Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fedgmul Structured version   Visualization version   GIF version

Theorem fedgmul 30631
 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 19251 . . . . . . . . . . 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 16392 . . . . . . . 8 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈) → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
102, 8, 9syl2anc 584 . . . . . . 7 (𝜑 → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
114oveq1i 7026 . . . . . . 7 (𝐹s 𝑉) = ((𝐸s 𝑈) ↾s 𝑉)
12 fedgmul.k . . . . . . 7 𝐾 = (𝐸s 𝑉)
1310, 11, 123eqtr4g 2856 . . . . . 6 (𝜑 → (𝐹s 𝑉) = 𝐾)
14 fedgmul.3 . . . . . 6 (𝜑𝐾 ∈ DivRing)
1513, 14eqeltrd 2883 . . . . 5 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
16 fedgmul.c . . . . . 6 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
17 eqid 2795 . . . . . 6 (𝐹s 𝑉) = (𝐹s 𝑉)
1816, 17sralvec 30594 . . . . 5 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
191, 15, 3, 18syl3anc 1364 . . . 4 (𝜑𝐶 ∈ LVec)
20 eqid 2795 . . . . 5 (LBasis‘𝐶) = (LBasis‘𝐶)
2120lbsex 19627 . . . 4 (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
2219, 21syl 17 . . 3 (𝜑 → (LBasis‘𝐶) ≠ ∅)
23 n0 4230 . . 3 ((LBasis‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
2422, 23sylib 219 . 2 (𝜑 → ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
25 fedgmul.1 . . . . . . 7 (𝜑𝐸 ∈ DivRing)
26 fedgmul.b . . . . . . . 8 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
2726, 4sralvec 30594 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
2825, 1, 2, 27syl3anc 1364 . . . . . 6 (𝜑𝐵 ∈ LVec)
29 eqid 2795 . . . . . . 7 (LBasis‘𝐵) = (LBasis‘𝐵)
3029lbsex 19627 . . . . . 6 (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
3128, 30syl 17 . . . . 5 (𝜑 → (LBasis‘𝐵) ≠ ∅)
32 n0 4230 . . . . 5 ((LBasis‘𝐵) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
3331, 32sylib 219 . . . 4 (𝜑 → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
3433adantr 481 . . 3 ((𝜑𝑥 ∈ (LBasis‘𝐶)) → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
35 drngring 19199 . . . . . . . . . . . . . . 15 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
3625, 35syl 17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ Ring)
3736ad4antr 728 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝐸 ∈ Ring)
38 simplr 765 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39 eqid 2795 . . . . . . . . . . . . . . . . . 18 (Base‘𝐶) = (Base‘𝐶)
4039, 20lbsss 19539 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (LBasis‘𝐶) → 𝑥 ⊆ (Base‘𝐶))
4138, 40syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐶))
42 eqid 2795 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐸) = (Base‘𝐸)
4342subrgss 19226 . . . . . . . . . . . . . . . . . . . . 21 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
442, 43syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑈 ⊆ (Base‘𝐸))
454, 42ressbas2 16384 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
4644, 45syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑈 = (Base‘𝐹))
4716a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐶 = ((subringAlg ‘𝐹)‘𝑉))
48 eqid 2795 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐹) = (Base‘𝐹)
4948subrgss 19226 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
503, 49syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑉 ⊆ (Base‘𝐹))
5147, 50srabase 19640 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
5246, 51eqtrd 2831 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐶))
5352, 44eqsstrrd 3927 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
5453ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐶) ⊆ (Base‘𝐸))
5541, 54sstrd 3899 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐸))
5655ad2antrr 722 . . . . . . . . . . . . . 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 3890 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . . 18 (Base‘𝐵) = (Base‘𝐵)
6160, 29lbsss 19539 . . . . . . . . . . . . . . . . 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 19640 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
6564ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐵))
6662, 65sseqtr4d 3929 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐸))
6766ad2antrr 722 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑦 ⊆ (Base‘𝐸))
68 simplr 765 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
6967, 68sseldd 3890 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗 ∈ (Base‘𝐸))
70 eqid 2795 . . . . . . . . . . . . . 14 (.r𝐸) = (.r𝐸)
