Theorem fedgmul 33683

Description: The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, we have [𝐸:𝐾] = [𝐸:𝐹][𝐹:𝐾]. Proposition 1.2 of [Lang], p. 224. Here (dim‘𝐴) is the degree of the extension 𝐸 of 𝐾, (dim‘𝐵) is the degree of the extension 𝐸 of 𝐹, and (dim‘𝐶) is the degree of the extension 𝐹 of 𝐾. This proof is valid for infinite dimensions, and is actually valid for division ring extensions, not just field extensions. (Contributed by Thierry Arnoux, 25-Jul-2023.)

Hypotheses

Ref	Expression
fedgmul.a	⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
fedgmul.b	⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
fedgmul.c	⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
fedgmul.f	⊢ 𝐹 = (𝐸 ↾s 𝑈)
fedgmul.k	⊢ 𝐾 = (𝐸 ↾s 𝑉)
fedgmul.1	⊢ (𝜑 → 𝐸 ∈ DivRing)
fedgmul.2	⊢ (𝜑 → 𝐹 ∈ DivRing)
fedgmul.3	⊢ (𝜑 → 𝐾 ∈ DivRing)
fedgmul.4	⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))
fedgmul.5	⊢ (𝜑 → 𝑉 ∈ (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.

Step	Hyp	Ref	Expression
1		fedgmul.2	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ DivRing)
2		fedgmul.4	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))
3		fedgmul.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))
4		fedgmul.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝐸 ↾s 𝑈)
5	4	subsubrg 20599	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈)))
6	5	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
7	2, 3, 6	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
8	7	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
9		ressabs 17295	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
10	2, 8, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
11	4	oveq1i 7442	. . . . . . 7 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
12		fedgmul.k	. . . . . . 7 ⊢ 𝐾 = (𝐸 ↾s 𝑉)
13	10, 11, 12	3eqtr4g 2801	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = 𝐾)
14		fedgmul.3	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ DivRing)
15	13, 14	eqeltrd 2840	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing)
16		fedgmul.c	. . . . . 6 ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
17		eqid 2736	. . . . . 6 ⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉)
18	16, 17	sralvec 33637	. . . . 5 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
19	1, 15, 3, 18	syl3anc 1372	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
20		eqid 2736	. . . . 5 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
21	20	lbsex 21168	. . . 4 ⊢ (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
22	19, 21	syl 17	. . 3 ⊢ (𝜑 → (LBasis‘𝐶) ≠ ∅)
23		n0 4352	. . 3 ⊢ ((LBasis‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
24	22, 23	sylib 218	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
25		fedgmul.1	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing)
26		fedgmul.b	. . . . . . . 8 ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
27	26, 4	sralvec 33637	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
28	25, 1, 2, 27	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ LVec)
29		eqid 2736	. . . . . . 7 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
30	29	lbsex 21168	. . . . . 6 ⊢ (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
31	28, 30	syl 17	. . . . 5 ⊢ (𝜑 → (LBasis‘𝐵) ≠ ∅)
32		n0 4352	. . . . 5 ⊢ ((LBasis‘𝐵) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
33	31, 32	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
35		drngring 20737	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
36	25, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
37	36	ad4antr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝐸 ∈ Ring)
38		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐶) = (Base‘𝐶)
40	39, 20	lbsss 21077	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (LBasis‘𝐶) → 𝑥 ⊆ (Base‘𝐶))
41	38, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐶))
42		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
43	42	subrgss 20573	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
44	2, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
45	4, 42	ressbas2 17284	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑈 = (Base‘𝐹))
47	16	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐶 = ((subringAlg ‘𝐹)‘𝑉))
48		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
49	48	subrgss 20573	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
50	3, 49	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
51	47, 50	srabase 21178	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
52	46, 51	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
53	52, 44	eqsstrrd 4018	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
54	53	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐶) ⊆ (Base‘𝐸))
55	41, 54	sstrd 3993	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐸))
56	55	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑥 ⊆ (Base‘𝐸))
57		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑖 ∈ 𝑥)
58	56, 57	sseldd 3983	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑖 ∈ (Base‘𝐸))
59		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LBasis‘𝐵))
60		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐵) = (Base‘𝐵)
61	60, 29	lbsss 21077	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (LBasis‘𝐵) → 𝑦 ⊆ (Base‘𝐵))
