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Theorem fedgmul 32326
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 20248 . . . . . . . . . . 11 (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈)))
65biimpa 477 . . . . . . . . . 10 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
72, 3, 6syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
87simprd 496 . . . . . . . 8 (𝜑𝑉𝑈)
9 ressabs 17130 . . . . . . . 8 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈) → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
102, 8, 9syl2anc 584 . . . . . . 7 (𝜑 → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
114oveq1i 7367 . . . . . . 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 2838 . . . . 5 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
16 fedgmul.c . . . . . 6 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
17 eqid 2736 . . . . . 6 (𝐹s 𝑉) = (𝐹s 𝑉)
1816, 17sralvec 32289 . . . . 5 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
191, 15, 3, 18syl3anc 1371 . . . 4 (𝜑𝐶 ∈ LVec)
20 eqid 2736 . . . . 5 (LBasis‘𝐶) = (LBasis‘𝐶)
2120lbsex 20626 . . . 4 (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
2219, 21syl 17 . . 3 (𝜑 → (LBasis‘𝐶) ≠ ∅)
23 n0 4306 . . 3 ((LBasis‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
2422, 23sylib 217 . 2 (𝜑 → ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
25 fedgmul.1 . . . . . . 7 (𝜑𝐸 ∈ DivRing)
26 fedgmul.b . . . . . . . 8 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
2726, 4sralvec 32289 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
2825, 1, 2, 27syl3anc 1371 . . . . . 6 (𝜑𝐵 ∈ LVec)
29 eqid 2736 . . . . . . 7 (LBasis‘𝐵) = (LBasis‘𝐵)
3029lbsex 20626 . . . . . 6 (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
3128, 30syl 17 . . . . 5 (𝜑 → (LBasis‘𝐵) ≠ ∅)
32 n0 4306 . . . . 5 ((LBasis‘𝐵) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
3331, 32sylib 217 . . . 4 (𝜑 → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
3433adantr 481 . . 3 ((𝜑𝑥 ∈ (LBasis‘𝐶)) → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
35 drngring 20192 . . . . . . . . . . . . . . 15 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
3625, 35syl 17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ Ring)
3736ad4antr 730 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝐸 ∈ Ring)
38 simplr 767 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39 eqid 2736 . . . . . . . . . . . . . . . . . 18 (Base‘𝐶) = (Base‘𝐶)
4039, 20lbsss 20538 . . . . . . . . . . . . . . . . 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 20223 . . . . . . . . . . . . . . . . . . . . 21 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
442, 43syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑈 ⊆ (Base‘𝐸))
454, 42ressbas2 17120 . . . . . . . . . . . . . . . . . . . 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 20223 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
503, 49syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑉 ⊆ (Base‘𝐹))
5147, 50srabase 20640 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
5246, 51eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐶))
5352, 44eqsstrrd 3983 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
5453ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐶) ⊆ (Base‘𝐸))
5541, 54sstrd 3954 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐸))
5655ad2antrr 724 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑥 ⊆ (Base‘𝐸))
57 simpr 485 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑖𝑥)
5856, 57sseldd 3945 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑖 ∈ (Base‘𝐸))
59 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LBasis‘𝐵))
60 eqid 2736 . . . . . . . . . . . . . . . . . 18 (Base‘𝐵) = (Base‘𝐵)
6160, 29lbsss 20538 . . . . . . . . . . . . . . . . 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 20640 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
6564ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐵))
6662, 65sseqtrrd 3985 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐸))
6766ad2antrr 724 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑦 ⊆ (Base‘𝐸))
68 simplr 767 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
6967, 68sseldd 3945 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗 ∈ (Base‘𝐸))
70 eqid 2736 . . . . . . . . . . . . . 14 (.r𝐸) = (.r𝐸)
7142, 70ringcl 19981 . . . . . . . . . . . . 13 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
7237, 58, 69, 71syl3anc 1371 . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 15 (𝜑𝑉 ∈ (SubRing‘𝐸))
7642subrgss 20223 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
7775, 76syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐸))
7874, 77srabase 20640 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
7978ad4antr 730 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (Base‘𝐸) = (Base‘𝐴))
8072, 79eleqtrd 2840 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
8180anasss 467 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗𝑦𝑖𝑥)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
8281ralrimivva 3197 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
83 oveq2 7365 . . . . . . . . . . 11 (𝑤 = 𝑗 → (𝑡(.r𝐸)𝑤) = (𝑡(.r𝐸)𝑗))
84 oveq1 7364 . . . . . . . . . . 11 (𝑡 = 𝑖 → (𝑡(.r𝐸)𝑗) = (𝑖(.r𝐸)𝑗))
8583, 84cbvmpov 7452 . . . . . . . . . 10 (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) = (𝑗𝑦, 𝑖𝑥 ↦ (𝑖(.r𝐸)𝑗))
8685fmpo 8000 . . . . . . . . 9 (∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴) ↔ (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
8782, 86sylib 217 . . . . . . . 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 724 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LVec)