7142, 70ringcl 19001 . . . . . . . . . . . . 13 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
7237, 58, 69, 71syl3anc 1364 . . . . . . . . . . . 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 19226 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
7775, 76syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐸))
7874, 77srabase 19640 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
7978ad4antr 728 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (Base‘𝐸) = (Base‘𝐴))
8072, 79eleqtrd 2885 . . . . . . . . . . 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 3158 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
83 oveq2 7024 . . . . . . . . . . 11 (𝑤 = 𝑗 → (𝑡(.r𝐸)𝑤) = (𝑡(.r𝐸)𝑗))
84 oveq1 7023 . . . . . . . . . . 11 (𝑡 = 𝑖 → (𝑡(.r𝐸)𝑗) = (𝑖(.r𝐸)𝑗))
8583, 84cbvmpov 7105 . . . . . . . . . 10 (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) = (𝑗𝑦, 𝑖𝑥 ↦ (𝑖(.r𝐸)𝑗))
8685fmpo 7622 . . . . . . . . 9 (∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴) ↔ (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
8782, 86sylib 219 . . . . . . . 8 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
88 eqid 2795 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
89 eqid 2795 . . . . . . . . . . . . . 14 ( ·𝑠𝐵) = ( ·𝑠𝐵)
90 eqid 2795 . . . . . . . . . . . . . 14 (+g𝐵) = (+g𝐵)
91 eqid 2795 . . . . . . . . . . . . . 14 (0g‘(Scalar‘𝐵)) = (0g‘(Scalar‘𝐵))
9228ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LVec)
9392ad5antr 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)) → 𝐵 ∈ LVec)
9429lbslinds 20659 . . . . . . . . . . . . . . . 16 (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
9594, 59sseldi 3887 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LIndS‘𝐵))
9695ad5antr 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)) → 𝑦 ∈ (LIndS‘𝐵))
9768ad3antrrr 726 . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . 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 19643 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
1004, 99syl5eq 2843 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 = (Scalar‘𝐵))
101100fveq2d 6542 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102101, 51eqtr3d 2833 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
103102ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
10441, 103sseqtr4d 3929 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
105104ad5antr 730 . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . 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 3890 . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . 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 3890 . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LVec)
111 eqid 2795 . . . . . . . . . . . . . . . . . . . . 21 (LSpan‘𝐶) = (LSpan‘𝐶)
11239, 20, 111islbs4 20658 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (LBasis‘𝐶) ↔ (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
11338, 112sylib 219 . . . . . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . . . 19 (0g𝐶) = (0g𝐶)
1161150nellinds 30583 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LIndS‘𝐶)) → ¬ (0g𝐶) ∈ 𝑥)
117110, 114, 116syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ¬ (0g𝐶) ∈ 𝑥)
118117ad5antr 730 . . . . . . . . . . . . . . . 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 3083 . . . . . . . . . . . . . . . 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 6542 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐵)))
12216, 1, 3drgext0g 30596 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐹) = (0g𝐶))
123121, 122eqtr3d 2833 . . . . . . . . . . . . . . . 16 (𝜑 → (0g‘(Scalar‘𝐵)) = (0g𝐶))
124123ad7antr 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)) → (0g‘(Scalar‘𝐵)) = (0g𝐶))
125120, 124neeqtrrd 3058 . . . . . . . . . . . . . 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 7050 . . . . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . . 17 ((𝑗𝑦𝑖𝑥 ∧ (𝑖(.r𝐸)𝑗) ∈ V) → (𝑗(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑖) = (𝑖(.r𝐸)𝑗))
12997, 106, 127, 128syl3anc 1364 . . . . . . . . . . . . . . . 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 30597 . . . . . . . . . . . . . . . . . 18 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
131130oveqd 7033 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑖(.r𝐸)𝑗) = (𝑖( ·𝑠𝐵)𝑗))
132131ad7antr 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)) → (𝑖(.r𝐸)𝑗) = (𝑖( ·𝑠𝐵)𝑗))
133129, 132eqtrd 2831 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑖 = 𝑢)
136 simprl 767 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → 𝑗 = 𝑣)
