . . . . . . . . . . . . . . . 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 2860 . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ⊆ (𝐹 “ { 0 }))
205106ad4antlr 720 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 “ { 0 }) ⊆ (Base‘𝑉))
206204, 205sstrd 3862 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → 𝑤 ⊆ (Base‘𝑉))
207 eqid 2772 . . . . . . . . . . . . . . . . . . . . . 22 (LSSum‘𝑉) = (LSSum‘𝑉)
20848, 52, 207lsmsp2 19575 . . . . . . . . . . . . . . . . . . . . 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 1351 . . . . . . . . . . . . . . . . . . . 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 6497 . . . . . . . . . . . . . . . . . . . 20 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏𝑤))) = ((LSpan‘𝑉)‘𝑏))
211209, 210, 1753eqtrrd 2813 . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . . . . . 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 18520 . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . 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 1353 . . . . . . . . . . . . . . . 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 3271 . . . . . . . . . . . . . . 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 6550 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦))
219218biimpa 469 . . . . . . . . . . . . . . . 16 ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹𝑥) = 𝑦)
22075, 219sylan 572 . . . . . . . . . . . . . . 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 3228 . . . . . . . . . . . . . 14 ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))))
22239, 221eqelssd 3872 . . . . . . . . . . . . 13 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝐹 “ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran 𝐹)
22337, 222syl5eqr 2822 . . . . . . . . . . . 12 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) = ran 𝐹)
224223f1oeq3d 6435 . . . . . . . . . . 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 224 . . . . . . . . . 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 576 . . . . . . . . . . . 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 6344 . . . . . . . . . . . . 13 (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
22929, 72ressbas2 16405 . . . . . . . . . . . . 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 6433 . . . . . . . . . 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 224 . . . . . . . . 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 2772 . . . . . . . . . 10 (Base‘𝐼) = (Base‘𝐼)
235145, 234islmim 19550 . . . . . . . . 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 575 . . . . . . . 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 19473 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ (𝑏𝑤) ⊆ (Base‘𝑉)) → (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)))
23842, 50, 237syl2anc 576 . . . . . . . . . 10 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤)))
23951, 55lsslinds 20671 . . . . . . . . . . 11 ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏𝑤))) → ((𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) ↔ (𝑏𝑤) ∈ (LIndS‘𝑉)))
240239biimpar 470 . . . . . . . . . 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 1353 . . . . . . . . 9 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LIndS‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
242 eqid 2772 . . . . . . . . . . . 12 (LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (LSpan‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
24355, 52, 242, 51lsslsp 19503 . . . . . . . . . . 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 1351 . . . . . . . . . 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 2810 . . . . . . . . 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 2772 . . . . . . . . . 10 (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))) = (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤))))
247145, 246, 242islbs4 20672 . . . . . . . . 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 575 . . . . . . . 8 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (𝑏𝑤) ∈ (LBasis‘(𝑉s ((LSpan‘𝑉)‘(𝑏𝑤)))))
249 eqid 2772 . . . . . . . . 9 (LBasis‘𝐼) = (LBasis‘𝐼)
250246, 249lmimlbs 20676 . . . . . . . 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 576 . . . . . . 7 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼))