9392ad5antr 732 . . . . . . . . . . . . . 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 21239 . . . . . . . . . . . . . . . 16 (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
9594, 59sselid 3942 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LIndS‘𝐵))
9695ad5antr 732 . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . 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 20646 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
1004, 99eqtrid 2788 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 = (Scalar‘𝐵))
101100fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102101, 51eqtr3d 2778 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
103102ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
10441, 103sseqtrrd 3985 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
105104ad5antr 732 . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LVec)
111 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (LSpan‘𝐶) = (LSpan‘𝐶)
11239, 20, 111islbs4 21238 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (LBasis‘𝐶) ↔ (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
11338, 112sylib 217 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
114113simpld 495 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LIndS‘𝐶))
115 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (0g𝐶) = (0g𝐶)
1161150nellinds 32159 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LIndS‘𝐶)) → ¬ (0g𝐶) ∈ 𝑥)
117110, 114, 116syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ¬ (0g𝐶) ∈ 𝑥)
118117ad5antr 732 . . . . . . . . . . . . . . . 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 3042 . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐵)))
12216, 1, 3drgext0g 32291 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g𝐶))
123121, 122eqtr3d 2778 . . . . . . . . . . . . . . . 16 (𝜑 → (0g‘(Scalar‘𝐵)) = (0g𝐶))
124123ad7antr 736 . . . . . . . . . . . . . . 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 3018 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 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 7392 . . . . . . . . . . . . . . . . 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 7502 . . . . . . . . . . . . . . . . 17 ((𝑗𝑦𝑖𝑥 ∧ (𝑖(.r𝐸)𝑗) ∈ V) → (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑖(.r𝐸)𝑗))
12997, 106, 127, 128syl3anc 1371 . . . . . . . . . . . . . . . 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 32292 . . . . . . . . . . . . . . . . . 18 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
131130oveqd 7374 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑖(.r𝐸)𝑗) = (𝑖( ·𝑠𝐵)𝑗))
132131ad7antr 736 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑖 = 𝑢)
136 simprl 769 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑗 = 𝑣)
137135, 136oveq12d 7375 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → (𝑖(.r𝐸)𝑗) = (𝑢(.r𝐸)𝑣))
138 simplr 767 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → 𝑣𝑦)
139 simpr 485 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → 𝑢𝑥)
140 ovexd 7392 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑢(.r𝐸)𝑣) ∈ V)
141134, 137, 138, 139, 140ovmpod 7507 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
142141adantllr 717 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
143142adantl3r 748 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
144143adantr 481 . . . . . . . . . . . . . . . 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 7374 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑢(.r𝐸)𝑣) = (𝑢( ·𝑠𝐵)𝑣))
146145ad7antr 736 . . . . . . . . . . . . . . . 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 32168 . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 3197 . . . . . . . . . 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 467 . . . . . . . . 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 3197 . . . . . . . 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 7413 . . . . . . . 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 7685 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158 f1rnen 31543 . . . . . . 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 14248 . . . . . 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 31709 . . . . . 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 32289 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
16625, 14, 75, 165syl3anc 1371 . . . . . 6 (𝜑𝐴 ∈ LVec)
167166ad2antrr 724 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168 lveclmod 20567 . . . . . . . . 9 (𝐴 ∈ LVec → 𝐴 ∈ LMod)
169166, 168syl 17 . . . . . . . 8 (𝜑𝐴 ∈ LMod)
170169ad2antrr 724 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LMod)
17125ad4antr 730 . . . . . . . . . 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 730 . . . . . . . . . 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 730 . . . . . . . . . 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 730 . . . . . . . . . 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 730 . . . . . . . . . 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 6842 . . . . . . . . . . . 12 (𝑤 = 𝑗 → (𝑓𝑤) = (𝑓𝑗))
177176fveq1d 6844 . . . . . . . . . . 11 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑗)‘𝑣))
178 fveq2 6842 . . . . . . . . . . 11 (𝑣 = 𝑖 → ((𝑓𝑗)‘𝑣) = ((𝑓𝑗)‘𝑖))
179177, 178cbvmpov 7452 . . . . . . . . . 10 (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) = (𝑗𝑦, 𝑖𝑥 ↦ ((𝑓𝑗)‘𝑖))