137135, 136oveq12d 7034 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) ∧ (𝑗 = 𝑣𝑖 = 𝑢)) → (𝑖(.r𝐸)𝑗) = (𝑢(.r𝐸)𝑣))
138 simplr 765 . . . . . . . . . . . . . . . . . . . 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 7050 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑢(.r𝐸)𝑣) ∈ V)
141134, 137, 138, 139, 140ovmpod 7158 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
142141adantllr 715 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑖𝑥) ∧ 𝑣𝑦) ∧ 𝑢𝑥) → (𝑣(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))𝑢) = (𝑢(.r𝐸)𝑣))
143142adantl3r 746 . . . . . . . . . . . . . . . . 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 7033 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑢(.r𝐸)𝑣) = (𝑢( ·𝑠𝐵)𝑣))
146145ad7antr 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)) → (𝑢(.r𝐸)𝑣) = (𝑢( ·𝑠𝐵)𝑣))
147144, 146eqtrd 2831 . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . 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 30587 . . . . . . . . . . . . 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 3158 . . . . . . . . . 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 3158 . . . . . . . 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 7069 . . . . . . . 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 7331 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158 f1rnen 30064 . . . . . . 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 13558 . . . . . 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 30213 . . . . . 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 2831 . . . 4 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = ((♯‘𝑦) ·e (♯‘𝑥)))
16573, 12sralvec 30594 . . . . . . 7 ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
16625, 14, 75, 165syl3anc 1364 . . . . . 6 (𝜑𝐴 ∈ LVec)
167166ad2antrr 722 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168 lveclmod 19568 . . . . . . . . 9 (𝐴 ∈ LVec → 𝐴 ∈ LMod)
169166, 168syl 17 . . . . . . . 8 (𝜑𝐴 ∈ LMod)
170169ad2antrr 722 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LMod)
17125ad4antr 728 . . . . . . . . . 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 728 . . . . . . . . . 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 728 . . . . . . . . . 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 728 . . . . . . . . . 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 728 . . . . . . . . . 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 6538 . . . . . . . . . . . 12 (𝑤 = 𝑗 → (𝑓𝑤) = (𝑓𝑗))
177176fveq1d 6540 . . . . . . . . . . 11 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑗)‘𝑣))
178 fveq2 6538 . . . . . . . . . . 11 (𝑣 = 𝑖 → ((𝑓𝑗)‘𝑣) = ((𝑓𝑗)‘𝑖))
179177, 178cbvmpov 7105 . . . . . . . . . 10 (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)) = (𝑗𝑦, 𝑖𝑥 ↦ ((𝑓𝑗)‘𝑖))
180 simp-4r 780 . . . . . . . . . 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 772 . . . . . . . . . 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 765 . . . . . . . . . 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 30630 . . . . . . . . 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 3149 . . . . . . 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 2795 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
188 eqid 2795 . . . . . . . . 9 (Scalar‘𝐴) = (Scalar‘𝐴)
189 eqid 2795 . . . . . . . . 9 ( ·𝑠𝐴) = ( ·𝑠𝐴)
190 eqid 2795 . . . . . . . . 9 (0g𝐴) = (0g𝐴)
191 eqid 2795 . . . . . . . . 9 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
192 eqid 2795 . . . . . . . . 9 (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥))) = (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))
193187, 188, 189, 190, 191, 192islindf4 20664 . . . . . . . 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 1366 . . . . . 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 3158 . . . . . . . . . 10 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗𝑦𝑖𝑥 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
19885rnmposs 30109 . . . . . . . . . 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 722 . . . . . . . . 9 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐴))
201199, 200sseqtrd 3928 . . . . . . . 8 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ⊆ (Base‘𝐴))
202 eqid 2795 . . . . . . . . 9 (LSpan‘𝐴) = (LSpan‘𝐴)
203187, 202lspssv 19445 . . . . . . . 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 728 . . . . . . . . . . . . . . 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 8278 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑𝑚 𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