62	59, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐵))
63	26	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 = ((subringAlg ‘𝐸)‘𝑈))
64	63, 44	srabase 21178	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
65	64	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐵))
66	62, 65	sseqtrrd 4020	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐸))
67	66	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑦 ⊆ (Base‘𝐸))
68		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ 𝑦)
69	67, 68	sseldd 3983	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ (Base‘𝐸))
70		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (.r‘𝐸) = (.r‘𝐸)
71	42, 70	ringcl 20248	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
72	37, 58, 69, 71	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
73		fedgmul.a	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
74	73	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝐸)‘𝑉))
75	7	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐸))
76	42	subrgss 20573	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
77	75, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
78	74, 77	srabase 21178	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
79	78	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → (Base‘𝐸) = (Base‘𝐴))
80	72, 79	eleqtrd 2842	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
81	80	anasss 466	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
82	81	ralrimivva 3201	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
83		oveq2 7440	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → (𝑡(.r‘𝐸)𝑤) = (𝑡(.r‘𝐸)𝑗))
84		oveq1 7439	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑖 → (𝑡(.r‘𝐸)𝑗) = (𝑖(.r‘𝐸)𝑗))
85	83, 84	cbvmpov 7529	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ (𝑖(.r‘𝐸)𝑗))
86	85	fmpo 8094	. . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
87	82, 86	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
88		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
89		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐵) = ( ·𝑠 ‘𝐵)
90		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐵) = (+g‘𝐵)
91		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0g‘(Scalar‘𝐵)) = (0g‘(Scalar‘𝐵))
92	28	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LVec)
93	92	ad5antr 734	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝐵 ∈ LVec)
94	29	lbslinds 21854	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
95	94, 59	sselid 3980	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LIndS‘𝐵))
96	95	ad5antr 734	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑦 ∈ (LIndS‘𝐵))
97	68	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑗 ∈ 𝑦)
98		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑣 ∈ 𝑦)
99	63, 44	srasca 21184	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
100	4, 99	eqtrid 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
101	100	fveq2d 6909	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102	101, 51	eqtr3d 2778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
103	102	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
104	41, 103	sseqtrrd 4020	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
105	104	ad5antr 734	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
106		simp-4r 783	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑖 ∈ 𝑥)
107	105, 106	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑖 ∈ (Base‘(Scalar‘𝐵)))
108		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑢 ∈ 𝑥)
109	105, 108	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑢 ∈ (Base‘(Scalar‘𝐵)))
110	19	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LVec)
111		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
112	39, 20, 111	islbs4 21853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (LBasis‘𝐶) ↔ (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
113	38, 112	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
114	113	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LIndS‘𝐶))
115		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘𝐶) = (0g‘𝐶)
116	115	0nellinds 33399	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LIndS‘𝐶)) → ¬ (0g‘𝐶) ∈ 𝑥)
117	110, 114, 116	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ¬ (0g‘𝐶) ∈ 𝑥)
118	117	ad5antr 734	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → ¬ (0g‘𝐶) ∈ 𝑥)
119		nelne2 3039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ 𝑥 ∧ ¬ (0g‘𝐶) ∈ 𝑥) → 𝑖 ≠ (0g‘𝐶))
120	106, 118, 119	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑖 ≠ (0g‘𝐶))
121	100	fveq2d 6909	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
122	16, 1, 3	drgext0g 33641	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
123	121, 122	eqtr3d 2778	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
124	123	ad7antr 738	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
125	120, 124	neeqtrrd 3014	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑖 ≠ (0g‘(Scalar‘𝐵)))
126		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢))
127		ovexd 7467	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑖(.r‘𝐸)𝑗) ∈ V)
128	85	ovmpt4g 7581	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥 ∧ (𝑖(.r‘𝐸)𝑗) ∈ V) → (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑖(.r‘𝐸)𝑗))
