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Theorem fedgmul 33683
Description: The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, we have [𝐸:𝐾] = [𝐸:𝐹][𝐹:𝐾]. Proposition 1.2 of [Lang], p. 224. Here (dim‘𝐴) is the degree of the extension 𝐸 of 𝐾, (dim‘𝐵) is the degree of the extension 𝐸 of 𝐹, and (dim‘𝐶) is the degree of the extension 𝐹 of 𝐾. This proof is valid for infinite dimensions, and is actually valid for division ring extensions, not just field extensions. (Contributed by Thierry Arnoux, 25-Jul-2023.)
Hypotheses
Ref Expression
fedgmul.a 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
fedgmul.b 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
fedgmul.c 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
fedgmul.f 𝐹 = (𝐸s 𝑈)
fedgmul.k 𝐾 = (𝐸s 𝑉)
fedgmul.1 (𝜑𝐸 ∈ DivRing)
fedgmul.2 (𝜑𝐹 ∈ DivRing)
fedgmul.3 (𝜑𝐾 ∈ DivRing)
fedgmul.4 (𝜑𝑈 ∈ (SubRing‘𝐸))
fedgmul.5 (𝜑𝑉 ∈ (SubRing‘𝐹))
Assertion
Ref Expression
fedgmul (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))

Proof of Theorem fedgmul
Dummy variables 𝑎 𝑐 𝑓 𝑢 𝑥 𝑦 𝑧 𝑖 𝑗 𝑤 𝑏 𝑣 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fedgmul.2 . . . . 5 (𝜑𝐹 ∈ DivRing)
2 fedgmul.4 . . . . . . . 8 (𝜑𝑈 ∈ (SubRing‘𝐸))
3 fedgmul.5 . . . . . . . . . 10 (𝜑𝑉 ∈ (SubRing‘𝐹))
4 fedgmul.f . . . . . . . . . . . 12 𝐹 = (𝐸s 𝑈)
54subsubrg 20599 . . . . . . . . . . 11 (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈)))
65biimpa 476 . . . . . . . . . 10 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
72, 3, 6syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
87simprd 495 . . . . . . . 8 (𝜑𝑉𝑈)
9 ressabs 17295 . . . . . . . 8 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈) → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
102, 8, 9syl2anc 584 . . . . . . 7 (𝜑 → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
114oveq1i 7442 . . . . . . 7 (𝐹s 𝑉) = ((𝐸s 𝑈) ↾s 𝑉)
12 fedgmul.k . . . . . . 7 𝐾 = (𝐸s 𝑉)
1310, 11, 123eqtr4g 2801 . . . . . 6 (𝜑 → (𝐹s 𝑉) = 𝐾)
14 fedgmul.3 . . . . . 6 (𝜑𝐾 ∈ DivRing)
1513, 14eqeltrd 2840 . . . . 5 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
16 fedgmul.c . . . . . 6 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
17 eqid 2736 . . . . . 6 (𝐹s 𝑉) = (𝐹s 𝑉)
1816, 17sralvec 33637 . . . . 5 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
191, 15, 3, 18syl3anc 1372 . . . 4 (𝜑𝐶 ∈ LVec)
20 eqid 2736 . . . . 5 (LBasis‘𝐶) = (LBasis‘𝐶)
2120lbsex 21168 . . . 4 (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
2219, 21syl 17 . . 3 (𝜑 → (LBasis‘𝐶) ≠ ∅)
23 n0 4352 . . 3 ((LBasis‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
2422, 23sylib 218 . 2 (𝜑 → ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
25 fedgmul.1 . . . . . . 7 (𝜑𝐸 ∈ DivRing)
26 fedgmul.b . . . . . . . 8 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
2726, 4sralvec 33637 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
2825, 1, 2, 27syl3anc 1372 . . . . . 6 (𝜑𝐵 ∈ LVec)
29 eqid 2736 . . . . . . 7 (LBasis‘𝐵) = (LBasis‘𝐵)
3029lbsex 21168 . . . . . 6 (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
3128, 30syl 17 . . . . 5 (𝜑 → (LBasis‘𝐵) ≠ ∅)
32 n0 4352 . . . . 5 ((LBasis‘𝐵) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
3331, 32sylib 218 . . . 4 (𝜑 → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
3433adantr 480 . . 3 ((𝜑𝑥 ∈ (LBasis‘𝐶)) → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
35 drngring 20737 . . . . . . . . . . . . . . 15 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
3625, 35syl 17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ Ring)
3736ad4antr 732 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝐸 ∈ Ring)
38 simplr 768 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39 eqid 2736 . . . . . . . . . . . . . . . . . 18 (Base‘𝐶) = (Base‘𝐶)
4039, 20lbsss 21077 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (LBasis‘𝐶) → 𝑥 ⊆ (Base‘𝐶))
4138, 40syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐶))
42 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐸) = (Base‘𝐸)
4342subrgss 20573 . . . . . . . . . . . . . . . . . . . . 21 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
442, 43syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑈 ⊆ (Base‘𝐸))
454, 42ressbas2 17284 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
4644, 45syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑈 = (Base‘𝐹))
4716a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐶 = ((subringAlg ‘𝐹)‘𝑉))
48 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐹) = (Base‘𝐹)
4948subrgss 20573 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
503, 49syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑉 ⊆ (Base‘𝐹))
5147, 50srabase 21178 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
5246, 51eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐶))
5352, 44eqsstrrd 4018 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
5453ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐶) ⊆ (Base‘𝐸))
5541, 54sstrd 3993 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐸))