208207ad4antlr 729 . . . . . . . . . . . . . . . . 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, 209ffvelrnd 6717 . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . . . 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 2886 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
217 eqid 2795 . . . . . . . . . . . . . . . . 17 (Scalar‘𝐶) = (Scalar‘𝐶)
218 eqid 2795 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
219 eqid 2795 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝐶) = ( ·𝑠𝐶)
220 lveclmod 19568 . . . . . . . . . . . . . . . . . . 19 (𝐶 ∈ LVec → 𝐶 ∈ LMod)
22119, 220syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐶 ∈ LMod)
222221ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LMod)
223111, 39, 216, 217, 218, 219, 222, 41ellspds 30581 . . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → (𝑎𝑤) = (𝑎𝑗))
228 fveq2 6538 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖 → (𝑏𝑣) = (𝑏𝑖))
229 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑖𝑣 = 𝑖)
230228, 229oveq12d 7034 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑖 → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = ((𝑏𝑖)( ·𝑠𝐶)𝑖))
231230cbvmptv 5061 . . . . . . . . . . . . . . . . . . . 20 (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖))
232231oveq2i 7027 . . . . . . . . . . . . . . . . . . 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 2810 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑗 → ((𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) ↔ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ ((𝑏𝑖)( ·𝑠𝐶)𝑖)))))
235234anbi2d 628 . . . . . . . . . . . . . . . 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 3260 . . . . . . . . . . . . . . 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))))))
237236cbvralv 3403 . . . . . . . . . . . . . 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 3440 . . . . . . . . . . . . . . 15 𝑦 ∈ V
239 breq1 4965 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑤) finSupp (0g‘(Scalar‘𝐶))))
240 fveq1 6537 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = (𝑓𝑤) → (𝑏𝑣) = ((𝑓𝑤)‘𝑣))
241240oveq1d 7031 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝑓𝑤) → ((𝑏𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))
242241mpteq2dv 5056 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓𝑤) → (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣)) = (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))
243242oveq2d 7032 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑓𝑤) → (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))))
244243eqeq2d 2805 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑤) → ((𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ ((𝑏𝑣)( ·𝑠𝐶)𝑣))) ↔ (𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)))))
245239, 244anbi12d 630 . . . . . . . . . . . . . . 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 9752 . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → 𝑗𝑦)
251249, 250ffvelrnd 6717 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥)) ∧ 𝑗𝑦) ∧ 𝑖𝑥) → (𝑓𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥))
252 elmapi 8278 . . . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → 𝑖𝑥)
256254, 255ffvelrnd 6717 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
25774, 77srasca 19643 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
25812, 257syl5eq 2843 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐴))
25947, 50srasca 19643 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
26013, 259eqtr3d 2833 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 = (Scalar‘𝐶))
261258, 260eqtr3d 2833 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262261fveq2d 6542 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
263262ad4antr 728 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
264256, 263eleqtrrd 2886 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥)) ∧ (𝑗𝑦𝑖𝑥)) → ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
265264ralrimivva 3158 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑥)) → ∀𝑗𝑦𝑖𝑥 ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
266179fmpo 7622 . . . . . . . . . . . . . . . . . . 19 (∀𝑗𝑦𝑖𝑥 ((𝑓𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ (𝑤𝑦, 𝑣𝑥 ↦ ((𝑓𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴)))