252249dimval 30630 . . . . . . 7 ((𝐼 ∈ LVec ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤)) ∈ (LBasis‘𝐼)) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))))
25331, 251, 252syl2anc 576 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))))
254 f1imaeng 8360 . . . . . . . 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 13517 . . . . . . . 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 1351 . . . . . 6 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏𝑤))) “ (𝑏𝑤))) = (♯‘(𝑏𝑤)))
258253, 257eqtrd 2808 . . . . 5 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝐼) = (♯‘(𝑏𝑤)))
25928, 258oveq12d 6988 . . . 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 2818 . . 3 (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤𝑏) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
2614lbslinds 20673 . . . . . 6 (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
262261, 94sseldi 3850 . . . . 5 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
26351, 2lsslinds 20671 . . . . . 6 ((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
264263biimpa 469 . . . . 5 (((𝑉 ∈ LMod ∧ (𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
26599, 102, 110, 262, 264syl31anc 1353 . . . 4 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
26623islinds4 20675 . . . . 5 (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏))
267266ad2antrr 713 . . . 4 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏))
268265, 267mpbid 224 . . 3 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ∃𝑏 ∈ (LBasis‘𝑉)𝑤𝑏)
269260, 268r19.29a 3228 . 2 (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
2708, 269exlimddv 1894 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 198  wa 387  w3a 1068   = wceq 1507  wex 1742  wcel 2050  wne 2961  wrex 3083  Vcvv 3409  cdif 3820  cun 3821  cin 3822  wss 3823  c0 4172  {csn 4435   class class class wbr 4923  ccnv 5400  ran crn 5402  cres 5403  cima 5404   Fn wfn 6177  wf 6178  1-1wf1 6179  1-1-ontowf1o 6181  cfv 6182  (class class class)co 6970  cen 8297   +𝑒 cxad 12316  chash 13499  Basecbs 16333  s cress 16334  +gcplusg 16415  0gc0g 16563  Mndcmnd 17756  Grpcgrp 17885  SubGrpcsubg 18051   GrpHom cghm 18120  LSSumclsm 18514  LModclmod 19350  LSubSpclss 19419  LSpanclspn 19459   LMHom clmhm 19507   LMIso clmim 19508  LBasisclbs 19562  LVecclvec 19590  LIndSclinds 20645  dimcldim 30628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-reg 8845  ax-inf2 8892  ax-ac2 9677  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-se 5361  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-rpss 7261  df-om 7391  df-1st 7495  df-2nd 7496  df-supp 7628  df-tpos 7689  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-ixp 8254  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-fsupp 8623  df-sup 8695  df-oi 8763  df-r1 8981  df-rank 8982  df-dju 9118  df-card 9156  df-acn 9159  df-ac 9330  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-2 11497  df-3 11498  df-4 11499  df-5 11500  df-6 11501  df-7 11502  df-8 11503  df-9 11504  df-n0 11702  df-xnn0 11774  df-z 11788  df-dec 11906  df-uz 12053  df-xadd 12319  df-fz 12703  df-fzo 12844  df-seq 13179  df-hash 13500  df-struct 16335  df-ndx 16336  df-slot 16337  df-base 16339  df-sets 16340  df-ress 16341  df-plusg 16428  df-mulr 16429  df-sca 16431  df-vsca 16432  df-ip 16433  df-tset 16434  df-ple 16435  df-ocomp 16436  df-ds 16437  df-hom 16439  df-cco 16440  df-0g 16565  df-gsum 16566  df-prds 16571  df-pws 16573  df-mre 16709  df-mrc 16710  df-mri 16711  df-acs 16712  df-proset 17390  df-drs 17391  df-poset 17408  df-ipo 17614  df-mgm 17704  df-sgrp 17746  df-mnd 17757  df-mhm 17797  df-submnd 17798  df-grp 17888  df-minusg 17889  df-sbg 17890  df-mulg 18006  df-subg 18054  df-ghm 18121  df-cntz 18212  df-lsm 18516  df-cmn 18662  df-abl 18663  df-mgp 18957  df-ur 18969  df-ring 19016  df-oppr 19090  df-dvdsr 19108  df-unit 19109  df-invr 19139  df-drng 19221  df-subrg 19250  df-lmod 19352  df-lss 19420  df-lsp 19460  df-lmhm 19510  df-lmim 19511  df-lbs 19563  df-lvec 19591  df-sra 19660  df-rgmod 19661  df-nzr 19746  df-dsmm 20572  df-frlm 20587  df-uvc 20623  df-lindf 20646  df-linds 20647  df-dim 30629
This theorem is referenced by:  qusdimsum  30653
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