5655ad2antrr 726 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑥 ⊆ (Base‘𝐸))
57 simpr 484 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑖𝑥)
5856, 57sseldd 3983 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑖 ∈ (Base‘𝐸))
59 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LBasis‘𝐵))
60 eqid 2736 . . . . . . . . . . . . . . . . . 18 (Base‘𝐵) = (Base‘𝐵)
6160, 29lbsss 21077 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (LBasis‘𝐵) → 𝑦 ⊆ (Base‘𝐵))
6259, 61syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐵))
6326a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 = ((subringAlg ‘𝐸)‘𝑈))
6463, 44srabase 21178 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
6564ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐵))
6662, 65sseqtrrd 4020 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐸))
6766ad2antrr 726 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑦 ⊆ (Base‘𝐸))
68 simplr 768 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
6967, 68sseldd 3983 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗 ∈ (Base‘𝐸))
70 eqid 2736 . . . . . . . . . . . . . 14 (.r𝐸) = (.r𝐸)
7142, 70ringcl 20248 . . . . . . . . . . . . 13 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
7237, 58, 69, 71syl3anc 1372 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
73 fedgmul.a . . . . . . . . . . . . . . 15 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
7473a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐴 = ((subringAlg ‘𝐸)‘𝑉))
757simpld 494 . . . . . . . . . . . . . . 15 (𝜑𝑉 ∈ (SubRing‘𝐸))
7642subrgss 20573 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
7775, 76syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐸))
7874, 77srabase 21178 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
7978ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (Base‘𝐸) = (Base‘𝐴))
8072, 79eleqtrd 2842 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
8180anasss 466 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗𝑦𝑖𝑥)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
8281ralrimivva 3201 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
83 oveq2 7440 . . . . . . . . . . 11 (𝑤 = 𝑗 → (𝑡(.r𝐸)𝑤) = (𝑡(.r𝐸)𝑗))
84 oveq1 7439 . . . . . . . . . . 11 (𝑡 = 𝑖 → (𝑡(.r𝐸)𝑗) = (𝑖(.r𝐸)𝑗))
8583, 84cbvmpov 7529 . . . . . . . . . 10 (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) = (𝑗𝑦, 𝑖𝑥 ↦ (𝑖(.r𝐸)𝑗))
8685fmpo 8094 . . . . . . . . 9 (∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴) ↔ (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
8782, 86sylib 218 . . . . . . . 8 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
88 eqid 2736 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
89 eqid 2736 . . . . . . . . . . . . . 14 ( ·𝑠𝐵) = ( ·𝑠𝐵)
90 eqid 2736 . . . . . . . . . . . . . 14 (+g𝐵) = (+g𝐵)
91 eqid 2736 . . . . . . . . . . . . . 14 (0g‘(Scalar‘𝐵)) = (0g‘(Scalar‘𝐵))
9228ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LVec)
9392ad5antr 734 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝐵 ∈ LVec)
9429lbslinds 21854 . . . . . . . . . . . . . . . 16 (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
9594, 59sselid 3980 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LIndS‘𝐵))
9695ad5antr 734 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑦 ∈ (LIndS‘𝐵))
9768ad3antrrr 730 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑗𝑦)
98 simpllr 775 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑣𝑦)
9963, 44srasca 21184 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
1004, 99eqtrid 2788 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 = (Scalar‘𝐵))
101100fveq2d 6909 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102101, 51eqtr3d 2778 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
103102ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
10441, 103sseqtrrd 4020 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
105104ad5antr 734 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
106 simp-4r 783 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑖𝑥)
107105, 106sseldd 3983 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑖 ∈ (Base‘(Scalar‘𝐵)))
108 simplr 768 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑢𝑥)
109105, 108sseldd 3983 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑢 ∈ (Base‘(Scalar‘𝐵)))
11019ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LVec)
111 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (LSpan‘𝐶) = (LSpan‘𝐶)
11239, 20, 111islbs4 21853 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (LBasis‘𝐶) ↔ (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
11338, 112sylib 218 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
114113simpld 494 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LIndS‘𝐶))
115 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (0g𝐶) = (0g𝐶)
1161150nellinds 33399 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LIndS‘𝐶)) → ¬ (0g𝐶) ∈ 𝑥)
117110, 114, 116syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ¬ (0g𝐶) ∈ 𝑥)