267265, 266sylib 219 . . . . . . . . . . . . . . . . . 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 6553 . . . . . . . . . . . . . . . . . . 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 8270 . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . 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 4972 . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . . . 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 7032 . . . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . 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 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))))) → 𝐸 ∈ DivRing)
2821ad8antr 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))))) → 𝐹 ∈ DivRing)
28314ad8antr 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))))) → 𝐾 ∈ DivRing)
2842ad8antr 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))))) → 𝑈 ∈ (SubRing‘𝐸))
2853ad8antr 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))))) → 𝑉 ∈ (SubRing‘𝐹))
28638ad6antr 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))))) → 𝑥 ∈ (LBasis‘𝐶))
28759ad6antr 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))))) → 𝑦 ∈ (LBasis‘𝐵))
288 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ (Base‘𝐴))
289288ad5antr 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)))) ∧ 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 731 . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . 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 7034 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑗 → ((𝑎𝑤)( ·𝑠𝐵)𝑤) = ((𝑎𝑗)( ·𝑠𝐵)𝑗))
295294cbvmptv 5061 . . . . . . . . . . . . . . . . 17 (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤)) = (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗))
296295oveq2i 7027 . . . . . . . . . . . . . . . 16 (𝐵 Σg (𝑤𝑦 ↦ ((𝑎𝑤)( ·𝑠𝐵)𝑤))) = (𝐵 Σg (𝑗𝑦 ↦ ((𝑎𝑗)( ·𝑠𝐵)𝑗)))
297292, 296syl6eq 2847 . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . 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 4972 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑗 → ((𝑓𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓𝑗) finSupp (0g‘(Scalar‘𝐶))))
301 fveq2 6538 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑖 → ((𝑓𝑤)‘𝑣) = ((𝑓𝑤)‘𝑖))
302301, 229oveq12d 7034 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑖 → (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣) = (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
303302cbvmptv 5061 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖))
304176fveq1d 6540 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑗 → ((𝑓𝑤)‘𝑖) = ((𝑓𝑗)‘𝑖))
305304oveq1d 7031 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑗 → (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖) = (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))
306305mpteq2dv 5056 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑗 → (𝑖𝑥 ↦ (((𝑓𝑤)‘𝑖)( ·𝑠𝐶)𝑖)) = (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
307303, 306syl5eq 2843 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑗 → (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣)) = (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
308307oveq2d 7032 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑗 → (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
309227, 308eqeq12d 2810 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑗 → ((𝑎𝑤) = (𝐶 Σg (𝑣𝑥 ↦ (((𝑓𝑤)‘𝑣)( ·𝑠𝐶)𝑣))) ↔ (𝑎𝑗) = (𝐶 Σg (𝑖𝑥 ↦ (((𝑓𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))))
310300, 309anbi12d 630 . . . . . . . . . . . . . . . . . . 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))))))
311310cbvralv 3403 . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . . . . 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 30629 . . . . . . . . . . . . . 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 3566 . . . . . . . . . . . . 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 1913 . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . 17 (LSpan‘𝐵) = (LSpan‘𝐵)
32260, 29, 321islbs4 20658 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (LBasis‘𝐵) ↔ (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
32359, 322sylib 219 . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . 14 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
327326ad3antrrr 726 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
328325, 327eqtr4d 2834 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐴))
329288, 328eleqtrrd 2886 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐵)‘𝑦))
330 eqid 2795 . . . . . . . . . . . . 13 (Scalar‘𝐵) = (Scalar‘𝐵)
331 lveclmod 19568 . . . . . . . . . . . . . . 15 (𝐵 ∈ LVec → 𝐵 ∈ LMod)
33228, 331syl 17 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ LMod)