129	97, 106, 127, 128	syl3anc 1372	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑖(.r‘𝐸)𝑗))
130	26, 25, 2	drgextvsca 33642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
131	130	oveqd 7449	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑖(.r‘𝐸)𝑗) = (𝑖( ·𝑠 ‘𝐵)𝑗))
132	131	ad7antr 738	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑖(.r‘𝐸)𝑗) = (𝑖( ·𝑠 ‘𝐵)𝑗))
133	129, 132	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑖( ·𝑠 ‘𝐵)𝑗))
134	85	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ (𝑖(.r‘𝐸)𝑗)))
135		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)) → 𝑖 = 𝑢)
136		simprl 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)) → 𝑗 = 𝑣)
137	135, 136	oveq12d 7450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)) → (𝑖(.r‘𝐸)𝑗) = (𝑢(.r‘𝐸)𝑣))
138		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → 𝑣 ∈ 𝑦)
139		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → 𝑢 ∈ 𝑥)
140		ovexd 7467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑢(.r‘𝐸)𝑣) ∈ V)
141	134, 137, 138, 139, 140	ovmpod 7586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) = (𝑢(.r‘𝐸)𝑣))
142	141	adantllr 719	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) = (𝑢(.r‘𝐸)𝑣))
143	142	adantl3r 750	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) = (𝑢(.r‘𝐸)𝑣))
144	143	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) = (𝑢(.r‘𝐸)𝑣))
145	130	oveqd 7449	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑢(.r‘𝐸)𝑣) = (𝑢( ·𝑠 ‘𝐵)𝑣))
146	145	ad7antr 738	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑢(.r‘𝐸)𝑣) = (𝑢( ·𝑠 ‘𝐵)𝑣))
147	144, 146	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) = (𝑢( ·𝑠 ‘𝐵)𝑣))
148	126, 133, 147	3eqtr3d 2784	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑖( ·𝑠 ‘𝐵)𝑗) = (𝑢( ·𝑠 ‘𝐵)𝑣))
149	88, 89, 90, 91, 93, 96, 97, 98, 107, 109, 125, 148	linds2eq 33410	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → (𝑗 = 𝑣 ∧ 𝑖 = 𝑢))
150	149	ex 412	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → ((𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) → (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)))
151	150	anasss 466	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ (𝑣 ∈ 𝑦 ∧ 𝑢 ∈ 𝑥)) → ((𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) → (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)))
152	151	ralrimivva 3201	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → ∀𝑣 ∈ 𝑦 ∀𝑢 ∈ 𝑥 ((𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) → (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)))
153	152	anasss 466	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → ∀𝑣 ∈ 𝑦 ∀𝑢 ∈ 𝑥 ((𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) → (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)))
154	153	ralrimivva 3201	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 ∀𝑣 ∈ 𝑦 ∀𝑢 ∈ 𝑥 ((𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) → (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)))
155		f1opr 7490	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)–1-1→(Base‘𝐴) ↔ ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴) ∧ ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 ∀𝑣 ∈ 𝑦 ∀𝑢 ∈ 𝑥 ((𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢) → (𝑗 = 𝑣 ∧ 𝑖 = 𝑢))))
156	87, 154, 155	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)–1-1→(Base‘𝐴))
157	59, 38	xpexd 7772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158		f1rnen 32640	. . . . . . 7 ⊢ (((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)–1-1→(Base‘𝐴) ∧ (𝑦 × 𝑥) ∈ V) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ≈ (𝑦 × 𝑥))
159	156, 157, 158	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ≈ (𝑦 × 𝑥))
160		hasheni 14388	. . . . . 6 ⊢ (ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ≈ (𝑦 × 𝑥) → (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (♯‘(𝑦 × 𝑥)))
161	159, 160	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (♯‘(𝑦 × 𝑥)))
162		hashxpe 32812	. . . . . 6 ⊢ ((𝑦 ∈ (LBasis‘𝐵) ∧ 𝑥 ∈ (LBasis‘𝐶)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
163	59, 38, 162	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
164	161, 163	eqtrd 2776	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = ((♯‘𝑦) ·e (♯‘𝑥)))
165	73, 12	sralvec 33637	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
166	25, 14, 75, 165	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ LVec)
167	166	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168		lveclmod 21106	. . . . . . . . 9 ⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
169	166, 168	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ LMod)
170	169	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LMod)
171	25	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝐸 ∈ DivRing)
172	1	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝐹 ∈ DivRing)
173	14	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝐾 ∈ DivRing)
174	2	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝑈 ∈ (SubRing‘𝐸))
175	3	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝑉 ∈ (SubRing‘𝐹))
176		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑗 → (𝑓‘𝑤) = (𝑓‘𝑗))
177	176	fveq1d 6907	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑗)‘𝑣))
178		fveq2 6905	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑗)‘𝑣) = ((𝑓‘𝑗)‘𝑖))