118117ad5antr 734 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → ¬ (0g𝐶) ∈ 𝑥)
119 nelne2 3039 . . . . . . . . . . . . . . . 16 ((𝑖𝑥 ∧ ¬ (0g𝐶) ∈ 𝑥) → 𝑖 ≠ (0g𝐶))
120106, 118, 119syl2anc 584 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑖 ≠ (0g𝐶))
121100fveq2d 6909 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐵)))
12216, 1, 3drgext0g 33641 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g𝐶))
123121, 122eqtr3d 2778 . . . . . . . . . . . . . . . 16 (𝜑 → (0g‘(Scalar‘𝐵)) = (0g𝐶))
124123ad7antr 738 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (0g‘(Scalar‘𝐵)) = (0g𝐶))
125120, 124neeqtrrd 3014 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → 𝑖 ≠ (0g‘(Scalar‘𝐵)))
126 simpr 484 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢))
127 ovexd 7467 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑖(.r𝐸)𝑗) ∈ V)
12885ovmpt4g 7581 . . . . . . . . . . . . . . . . 17 ((𝑗𝑦𝑖𝑥 ∧ (𝑖(.r𝐸)𝑗) ∈ V) → (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑖(.r𝐸)𝑗))
12997, 106, 127, 128syl3anc 1372 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑖(.r𝐸)𝑗))
13026, 25, 2drgextvsca 33642 . . . . . . . . . . . . . . . . . 18 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
131130oveqd 7449 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑖(.r𝐸)𝑗) = (𝑖( ·𝑠𝐵)𝑗))
132131ad7antr 738 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑖(.r𝐸)𝑗) = (𝑖( ·𝑠𝐵)𝑗))
133129, 132eqtrd 2776 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑖( ·𝑠𝐵)𝑗))
13485a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) = (𝑗𝑦, 𝑖𝑥 ↦ (𝑖(.r𝐸)𝑗)))
135 simprr 772 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑖 = 𝑢)
136 simprl 770 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑗 = 𝑣)
137135, 136oveq12d 7450 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → (𝑖(.r𝐸)𝑗) = (𝑢(.r𝐸)𝑣))
138 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → 𝑣𝑦)
139 simpr 484 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → 𝑢𝑥)
140 ovexd 7467 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑢(.r𝐸)𝑣) ∈ V)
141134, 137, 138, 139, 140ovmpod 7586 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
142141adantllr 719 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
143142adantl3r 750 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
144143adantr 480 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
145130oveqd 7449 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑢(.r𝐸)𝑣) = (𝑢( ·𝑠𝐵)𝑣))
146145ad7antr 738 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑢(.r𝐸)𝑣) = (𝑢( ·𝑠𝐵)𝑣))
147144, 146eqtrd 2776 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢( ·𝑠𝐵)𝑣))
148126, 133, 1473eqtr3d 2784 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑖( ·𝑠𝐵)𝑗) = (𝑢( ·𝑠𝐵)𝑣))
14988, 89, 90, 91, 93, 96, 97, 98, 107, 109, 125, 148linds2eq 33410 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢)) → (𝑗 = 𝑣𝑖 = 𝑢))
150149ex 412 . . . . . . . . . . . 12 (((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → ((𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) → (𝑗 = 𝑣𝑖 = 𝑢)))
151150anasss 466 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ (𝑣𝑦𝑢𝑥)) → ((𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) → (𝑗 = 𝑣𝑖 = 𝑢)))
152151ralrimivva 3201 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → ∀𝑣𝑦𝑢𝑥 ((𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) → (𝑗 = 𝑣𝑖 = 𝑢)))
153152anasss 466 . . . . . . . . 9 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗𝑦𝑖𝑥)) → ∀𝑣𝑦𝑢𝑥 ((𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) → (𝑗 = 𝑣𝑖 = 𝑢)))
154153ralrimivva 3201 . . . . . . . 8 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥𝑣𝑦𝑢𝑥 ((𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) → (𝑗 = 𝑣𝑖 = 𝑢)))
155 f1opr 7490 . . . . . . . 8 ((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)–1-1→(Base‘𝐴) ↔ ((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴) ∧ ∀𝑗𝑦𝑖𝑥𝑣𝑦𝑢𝑥 ((𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) → (𝑗 = 𝑣𝑖 = 𝑢))))
15687, 154, 155sylanbrc 583 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)–1-1→(Base‘𝐴))
15759, 38xpexd 7772 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158 f1rnen 32640 . . . . . . 7 (((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)–1-1→(Base‘𝐴) ∧ (𝑦 × 𝑥) ∈ V) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ≈ (𝑦 × 𝑥))
159156, 157, 158syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ≈ (𝑦 × 𝑥))
160 hasheni 14388 . . . . . 6 (ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ≈ (𝑦 × 𝑥) → (♯‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (♯‘(𝑦 × 𝑥)))
161159, 160syl 17 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (♯‘(𝑦 × 𝑥)))
162 hashxpe 32812 . . . . . 6 ((𝑦 ∈ (LBasis‘𝐵) ∧ 𝑥 ∈ (LBasis‘𝐶)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
16359, 38, 162syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
164161, 163eqtrd 2776 . . . 4 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = ((♯‘𝑦) ·e (♯‘𝑥)))
16573, 12sralvec 33637 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
16625, 14, 75, 165syl3anc 1372 . . . . . 6 (𝜑𝐴 ∈ LVec)