180 simp-4r 782 . . . . . . . . . 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 774 . . . . . . . . . 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 767 . . . . . . . . . 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 485 . . . . . . . . . 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 32325 . . . . . . . . 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 413 . . . . . . . 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 3143 . . . . . . 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 21244 . . . . . . . 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 478 . . . . . . 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 1373 . . . . . 6 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) LIndF 𝐴)
19672anasss 467 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗𝑦𝑖𝑥)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
197196ralrimivva 3197 . . . . . . . . . 10 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
19885rnmposs 31590 . . . . . . . . . 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 724 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐴))
201199, 200sseqtrd 3984 . . . . . . . 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 20444 . . . . . . . 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 483 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)))
206205ad4antr 730 . . . . . . . . . . . . . . 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 8787 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
208207ad4antlr 731 . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . 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 7036 . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 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 20567 . . . . . . . . . . . . . . . . . . 19 (𝐶 ∈ LVec → 𝐶 ∈ LMod)
22119, 220syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐶 ∈ LMod)
222221ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LMod)
223111, 39, 216, 217, 218, 219, 222, 41ellspds 32157 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → (𝑎𝑤) = (𝑎𝑗))
228 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖 → (𝑏𝑣) = (𝑏𝑖))
229 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖𝑣 = 𝑖)
230228, 229oveq12d 7375 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑖 → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = ((𝑏𝑖)( ·𝑠𝐶)𝑖))
231230cbvmptv 5218 . . . . . . . . . . . . . . . . . . . 20 (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))
232231oveq2i 7368 . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . . . 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 3225 . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . 15 𝑦 ∈ V
239 breq1 5108 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑤) finSupp (0g‘(Scalar‘𝐶))))
240 fveq1 6841 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = (𝑓𝑤) → (𝑏𝑣) = ((𝑓𝑤)‘𝑣))
241240oveq1d 7372 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝑓𝑤) → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))
242241mpteq2dv 5207 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓𝑤) → (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))
243242oveq2d 7373 . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . 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 10420 . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
251249, 250ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑓𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥))
252 elmapi 8787 . . . . . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → (𝑓𝑗):𝑥⟶(Base‘(Scalar‘𝐶)))
255 simprr 771 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → 𝑖𝑥)
256254, 255ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
25774, 77srasca 20646 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
25812, 257eqtrid 2788 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐴))
25947, 50srasca 20646 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
26013, 259eqtr3d 2778 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐶))
261258, 260eqtr3d 2778 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262261fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
263262ad4antr 730 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
264256, 263eleqtrrd 2841 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
265264ralrimivva 3197 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → ∀𝑗𝑦𝑖𝑥 ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
266179fmpo 8000 . . . . . . . . . . . . . . . . . . 19 (∀𝑗𝑦𝑖𝑥 ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴)))
267265, 266sylib 217 . . . . . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (Base‘(Scalar‘𝐴)) ∈ V)
269157adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑦 × 𝑥) ∈ V)
270268, 269elmapd 8779 . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 748 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 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 5115 . . . . . . . . . . . . . . 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 7372 . . . . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 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))))) → 𝑥 ∈ (LBasis‘𝐶))
28759ad6antr 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))))) → 𝑦 ∈ (LBasis‘𝐵))
288 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ (Base‘𝐴))
289288ad5antr 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)))) ∧ 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 733 . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . 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 7375 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → ((𝑎𝑤)( ·𝑠𝐵)𝑤) = ((𝑎𝑗)( ·𝑠𝐵)𝑗))
295294cbvmptv 5218 . . . . . . . . . . . . . . . . 17 (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)) = (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗))