179	177, 178	cbvmpov 7529	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ ((𝑓‘𝑗)‘𝑖))
180		simp-4r 783	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝑥 ∈ (LBasis‘𝐶))
181		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝑦 ∈ (LBasis‘𝐵))
182		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥))))
183		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴))
184	73, 26, 16, 4, 12, 171, 172, 173, 174, 175, 85, 179, 180, 181, 182, 183	fedgmullem2 33682	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) ∧ (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴)) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))}))
185	184	ex 412	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))) → ((𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))})))
186	185	ralrimiva 3145	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))((𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴) → 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 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
190		eqid 2736	. . . . . . . . 9 ⊢ (0g‘𝐴) = (0g‘𝐴)
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)))
193	187, 188, 189, 190, 191, 192	islindf4 21859	. . . . . . . 8 ⊢ ((𝐴 ∈ LMod ∧ (𝑦 × 𝑥) ∈ V ∧ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴)) → ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) LIndF 𝐴 ↔ ∀𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))((𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))}))))
194	193	biimpar 477	. . . . . . 7 ⊢ (((𝐴 ∈ LMod ∧ (𝑦 × 𝑥) ∈ V ∧ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴)) ∧ ∀𝑐 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))((𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (0g‘𝐴) → 𝑐 = ((𝑦 × 𝑥) × {(0g‘(Scalar‘𝐴))}))) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) LIndF 𝐴)
195	170, 157, 87, 186, 194	syl31anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) LIndF 𝐴)
196	72	anasss 466	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
197	196	ralrimivva 3201	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
198	85	rnmposs 32685	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ⊆ (Base‘𝐸))
199	197, 198	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ⊆ (Base‘𝐸))
200	78	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐴))
201	199, 200	sseqtrd 4019	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ⊆ (Base‘𝐴))
202		eqid 2736	. . . . . . . . 9 ⊢ (LSpan‘𝐴) = (LSpan‘𝐴)
203	187, 202	lspssv 20982	. . . . . . . 8 ⊢ ((𝐴 ∈ LMod ∧ ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ⊆ (Base‘𝐴)) → ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) ⊆ (Base‘𝐴))
204	170, 201, 203	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) ⊆ (Base‘𝐴))
205		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)))
206	205	ad4antr 732	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)))
207		elmapi 8890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
208	207	ad4antlr 733	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
209		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → 𝑗 ∈ 𝑦)
210	208, 209	ffvelcdmd 7104	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → (𝑎‘𝑗) ∈ (Base‘(Scalar‘𝐵)))
211	113	simprd 495	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶))
212	206, 211	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶))
213	102	ad7antr 738	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
214	212, 213	eqtr4d 2779	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → ((LSpan‘𝐶)‘𝑥) = (Base‘(Scalar‘𝐵)))
215	210, 214	eleqtrrd 2843	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → (𝑎‘𝑗) ∈ ((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 ⊢ ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘𝐶)
220		lveclmod 21106	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
221	19, 220	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ LMod)
222	221	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LMod)
223	111, 39, 216, 217, 218, 219, 222, 41	ellspds 33397	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((𝑎‘𝑗) ∈ ((LSpan‘𝐶)‘𝑥) ↔ ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))))))
224	223	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ (𝑎‘𝑗) ∈ ((LSpan‘𝐶)‘𝑥)) → ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
225	206, 215, 224	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
226	225	ralrimiva 3145	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) → ∀𝑗 ∈ 𝑦 ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
227		fveq2 6905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → (𝑎‘𝑤) = (𝑎‘𝑗))
228		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → (𝑏‘𝑣) = (𝑏‘𝑖))
229		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → 𝑣 = 𝑖)
230	228, 229	oveq12d 7450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑖 → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
231	230	cbvmptv 5254	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
232	231	oveq2i 7443	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
233	232	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
234	227, 233	eqeq12d 2752	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) ↔ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
235	234	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑗 → ((𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ (𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))))))