167166ad2antrr 726 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168 lveclmod 21106 . . . . . . . . 9 (𝐴 ∈ LVec → 𝐴 ∈ LMod)
169166, 168syl 17 . . . . . . . 8 (𝜑𝐴 ∈ LMod)
170169ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LMod)
17125ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝐸 ∈ DivRing)
1721ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝐹 ∈ DivRing)
17314ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝐾 ∈ DivRing)
1742ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝑈 ∈ (SubRing‘𝐸))
1753ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝑉 ∈ (SubRing‘𝐹))
176 fveq2 6905 . . . . . . . . . . . 12 (𝑤 = 𝑗 → (𝑓𝑤) = (𝑓𝑗))
177176fveq1d 6907 . . . . . . . . . . 11 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑗)‘𝑣))
178 fveq2 6905 . . . . . . . . . . 11 (𝑣 = 𝑖 → ((𝑓𝑗)‘𝑣) = ((𝑓𝑗)‘𝑖))
179177, 178cbvmpov 7529 . . . . . . . . . 10 (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) = (𝑗𝑦, 𝑖𝑥 ↦ ((𝑓𝑗)‘𝑖))
180 simp-4r 783 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝑥 ∈ (LBasis‘𝐶))
181 simpllr 775 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝑦 ∈ (LBasis‘𝐵))
182 simplr 768 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥))))
183 simpr 484 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴))
18473, 26, 16, 4, 12, 171, 172, 173, 174, 175, 85, 179, 180, 181, 182, 183fedgmullem2 33682 . . . . . . . . 9 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴)) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))}))
185184ex 412 . . . . . . . 8 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) → ((𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))})))
186185ralrimiva 3145 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))((𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))})))
187 eqid 2736 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
188 eqid 2736 . . . . . . . . 9 (Scalar‘𝐴) = (Scalar‘𝐴)
189 eqid 2736 . . . . . . . . 9 ( ·𝑠𝐴) = ( ·𝑠𝐴)
190 eqid 2736 . . . . . . . . 9 (0g𝐴) = (0g𝐴)
191 eqid 2736 . . . . . . . . 9 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
192 eqid 2736 . . . . . . . . 9 (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥))) = (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))
193187, 188, 189, 190, 191, 192islindf4 21859 . . . . . . . 8 ((𝐴 ∈ LMod ∧ (𝑦 × 𝑥) ∈ V ∧ (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴)) → ((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) LIndF 𝐴 ↔ ∀𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))((𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))}))))
194193biimpar 477 . . . . . . 7 (((𝐴 ∈ LMod ∧ (𝑦 × 𝑥) ∈ V ∧ (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴)) ∧ ∀𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))((𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (0g𝐴) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))}))) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) LIndF 𝐴)
195170, 157, 87, 186, 194syl31anc 1374 . . . . . 6 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) LIndF 𝐴)
19672anasss 466 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗𝑦𝑖𝑥)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
197196ralrimivva 3201 . . . . . . . . . 10 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
19885rnmposs 32685 . . . . . . . . . 10 (∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ⊆ (Base‘𝐸))
199197, 198syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ⊆ (Base‘𝐸))
20078ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐴))
201199, 200sseqtrd 4019 . . . . . . . 8 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ⊆ (Base‘𝐴))
202 eqid 2736 . . . . . . . . 9 (LSpan‘𝐴) = (LSpan‘𝐴)
203187, 202lspssv 20982 . . . . . . . 8 ((𝐴 ∈ LMod ∧ ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ⊆ (Base‘𝐴)) → ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) ⊆ (Base‘𝐴))
204170, 201, 203syl2anc 584 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) ⊆ (Base‘𝐴))
205 simpl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)))
206205ad4antr 732 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → ((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)))
207 elmapi 8890 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
208207ad4antlr 733 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
209 simpr 484 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → 𝑗𝑦)
210208, 209ffvelcdmd 7104 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → (𝑎𝑗) ∈ (Base‘(Scalar‘𝐵)))
211113simprd 495 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶))
212206, 211syl 17 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶))
213102ad7antr 738 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
214212, 213eqtr4d 2779 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → ((LSpan‘𝐶)‘𝑥) = (Base‘(Scalar‘𝐵)))
215210, 214eleqtrrd 2843 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → (𝑎𝑗) ∈ ((LSpan‘𝐶)‘𝑥))
216 eqid 2736 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
217 eqid 2736 . . . . . . . . . . . . . . . . 17 (Scalar‘𝐶) = (Scalar‘𝐶)
