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Theorem fedgmul 33659
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 20615 . . . . . . . . . . 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 7441 . . . . . . 7 (𝐹s 𝑉) = ((𝐸s 𝑈) ↾s 𝑉)
12 fedgmul.k . . . . . . 7 𝐾 = (𝐸s 𝑉)
1310, 11, 123eqtr4g 2800 . . . . . 6 (𝜑 → (𝐹s 𝑉) = 𝐾)
14 fedgmul.3 . . . . . 6 (𝜑𝐾 ∈ DivRing)
1513, 14eqeltrd 2839 . . . . 5 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
16 fedgmul.c . . . . . 6 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
17 eqid 2735 . . . . . 6 (𝐹s 𝑉) = (𝐹s 𝑉)
1816, 17sralvec 33615 . . . . 5 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
191, 15, 3, 18syl3anc 1370 . . . 4 (𝜑𝐶 ∈ LVec)
20 eqid 2735 . . . . 5 (LBasis‘𝐶) = (LBasis‘𝐶)
2120lbsex 21185 . . . 4 (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
2219, 21syl 17 . . 3 (𝜑 → (LBasis‘𝐶) ≠ ∅)
23 n0 4359 . . 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 33615 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
2825, 1, 2, 27syl3anc 1370 . . . . . 6 (𝜑𝐵 ∈ LVec)
29 eqid 2735 . . . . . . 7 (LBasis‘𝐵) = (LBasis‘𝐵)
3029lbsex 21185 . . . . . 6 (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
3128, 30syl 17 . . . . 5 (𝜑 → (LBasis‘𝐵) ≠ ∅)
32 n0 4359 . . . . 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 20753 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39 eqid 2735 . . . . . . . . . . . . . . . . . 18 (Base‘𝐶) = (Base‘𝐶)
4039, 20lbsss 21094 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (LBasis‘𝐶) → 𝑥 ⊆ (Base‘𝐶))
4138, 40syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐶))
42 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐸) = (Base‘𝐸)
4342subrgss 20589 . . . . . . . . . . . . . . . . . . . . 21 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
442, 43syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑈 ⊆ (Base‘𝐸))
454, 42ressbas2 17283 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
4644, 45syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑈 = (Base‘𝐹))
4716a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐶 = ((subringAlg ‘𝐹)‘𝑉))
48 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐹) = (Base‘𝐹)
4948subrgss 20589 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
503, 49syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑉 ⊆ (Base‘𝐹))
5147, 50srabase 21195 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
5246, 51eqtrd 2775 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐶))
5352, 44eqsstrrd 4035 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
5453ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐶) ⊆ (Base‘𝐸))
5541, 54sstrd 4006 . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . 18 (Base‘𝐵) = (Base‘𝐵)
6160, 29lbsss 21094 . . . . . . . . . . . . . . . . 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 21195 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
6564ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐵))
6662, 65sseqtrrd 4037 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
6967, 68sseldd 3996 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗 ∈ (Base‘𝐸))
70 eqid 2735 . . . . . . . . . . . . . 14 (.r𝐸) = (.r𝐸)
7142, 70ringcl 20268 . . . . . . . . . . . . 13 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
7237, 58, 69, 71syl3anc 1370 . . . . . . . . . . . 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 20589 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
7775, 76syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐸))
7874, 77srabase 21195 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
7978ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (Base‘𝐸) = (Base‘𝐴))
8072, 79eleqtrd 2841 . . . . . . . . . . 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 3200 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
83 oveq2 7439 . . . . . . . . . . 11 (𝑤 = 𝑗 → (𝑡(.r𝐸)𝑤) = (𝑡(.r𝐸)𝑗))
84 oveq1 7438 . . . . . . . . . . 11 (𝑡 = 𝑖 → (𝑡(.r𝐸)𝑗) = (𝑖(.r𝐸)𝑗))
8583, 84cbvmpov 7528 . . . . . . . . . 10 (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) = (𝑗𝑦, 𝑖𝑥 ↦ (𝑖(.r𝐸)𝑗))
8685fmpo 8092 . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
89 eqid 2735 . . . . . . . . . . . . . 14 ( ·𝑠𝐵) = ( ·𝑠𝐵)
90 eqid 2735 . . . . . . . . . . . . . 14 (+g𝐵) = (+g𝐵)
91 eqid 2735 . . . . . . . . . . . . . 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 21871 . . . . . . . . . . . . . . . 16 (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
9594, 59sselid 3993 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 21201 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
1004, 99eqtrid 2787 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 = (Scalar‘𝐵))
101100fveq2d 6911 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102101, 51eqtr3d 2777 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
103102ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
10441, 103sseqtrrd 4037 . . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . . 21 (LSpan‘𝐶) = (LSpan‘𝐶)
11239, 20, 111islbs4 21870 . . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . 19 (0g𝐶) = (0g𝐶)
1161150nellinds 33378 . . . . . . . . . . . . . . . . . 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 3038 . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐵)))
12216, 1, 3drgext0g 33619 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g𝐶))
123121, 122eqtr3d 2777 . . . . . . . . . . . . . . . 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 3013 . . . . . . . . . . . . . 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 7466 . . . . . . . . . . . . . . . . 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 7580 . . . . . . . . . . . . . . . . 17 ((𝑗𝑦𝑖𝑥 ∧ (𝑖(.r𝐸)𝑗) ∈ V) → (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑖(.r𝐸)𝑗))
12997, 106, 127, 128syl3anc 1370 . . . . . . . . . . . . . . . 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 33620 . . . . . . . . . . . . . . . . . 18 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
131130oveqd 7448 . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑖 = 𝑢)
136 simprl 771 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑗 = 𝑣)
137135, 136oveq12d 7449 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → (𝑖(.r𝐸)𝑗) = (𝑢(.r𝐸)𝑣))
138 simplr 769 . . . . . . . . . . . . . . . . . . . 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 7466 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑢(.r𝐸)𝑣) ∈ V)
141134, 137, 138, 139, 140ovmpod 7585 . . . . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . 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 33389 . . . . . . . . . . . . 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 3200 . . . . . . . . . 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 3200 . . . . . . . 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 7489 . . . . . . . 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 7770 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158 f1rnen 32646 . . . . . . 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 14384 . . . . . 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 32817 . . . . . 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 2775 . . . 4 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = ((♯‘𝑦) ·e (♯‘𝑥)))
16573, 12sralvec 33615 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
16625, 14, 75, 165syl3anc 1370 . . . . . 6 (𝜑𝐴 ∈ LVec)
167166ad2antrr 726 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168 lveclmod 21123 . . . . . . . . 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 6907 . . . . . . . . . . . 12 (𝑤 = 𝑗 → (𝑓𝑤) = (𝑓𝑗))
177176fveq1d 6909 . . . . . . . . . . 11 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑗)‘𝑣))
178 fveq2 6907 . . . . . . . . . . 11 (𝑣 = 𝑖 → ((𝑓𝑗)‘𝑣) = ((𝑓𝑗)‘𝑖))
179177, 178cbvmpov 7528 . . . . . . . . . 10 (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) = (𝑗𝑦, 𝑖𝑥 ↦ ((𝑓𝑗)‘𝑖))
180 simp-4r 784 . . . . . . . . . 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 776 . . . . . . . . . 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 769 . . . . . . . . . 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 33658 . . . . . . . . 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 3144 . . . . . . 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 2735 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
188 eqid 2735 . . . . . . . . 9 (Scalar‘𝐴) = (Scalar‘𝐴)
189 eqid 2735 . . . . . . . . 9 ( ·𝑠𝐴) = ( ·𝑠𝐴)
190 eqid 2735 . . . . . . . . 9 (0g𝐴) = (0g𝐴)
191 eqid 2735 . . . . . . . . 9 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
192 eqid 2735 . . . . . . . . 9 (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥))) = (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))
193187, 188, 189, 190, 191, 192islindf4 21876 . . . . . . . 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 1372 . . . . . 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 3200 . . . . . . . . . 10 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
19885rnmposs 32691 . . . . . . . . . 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 4036 . . . . . . . 8 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ⊆ (Base‘𝐴))
202 eqid 2735 . . . . . . . . 9 (LSpan‘𝐴) = (LSpan‘𝐴)
203187, 202lspssv 20999 . . . . . . . 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 8888 . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
217 eqid 2735 . . . . . . . . . . . . . . . . 17 (Scalar‘𝐶) = (Scalar‘𝐶)
218 eqid 2735 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
219 eqid 2735 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝐶) = ( ·𝑠𝐶)
220 lveclmod 21123 . . . . . . . . . . . . . . . . . . 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 33376 . . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → (𝑎𝑤) = (𝑎𝑗))
228 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖 → (𝑏𝑣) = (𝑏𝑖))
229 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖𝑣 = 𝑖)
230228, 229oveq12d 7449 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑖 → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = ((𝑏𝑖)( ·𝑠𝐶)𝑖))
231230cbvmptv 5261 . . . . . . . . . . . . . . . . . . . 20 (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))
232231oveq2i 7442 . . . . . . . . . . . . . . . . . . 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 2751 . . . . . . . . . . . . . . . . 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 3177 . . . . . . . . . . . . . . 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 3235 . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . 15 𝑦 ∈ V
239 breq1 5151 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑤) finSupp (0g‘(Scalar‘𝐶))))
240 fveq1 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = (𝑓𝑤) → (𝑏𝑣) = ((𝑓𝑤)‘𝑣))
241240oveq1d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝑓𝑤) → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))
242241mpteq2dv 5250 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓𝑤) → (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))
243242oveq2d 7447 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑓𝑤) → (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))
244243eqeq2d 2746 . . . . . . . . . . . . . . . 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 10522 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
251249, 250ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑓𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥))
252 elmapi 8888 . . . . . . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → 𝑖𝑥)
256254, 255ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
25774, 77srasca 21201 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
25812, 257eqtrid 2787 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐴))
25947, 50srasca 21201 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
26013, 259eqtr3d 2777 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐶))
261258, 260eqtr3d 2777 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262261fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
265264ralrimivva 3200 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → ∀𝑗𝑦𝑖𝑥 ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