296295oveq2i 7368 . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 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 5115 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑗 → ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑗) finSupp (0g‘(Scalar‘𝐶))))
301 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑖 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑤)‘𝑖))
302301, 229oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑖 → (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
303302cbvmptv 5218 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
304176fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑖) = ((𝑓𝑗)‘𝑖))
305304oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑗 → (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖) = (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))
306305mpteq2dv 5207 . . . . . . . . . . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . . 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 3225 . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . . . 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 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) → (𝑓𝑗) finSupp (0g‘(Scalar‘𝐶)))
315313simprd 496 . . . . . . . . . . . . . . 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 32324 . . . . . . . . . . . . . 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 3583 . . . . . . . . . . . . 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 467 . . . . . . . . . . . 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 1938 . . . . . . . . . . 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 467 . . . . . . . . . 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 21238 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (LBasis‘𝐵) ↔ (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
32359, 322sylib 217 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
324323simprd 496 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵))
325324adantr 481 . . . . . . . . . . . . 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 728 . . . . . . . . . . . . 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 2841 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐵)‘𝑦))
330 eqid 2736 . . . . . . . . . . . . 13 (Scalar‘𝐵) = (Scalar‘𝐵)
331 lveclmod 20567 . . . . . . . . . . . . . . 15 (𝐵 ∈ LVec → 𝐵 ∈ LMod)
33228, 331syl 17 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ LMod)
333332ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LMod)
334321, 60, 88, 330, 91, 89, 333, 62ellspds 32157 . . . . . . . . . . . 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 477 . . . . . . . . . . 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 3159 . . . . . . . . 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 21208 . . . . . . . . . 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 481 . . . . . . . . 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 256 . . . . . . . 8 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥))))
34287ffnd 6669 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) Fn (𝑦 × 𝑥))
343342adantr 481 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) Fn (𝑦 × 𝑥))
344 fnima 6631 . . . . . . . . . 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 6846 . . . . . . . 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 2840 . . . . . . 7 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
348204, 347eqelssd 3965 . . . . . 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 20732 . . . . . . . . . 10 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
35114, 350syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ NzRing)
352258, 351eqeltrrd 2839 . . . . . . . 8 (𝜑 → (Scalar‘𝐴) ∈ NzRing)
353352ad2antrr 724 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Scalar‘𝐴) ∈ NzRing)
354187, 349, 188, 189, 190, 191, 202, 170, 353, 157, 156lindflbs 32167 . . . . . 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 711 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ∈ (LBasis‘𝐴))
356 eqid 2736 . . . . . 6 (LBasis‘𝐴) = (LBasis‘𝐴)
357356dimval 32300 . . . . 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 32300 . . . . . 6 ((𝐵 ∈ LVec ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36092, 59, 359syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36120dimval 32300 . . . . . 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 7375 . . . 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 1938 . 2 ((𝜑𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
36624, 365exlimddv 1938 1 (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615  m cmap 8765  cen 8880   finSupp cfsupp 9305   ·e cxmu 13032  chash 14230  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Ringcrg 19964  DivRingcdr 20185  SubRingcsubrg 20218  LModclmod 20322  LSpanclspn 20432  LBasisclbs 20535  LVecclvec 20563  subringAlg csra 20629  NzRingcnzr 20727   freeLMod cfrlm 21152   LIndF clindf 21210  LIndSclinds 21211  dimcldim 32298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-rpss 7660  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-r1 9700  df-rank 9701  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-xmul 13035  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ocomp 17154  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-mri 17468  df-acs 17469  df-proset 18184  df-drs 18185  df-poset 18202  df-ipo 18417  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-drng 20187  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lmhm 20483  df-lbs 20536  df-lvec 20564  df-sra 20633  df-rgmod 20634  df-nzr 20728  df-dsmm 21138  df-frlm 21153  df-uvc 21189  df-lindf 21212  df-linds 21213  df-dim 32299
This theorem is referenced by:  extdgmul  32350
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