218 eqid 2736 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
219 eqid 2736 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝐶) = ( ·𝑠𝐶)
220 lveclmod 21106 . . . . . . . . . . . . . . . . . . 19 (𝐶 ∈ LVec → 𝐶 ∈ LMod)
22119, 220syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐶 ∈ LMod)
222221ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LMod)
223111, 39, 216, 217, 218, 219, 222, 41ellspds 33397 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((𝑎𝑗) ∈ ((LSpan‘𝐶)‘𝑥) ↔ ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))))))
224223biimpa 476 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑎𝑗) ∈ ((LSpan‘𝐶)‘𝑥)) → ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))))
225206, 215, 224syl2anc 584 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑗𝑦) → ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))))
226225ralrimiva 3145 . . . . . . . . . . . . 13 (((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) → ∀𝑗𝑦𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))))
227 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → (𝑎𝑤) = (𝑎𝑗))
228 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖 → (𝑏𝑣) = (𝑏𝑖))
229 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖𝑣 = 𝑖)
230228, 229oveq12d 7450 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑖 → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = ((𝑏𝑖)( ·𝑠𝐶)𝑖))
231230cbvmptv 5254 . . . . . . . . . . . . . . . . . . . 20 (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))
232231oveq2i 7443 . . . . . . . . . . . . . . . . . . 19 (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))
233232a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))))
234227, 233eqeq12d 2752 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑗 → ((𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) ↔ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))))
235234anbi2d 630 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑗 → ((𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)))) ↔ (𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))))))
236235rexbidv 3178 . . . . . . . . . . . . . . 15 (𝑤 = 𝑗 → (∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)))) ↔ ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))))))
237236cbvralvw 3236 . . . . . . . . . . . . . 14 (∀𝑤𝑦𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)))) ↔ ∀𝑗𝑦𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))))
238 vex 3483 . . . . . . . . . . . . . . 15 𝑦 ∈ V
239 breq1 5145 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑤) finSupp (0g‘(Scalar‘𝐶))))
240 fveq1 6904 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = (𝑓𝑤) → (𝑏𝑣) = ((𝑓𝑤)‘𝑣))
241240oveq1d 7447 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝑓𝑤) → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))
242241mpteq2dv 5243 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓𝑤) → (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))
243242oveq2d 7448 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑓𝑤) → (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))
244243eqeq2d 2747 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑤) → ((𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) ↔ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))))
245239, 244anbi12d 632 . . . . . . . . . . . . . . 15 (𝑏 = (𝑓𝑤) → ((𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)))) ↔ ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))))
246238, 245ac6s 10525 . . . . . . . . . . . . . 14 (∀𝑤𝑦𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)))) → ∃𝑓(𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))))
247237, 246sylbir 235 . . . . . . . . . . . . 13 (∀𝑗𝑦𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))) → ∃𝑓(𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))))
248226, 247syl 17 . . . . . . . . . . . 12 (((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) → ∃𝑓(𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))))
249 simpllr 775 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥))
250 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
251249, 250ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑓𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥))
252 elmapi 8890 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥) → (𝑓𝑗):𝑥⟶(Base‘(Scalar‘𝐶)))
253251, 252syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑓𝑗):𝑥⟶(Base‘(Scalar‘𝐶)))
254253anasss 466 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → (𝑓𝑗):𝑥⟶(Base‘(Scalar‘𝐶)))
255 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → 𝑖𝑥)
256254, 255ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
25774, 77srasca 21184 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
25812, 257eqtrid 2788 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐴))
25947, 50srasca 21184 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
26013, 259eqtr3d 2778 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐶))