236	235	rexbidv 3178	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑗 → (∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))))))
237	236	cbvralvw 3236	. . . . . . . . . . . . . 14 ⊢ (∀𝑤 ∈ 𝑦 ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ ∀𝑗 ∈ 𝑦 ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
238		vex 3483	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
239		breq1 5145	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶))))
240		fveq1 6904	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏‘𝑣) = ((𝑓‘𝑤)‘𝑣))
241	240	oveq1d 7447	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))
242	241	mpteq2dv 5243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))
243	242	oveq2d 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑓‘𝑤) → (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))
244	243	eqeq2d 2747	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) ↔ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))))
245	239, 244	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))))
246	238, 245	ac6s 10525	. . . . . . . . . . . . . 14 ⊢ (∀𝑤 ∈ 𝑦 ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) → ∃𝑓(𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))))
247	237, 246	sylbir 235	. . . . . . . . . . . . 13 ⊢ (∀𝑗 ∈ 𝑦 ∃𝑏 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥)(𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))) → ∃𝑓(𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))))
248	226, 247	syl 17	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) → ∃𝑓(𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))))
249		simpllr 775	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥))
250		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ 𝑦)
251	249, 250	ffvelcdmd 7104	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → (𝑓‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥))
252		elmapi 8890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑥) → (𝑓‘𝑗):𝑥⟶(Base‘(Scalar‘𝐶)))
253	251, 252	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → (𝑓‘𝑗):𝑥⟶(Base‘(Scalar‘𝐶)))
254	253	anasss 466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → (𝑓‘𝑗):𝑥⟶(Base‘(Scalar‘𝐶)))
255		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → 𝑖 ∈ 𝑥)
256	254, 255	ffvelcdmd 7104	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → ((𝑓‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
257	74, 77	srasca 21184	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
258	12, 257	eqtrid 2788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
259	47, 50	srasca 21184	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
260	13, 259	eqtr3d 2778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
261	258, 260	eqtr3d 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262	261	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
263	262	ad4antr 732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
264	256, 263	eleqtrrd 2843	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → ((𝑓‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
265	264	ralrimivva 3201	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 ((𝑓‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
266	179	fmpo 8094	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 ((𝑓‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴)))
267	265, 266	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴)))
268		fvexd 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (Base‘(Scalar‘𝐴)) ∈ V)
269	157	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑦 × 𝑥) ∈ V)
270	268, 269	elmapd 8881	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)) ↔ (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)):(𝑦 × 𝑥)⟶(Base‘(Scalar‘𝐴))))
271	267, 270	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
272	271	ad5ant15 758	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) → (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
273	272	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
274	273	adantl3r 750	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥)))
275		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)))
276	275	breq1d 5152	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → (𝑐 finSupp (0g‘(Scalar‘𝐴)) ↔ (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) finSupp (0g‘(Scalar‘𝐴))))
277	275	oveq1d 7447	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))
278	277	oveq2d 7448	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (𝐴 Σg ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))))
279	278	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → (𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) ↔ 𝑧 = (𝐴 Σg ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
280	276, 279	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → ((𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))) ↔ ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))))))
281	25	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝐸 ∈ DivRing)
282	1	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝐹 ∈ DivRing)
283	14	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝐾 ∈ DivRing)
284	2	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑈 ∈ (SubRing‘𝐸))
285	3	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑉 ∈ (SubRing‘𝐹))