333332ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LMod)
334321, 60, 88, 330, 91, 89, 333, 62ellspds 30581 . . . . . . . . . . . 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 3252 . . . . . . . . 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 2795 . . . . . . . . . . 11 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
339202, 187, 338, 188, 191, 189, 87, 170, 157ellspd 20628 . . . . . . . . . 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 258 . . . . . . . 8 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘((𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) “ (𝑦 × 𝑥))))
34287ffnd 6383 . . . . . . . . . . 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 6346 . . . . . . . . . 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 6542 . . . . . . . 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 2885 . . . . . . 7 ((((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))))
348204, 347eqelssd 3909 . . . . . 6 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (Base‘𝐴))
349 eqid 2795 . . . . . . 7 (Base‘(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤))) = (Base‘(𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)))
350 drngnzr 19724 . . . . . . . . . 10 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
35114, 350syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ NzRing)
352258, 351eqeltrrd 2884 . . . . . . . 8 (𝜑 → (Scalar‘𝐴) ∈ NzRing)
353352ad2antrr 722 . . . . . . 7 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Scalar‘𝐴) ∈ NzRing)
354187, 349, 188, 189, 190, 191, 202, 170, 353, 157, 156lindflbs 30586 . . . . . 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 709 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤𝑦, 𝑡𝑥 ↦ (𝑡(.r𝐸)𝑤)) ∈ (LBasis‘𝐴))
356 eqid 2795 . . . . . 6 (LBasis‘𝐴) = (LBasis‘𝐴)
357356dimval 30605 . . . . 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 30605 . . . . . 6 ((𝐵 ∈ LVec ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36092, 59, 359syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
36120dimval 30605 . . . . . 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 7034 . . . 4 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((dim‘𝐵) ·e (dim‘𝐶)) = ((♯‘𝑦) ·e (♯‘𝑥)))
364164, 358, 3633eqtr4d 2841 . . 3 (((𝜑𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
36534, 364exlimddv 1913 . 2 ((𝜑𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
36624, 365exlimddv 1913 1 (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  ∃wrex 3106  Vcvv 3437   ⊆ wss 3859  ∅c0 4211  {csn 4472   class class class wbr 4962   ↦ cmpt 5041   × cxp 5441  ran crn 5444   “ cima 5446   Fn wfn 6220  ⟶wf 6221  –1-1→wf1 6222  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018   ∘𝑓 cof 7265   ↑𝑚 cmap 8256   ≈ cen 8354   finSupp cfsupp 8679   ·e cxmu 12356  ♯chash 13540  Basecbs 16312   ↾s cress 16313  +gcplusg 16394  .rcmulr 16395  Scalarcsca 16397   ·𝑠 cvsca 16398  0gc0g 16542   Σg cgsu 16543  Ringcrg 18987  DivRingcdr 19192  SubRingcsubrg 19221  LModclmod 19324  LSpanclspn 19433  LBasisclbs 19536  LVecclvec 19564  subringAlg csra 19630  NzRingcnzr 19719   freeLMod cfrlm 20572   LIndF clindf 20630  LIndSclinds 20631  dimcldim 30603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-reg 8902  ax-inf2 8950  ax-ac2 9731  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-rpss 7307  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-sup 8752  df-oi 8820  df-r1 9039  df-rank 9040  df-dju 9176  df-card 9214  df-acn 9217  df-ac 9388  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-xnn0 11816  df-z 11830  df-dec 11948  df-uz 12094  df-xmul 12359  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ocomp 16415  df-ds 16416  df-hom 16418  df-cco 16419  df-0g 16544  df-gsum 16545  df-prds 16550  df-pws 16552  df-mre 16686  df-mrc 16687  df-mri 16688  df-acs 16689  df-proset 17367  df-drs 17368  df-poset 17385  df-ipo 17591  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-mhm 17774  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-mulg 17982  df-subg 18030  df-ghm 18097  df-cntz 18188  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-ring 18989  df-oppr 19063  df-dvdsr 19081  df-unit 19082  df-invr 19112  df-drng 19194  df-subrg 19223  df-lmod 19326  df-lss 19394  df-lsp 19434  df-lmhm 19484  df-lbs 19537  df-lvec 19565  df-sra 19634  df-rgmod 19635  df-nzr 19720  df-dsmm 20558  df-frlm 20573  df-uvc 20609  df-lindf 20632  df-linds 20633  df-dim 30604 This theorem is referenced by:  extdgmul  30655
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