261258, 260eqtr3d 2778 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262261fveq2d 6909 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
263262ad4antr 732 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
264256, 263eleqtrrd 2843 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
265264ralrimivva 3201 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → ∀𝑗𝑦𝑖𝑥 ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
266179fmpo 8094 . . . . . . . . . . . . . . . . . . 19 (∀𝑗𝑦𝑖𝑥 ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴)))
267265, 266sylib 218 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴)))
268 fvexd 6920 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (Base‘(Scalar‘𝐴)) ∈ V)
269157adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑦 × 𝑥) ∈ V)
270268, 269elmapd 8881 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)) ↔ (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴))))
271267, 270mpbird 257 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
272271ad5ant15 758 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
273272adantr 480 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
274273adantl3r 750 . . . . . . . . . . . . . 14 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
275 simpr 484 . . . . . . . . . . . . . . . 16 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑐 = (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣))) → 𝑐 = (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)))
276275breq1d 5152 . . . . . . . . . . . . . . 15 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑐 = (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣))) → (𝑐 finSupp (0g‘(Scalar‘𝐴)) ↔ (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) finSupp (0g‘(Scalar‘𝐴))))
277275oveq1d 7447 . . . . . . . . . . . . . . . . 17 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑐 = (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣))) → (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∘f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
278277oveq2d 7448 . . . . . . . . . . . . . . . 16 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑐 = (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣))) → (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) = (𝐴 Σg ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∘f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))))
279278eqeq2d 2747 . . . . . . . . . . . . . . 15 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑐 = (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣))) → (𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))) ↔ 𝑧 = (𝐴 Σg ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∘f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))))
280276, 279anbi12d 632 . . . . . . . . . . . . . 14 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑐 = (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣))) → ((𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))) ↔ ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∘f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))))))
28125ad8antr 740 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝐸 ∈ DivRing)
2821ad8antr 740 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝐹 ∈ DivRing)
28314ad8antr 740 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝐾 ∈ DivRing)
2842ad8antr 740 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑈 ∈ (SubRing‘𝐸))
2853ad8antr 740 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑉 ∈ (SubRing‘𝐹))
28638ad6antr 736 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑥 ∈ (LBasis‘𝐶))
28759ad6antr 736 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑦 ∈ (LBasis‘𝐵))
288 simpr 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ (Base‘𝐴))
289288ad5antr 734 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑧 ∈ (Base‘𝐴))
290207ad5antlr 735 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
291 simp-4r 783 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑎 finSupp (0g‘(Scalar‘𝐵)))
292 simpllr 775 . . . . . . . . . . . . . . . 16 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤))))
293 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑗𝑤 = 𝑗)
294227, 293oveq12d 7450 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → ((𝑎𝑤)( ·𝑠𝐵)𝑤) = ((𝑎𝑗)( ·𝑠𝐵)𝑗))
295294cbvmptv 5254 . . . . . . . . . . . . . . . . 17 (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)) = (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗))
296295oveq2i 7443 . . . . . . . . . . . . . . . 16 (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤))) = (𝐵 Σg (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗)))
297292, 296eqtrdi 2792 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑧 = (𝐵 Σg (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗))))
298 simplr 768 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥))
299 simpr 484 . . . . . . . . . . . . . . . . . 18 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))))
300176breq1d 5152 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑗 → ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑗) finSupp (0g‘(Scalar‘𝐶))))