266179fmpo 8092 . . . . . . . . . . . . . . . . . . 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 6922 . . . . . . . . . . . . . . . . . . 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 8879 . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . 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 5158 . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → ((𝑎𝑤)( ·𝑠𝐵)𝑤) = ((𝑎𝑗)( ·𝑠𝐵)𝑗))
295294cbvmptv 5261 . . . . . . . . . . . . . . . . 17 (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)) = (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗))
296295oveq2i 7442 . . . . . . . . . . . . . . . 16 (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤))) = (𝐵 Σg (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗)))
297292, 296eqtrdi 2791 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 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 5158 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑗 → ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑗) finSupp (0g‘(Scalar‘𝐶))))
301 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑖 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑤)‘𝑖))
302301, 229oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑖 → (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
303302cbvmptv 5261 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
304176fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑖) = ((𝑓𝑗)‘𝑖))
305304oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑗 → (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖) = (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))
306305mpteq2dv 5250 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑗 → (𝑖𝑥 ↦ (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖)) = (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
307303, 306eqtrid 2787 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑗 → (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
308307oveq2d 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑗 → (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
309227, 308eqeq12d 2751 . . . . . . . . . . . . . . . . . . . 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 3235 . . . . . . . . . . . . . . . . . 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 3249 . . . . . . . . . . . . . . . 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 33657 . . . . . . . . . . . . . 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 3624 . . . . . . . . . . . . 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 1933 . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (LSpan‘𝐵) = (LSpan‘𝐵)
32260, 29, 321islbs4 21870 . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . 14 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
327326ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
328325, 327eqtr4d 2778 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐴))
329288, 328eleqtrrd 2842 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐵)‘𝑦))
330 eqid 2735 . . . . . . . . . . . . 13 (Scalar‘𝐵) = (Scalar‘𝐵)
331 lveclmod 21123 . . . . . . . . . . . . . . 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 33376 . . . . . . . . . . . 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 3160 . . . . . . . . 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 2735 . . . . . . . . . . 11 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
339202, 187, 338, 188, 191, 189, 87, 170, 157ellspd 21840 . . . . . . . . . 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 6738 . . . . . . . . . . 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 6699 . . . . . . . . . 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 6911 . . . . . . . 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 2841 . . . . . . 7 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
348204, 347eqelssd 4017 . . . . . 6 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (Base‘𝐴))
349 eqid 2735 . . . . . . 7 (Base‘(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (Base‘(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))
350 drngnzr 20765 . . . . . . . . . 10 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
35114, 350syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ NzRing)
352258, 351eqeltrrd 2840 . . . . . . . 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 33387 . . . . . 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 2735 . . . . . 6 (LBasis‘𝐴) = (LBasis‘𝐴)
357356dimval 33628 . . . . 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 33628 . . . . . 6 ((𝐵 ∈ LVec ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36092, 59, 359syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36120dimval 33628 . . . . . 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 7449 . . . 4 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((dim‘𝐵) ·e (dim‘𝐶)) = ((♯‘𝑦) ·e (♯‘𝑥)))
364164, 358, 3633eqtr4d 2785 . . 3 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
36534, 364exlimddv 1933 . 2 ((𝜑𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
36624, 365exlimddv 1933 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 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  1-1wf1 6560  cfv 6563  (class class class)co 7431  cmpo 7433  f cof 7695  m cmap 8865  cen 8981   finSupp cfsupp 9399   ·e cxmu 13151  chash 14366  Basecbs 17245  s cress 17274  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486   Σg cgsu 17487  Ringcrg 20251  NzRingcnzr 20529  SubRingcsubrg 20586  DivRingcdr 20746  LModclmod 20875  LSpanclspn 20987  LBasisclbs 21091  LVecclvec 21119  subringAlg csra 21188   freeLMod cfrlm 21784   LIndF clindf 21842  LIndSclinds 21843  dimcldim 33626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-xmul 13154  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  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 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-mri 17633  df-acs 17634  df-proset 18352  df-drs 18353  df-poset 18371  df-ipo 18586  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-drng 20748  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-lindf 21844  df-linds 21845  df-dim 33627
This theorem is referenced by:  extdgmul  33689
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