286	38	ad6antr 736	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑥 ∈ (LBasis‘𝐶))
287	59	ad6antr 736	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑦 ∈ (LBasis‘𝐵))
288		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ (Base‘𝐴))
289	288	ad5antr 734	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑧 ∈ (Base‘𝐴))
290	207	ad5antlr 735	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
291		simp-4r 783	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑎 finSupp (0g‘(Scalar‘𝐵)))
292		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))))
293		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑗 → 𝑤 = 𝑗)
294	227, 293	oveq12d 7450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤) = ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
295	294	cbvmptv 5254	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)) = (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
296	295	oveq2i 7443	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))) = (𝐵 Σg (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗)))
297	292, 296	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑧 = (𝐵 Σg (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
298		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥))
299		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))))
300	176	breq1d 5152	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
301		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑤)‘𝑖))
302	301, 229	oveq12d 7450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑖 → (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
303	302	cbvmptv 5254	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
304	176	fveq1d 6907	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑖) = ((𝑓‘𝑗)‘𝑖))
305	304	oveq1d 7447	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑗 → (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
306	305	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑗 → (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
307	303, 306	eqtrid 2788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑗 → (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
308	307	oveq2d 7448	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑗 → (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
309	227, 308	eqeq12d 2752	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))) ↔ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
310	300, 309	anbi12d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑗 → (((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ ((𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))))
311	310	cbvralvw 3236	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ ∀𝑗 ∈ 𝑦 ((𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
312	299, 311	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → ∀𝑗 ∈ 𝑦 ((𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
313	312	r19.21bi 3250	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑗 ∈ 𝑦) → ((𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
314	313	simpld 494	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑗 ∈ 𝑦) → (𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶)))
315	313	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑗 ∈ 𝑦) → (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
316	73, 26, 16, 4, 12, 281, 282, 283, 284, 285, 85, 179, 286, 287, 289, 290, 291, 297, 298, 314, 315	fedgmullem1 33681	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
317	274, 280, 316	rspcedvd 3623	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
318	317	anasss 466	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ (𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
319	248, 318	exlimddv 1934	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
320	319	anasss 466	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ (𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))))) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
321		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (LSpan‘𝐵) = (LSpan‘𝐵)
322	60, 29, 321	islbs4 21853	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (LBasis‘𝐵) ↔ (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
323	59, 322	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
324	323	simprd 495	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵))
325	324	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵))
326	78, 64	eqtr3d 2778	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
327	326	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
328	325, 327	eqtr4d 2779	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐴))
329	288, 328	eleqtrrd 2843	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐵)‘𝑦))
330		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐵) = (Scalar‘𝐵)
331		lveclmod 21106	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ LVec → 𝐵 ∈ LMod)
332	28, 331	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ LMod)
333	332	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LMod)
334	321, 60, 88, 330, 91, 89, 333, 62	ellspds 33397	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑧 ∈ ((LSpan‘𝐵)‘𝑦) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)(𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))))))