301 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑖 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑤)‘𝑖))
302301, 229oveq12d 7450 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑖 → (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
303302cbvmptv 5254 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
304176fveq1d 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑖) = ((𝑓𝑗)‘𝑖))
305304oveq1d 7447 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑗 → (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖) = (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))
306305mpteq2dv 5243 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑗 → (𝑖𝑥 ↦ (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖)) = (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
307303, 306eqtrid 2788 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑗 → (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
308307oveq2d 7448 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑗 → (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
309227, 308eqeq12d 2752 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑗 → ((𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))) ↔ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))))
310300, 309anbi12d 632 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑗 → (((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))) ↔ ((𝑓𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))))
311310cbvralvw 3236 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))) ↔ ∀𝑗𝑦 ((𝑓𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))))
312299, 311sylib 218 . . . . . . . . . . . . . . . . 17 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → ∀𝑗𝑦 ((𝑓𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))))
313312r19.21bi 3250 . . . . . . . . . . . . . . . 16 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑗𝑦) → ((𝑓𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))))
314313simpld 494 . . . . . . . . . . . . . . 15 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑗𝑦) → (𝑓𝑗) finSupp (0g‘(Scalar‘𝐶)))
315313simprd 495 . . . . . . . . . . . . . . 15 ((((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) ∧ 𝑗𝑦) → (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
31673, 26, 16, 4, 12, 281, 282, 283, 284, 285, 85, 179, 286, 287, 289, 290, 291, 297, 298, 314, 315fedgmullem1 33681 . . . . . . . . . . . . . 14 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg ((𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) ∘f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))))
317274, 280, 316rspcedvd 3623 . . . . . . . . . . . . 13 (((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))))
318317anasss 466 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) ∧ (𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤𝑦 ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))))
319248, 318exlimddv 1934 . . . . . . . . . . 11 (((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))))
320319anasss 466 . . . . . . . . . 10 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ (𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤))))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))))
321 eqid 2736 . . . . . . . . . . . . . . . . 17 (LSpan‘𝐵) = (LSpan‘𝐵)
32260, 29, 321islbs4 21853 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (LBasis‘𝐵) ↔ (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
32359, 322sylib 218 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
324323simprd 495 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵))
325324adantr 480 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵))
32678, 64eqtr3d 2778 . . . . . . . . . . . . . 14 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
327326ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
328325, 327eqtr4d 2779 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐴))
329288, 328eleqtrrd 2843 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐵)‘𝑦))
330 eqid 2736 . . . . . . . . . . . . 13 (Scalar‘𝐵) = (Scalar‘𝐵)
331 lveclmod 21106 . . . . . . . . . . . . . . 15 (𝐵 ∈ LVec → 𝐵 ∈ LMod)
33228, 331syl 17 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ LMod)
333332ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LMod)
334321, 60, 88, 330, 91, 89, 333, 62ellspds 33397 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑧 ∈ ((LSpan‘𝐵)‘𝑦) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)(𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤))))))
335334biimpa 476 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ ((LSpan‘𝐵)‘𝑦)) → ∃𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)(𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))))
336205, 329, 335syl2anc 584 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ∃𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)(𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)))))
337320, 336r19.29a 3161 . . . . . . . . 9 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))))