335	334	biimpa 476	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ ((LSpan‘𝐵)‘𝑦)) → ∃𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)(𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))))
336	205, 329, 335	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ∃𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)(𝑎 finSupp (0g‘(Scalar‘𝐵)) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))))
337	320, 336	r19.29a 3161	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
338		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
339	202, 187, 338, 188, 191, 189, 87, 170, 157	ellspd 21823	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑧 ∈ ((LSpan‘𝐴)‘((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) “ (𝑦 × 𝑥))) ↔ ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))))))
340	339	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (𝑧 ∈ ((LSpan‘𝐴)‘((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) “ (𝑦 × 𝑥))) ↔ ∃𝑐 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑦 × 𝑥))(𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))))))
341	337, 340	mpbird 257	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) “ (𝑦 × 𝑥))))
342	87	ffnd 6736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) Fn (𝑦 × 𝑥))
343	342	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) Fn (𝑦 × 𝑥))
344		fnima 6697	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) Fn (𝑦 × 𝑥) → ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) “ (𝑦 × 𝑥)) = ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))
345	343, 344	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) “ (𝑦 × 𝑥)) = ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))
346	345	fveq2d 6909	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐴)‘((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) “ (𝑦 × 𝑥))) = ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))
347	341, 346	eleqtrd 2842	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))
348	204, 347	eqelssd 4004	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘𝐴))
349		eqid 2736	. . . . . . 7 ⊢ (Base‘(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))
350		drngnzr 20749	. . . . . . . . . 10 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
351	14, 350	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ NzRing)
352	258, 351	eqeltrrd 2841	. . . . . . . 8 ⊢ (𝜑 → (Scalar‘𝐴) ∈ NzRing)
353	352	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Scalar‘𝐴) ∈ NzRing)
354	187, 349, 188, 189, 190, 191, 202, 170, 353, 157, 156	lindflbs 33408	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ∈ (LBasis‘𝐴) ↔ ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) LIndF 𝐴 ∧ ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘𝐴))))
355	195, 348, 354	mpbir2and 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ∈ (LBasis‘𝐴))
356		eqid 2736	. . . . . 6 ⊢ (LBasis‘𝐴) = (LBasis‘𝐴)
357	356	dimval 33652	. . . . 5 ⊢ ((𝐴 ∈ LVec ∧ ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ∈ (LBasis‘𝐴)) → (dim‘𝐴) = (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))
358	167, 355, 357	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))
359	29	dimval 33652	. . . . . 6 ⊢ ((𝐵 ∈ LVec ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
360	92, 59, 359	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
361	20	dimval 33652	. . . . . 6 ⊢ ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐶) = (♯‘𝑥))
362	110, 38, 361	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐶) = (♯‘𝑥))
363	360, 362	oveq12d 7450	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((dim‘𝐵) ·e (dim‘𝐶)) = ((♯‘𝑦) ·e (♯‘𝑥)))
364	164, 358, 363	3eqtr4d 2786	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
365	34, 364	exlimddv 1934	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
366	24, 365	exlimddv 1934	1 ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ⊆ wss 3950  ∅c0 4332  {csn 4625   class class class wbr 5142   ↦ cmpt 5224   × cxp 5682  ran crn 5685   “ cima 5687   Fn wfn 6555  ⟶wf 6556  –1-1→wf1 6557  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434   ∘_f cof 7696   ↑_m cmap 8867   ≈ cen 8983   finSupp cfsupp 9402   ·_e cxmu 13154  ♯chash 14370  Basecbs 17248   ↾_s cress 17275  +_gcplusg 17298  ._rcmulr 17299  Scalarcsca 17301   ·_𝑠 cvsca 17302  0_gc0g 17485   Σ_g cgsu 17486  Ringcrg 20231  NzRingcnzr 20513  SubRingcsubrg 20570  DivRingcdr 20730  LModclmod 20859  LSpanclspn 20970  LBasisclbs 21074  LVecclvec 21102  subringAlg csra 21171   freeLMod cfrlm 21767   LIndF clindf 21825  LIndSclinds 21826  dimcldim 33650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682  ax-ac2 10504  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-rpss 7744  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-r1 9805  df-rank 9806  df-dju 9942  df-card 9980  df-acn 9983  df-ac 10157  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12736  df-uz 12880  df-xmul 13157  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-mri 17632  df-acs 17633  df-proset 18341  df-drs 18342  df-poset 18360  df-ipo 18574  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-nzr 20514  df-subrng 20547  df-subrg 20571  df-drng 20732  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-lindf 21827  df-linds 21828  df-dim 33651

This theorem is referenced by:  extdgmul  33715