338 eqid 2736 . . . . . . . . . . 11 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
339202, 187, 338, 188, 191, 189, 87, 170, 157ellspd 21823 . . . . . . . . . 10 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑧 ∈ ((LSpan‘𝐴)‘((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥))) ↔ ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))))))
340339adantr 480 . . . . . . . . 9 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (𝑧 ∈ ((LSpan‘𝐴)‘((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥))) ↔ ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐f ( ·𝑠𝐴)(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))))))
341337, 340mpbird 257 . . . . . . . 8 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥))))
34287ffnd 6736 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) Fn (𝑦 × 𝑥))
343342adantr 480 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) Fn (𝑦 × 𝑥))
344 fnima 6697 . . . . . . . . . 10 ((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) Fn (𝑦 × 𝑥) → ((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥)) = ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))
345343, 344syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥)) = ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))
346345fveq2d 6909 . . . . . . . 8 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐴)‘((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥))) = ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
347341, 346eleqtrd 2842 . . . . . . 7 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
348204, 347eqelssd 4004 . . . . . 6 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (Base‘𝐴))
349 eqid 2736 . . . . . . 7 (Base‘(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (Base‘(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))
350 drngnzr 20749 . . . . . . . . . 10 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
35114, 350syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ NzRing)
352258, 351eqeltrrd 2841 . . . . . . . 8 (𝜑 → (Scalar‘𝐴) ∈ NzRing)
353352ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Scalar‘𝐴) ∈ NzRing)
354187, 349, 188, 189, 190, 191, 202, 170, 353, 157, 156lindflbs 33408 . . . . . 6 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ∈ (LBasis‘𝐴) ↔ ((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) LIndF 𝐴 ∧ ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (Base‘𝐴))))
355195, 348, 354mpbir2and 713 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ∈ (LBasis‘𝐴))
356 eqid 2736 . . . . . 6 (LBasis‘𝐴) = (LBasis‘𝐴)
357356dimval 33652 . . . . 5 ((𝐴 ∈ LVec ∧ ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ∈ (LBasis‘𝐴)) → (dim‘𝐴) = (♯‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
358167, 355, 357syl2anc 584 . . . 4 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = (♯‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
35929dimval 33652 . . . . . 6 ((𝐵 ∈ LVec ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36092, 59, 359syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36120dimval 33652 . . . . . 6 ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐶) = (♯‘𝑥))
362110, 38, 361syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐶) = (♯‘𝑥))
363360, 362oveq12d 7450 . . . 4 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((dim‘𝐵) ·e (dim‘𝐶)) = ((♯‘𝑦) ·e (♯‘𝑥)))
364164, 358, 3633eqtr4d 2786 . . 3 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
36534, 364exlimddv 1934 . 2 ((𝜑𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
36624, 365exlimddv 1934 1 (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  wss 3950  c0 4332  {csn 4625   class class class wbr 5142  cmpt 5224   × cxp 5682  ran crn 5685  cima 5687   Fn wfn 6555  wf 6556  1-1wf1 6557  cfv 6560  (class class class)co 7432  cmpo 7434  f cof 7696  m cmap 8867  cen 8983   finSupp cfsupp 9402   ·e cxmu 13154  chash 14370  Basecbs 17248  s cress 17275  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17485   Σg cgsu 17486  Ringcrg 20231  NzRingcnzr 20513  SubRingcsubrg 20570  DivRingcdr 20730  LModclmod 20859  LSpanclspn 20970  LBasisclbs 21074  LVecclvec 21102  subringAlg csra 21171   freeLMod cfrlm 21767   LIndF clindf 21825  LIndSclinds 21826  dimcldim 33650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682  ax-ac2 10504  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-rpss 7744  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-r1 9805  df-rank 9806  df-dju 9942  df-card 9980  df-acn 9983  df-ac 10157  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12736  df-uz 12880  df-xmul 13157  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-mri 17632  df-acs 17633  df-proset 18341  df-drs 18342  df-poset 18360  df-ipo 18574  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-nzr 20514  df-subrng 20547  df-subrg 20571  df-drng 20732  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-lindf 21827  df-linds 21828  df-dim 33651
This theorem is referenced by:  extdgmul  33715
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