Theorem fedgmul 33775

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 20575	. . . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
8	7	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
9		ressabs 17218	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
10	2, 8, 9	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
11	4	oveq1i 7377	. . . . . . 7 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
12		fedgmul.k	. . . . . . 7 ⊢ 𝐾 = (𝐸 ↾s 𝑉)
13	10, 11, 12	3eqtr4g 2796	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = 𝐾)
14		fedgmul.3	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ DivRing)
15	13, 14	eqeltrd 2836	. . . . 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 33729	. . . . 5 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
19	1, 15, 3, 18	syl3anc 1374	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
20		eqid 2736	. . . . 5 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
21	20	lbsex 21163	. . . 4 ⊢ (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
22	19, 21	syl 17	. . 3 ⊢ (𝜑 → (LBasis‘𝐶) ≠ ∅)
23		n0 4293	. . 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 33729	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
28	25, 1, 2, 27	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ LVec)
29		eqid 2736	. . . . . . 7 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
30	29	lbsex 21163	. . . . . 6 ⊢ (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
31	28, 30	syl 17	. . . . 5 ⊢ (𝜑 → (LBasis‘𝐵) ≠ ∅)
32		n0 4293	. . . . 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 20713	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
36	25, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
37	36	ad4antr 733	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝐸 ∈ Ring)
38		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐶) = (Base‘𝐶)
40	39, 20	lbsss 21072	. . . . . . . . . . . . . . . . 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 20549	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
44	2, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
45	4, 42	ressbas2 17208	. . . . . . . . . . . . . . . . . . . 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 20549	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
50	3, 49	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
51	47, 50	srabase 21172	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
52	46, 51	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
53	52, 44	eqsstrrd 3957	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
54	53	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐶) ⊆ (Base‘𝐸))
55	41, 54	sstrd 3932	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐸))
56	55	ad2antrr 727	. . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . 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 21072	. . . . . . . . . . . . . . . . 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 21172	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
65	64	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐵))
66	62, 65	sseqtrrd 3959	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐸))
67	66	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑦 ⊆ (Base‘𝐸))
68		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ 𝑦)
69	67, 68	sseldd 3922	. . . . . . . . . . . . 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 20231	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
72	37, 58, 69, 71	syl3anc 1374	. . . . . . . . . . . 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 20549	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
77	75, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
78	74, 77	srabase 21172	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
79	78	ad4antr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → (Base‘𝐸) = (Base‘𝐴))
80	72, 79	eleqtrd 2838	. . . . . . . . . . 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 3180	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
83		oveq2 7375	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → (𝑡(.r‘𝐸)𝑤) = (𝑡(.r‘𝐸)𝑗))
84		oveq1 7374	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑖 → (𝑡(.r‘𝐸)𝑗) = (𝑖(.r‘𝐸)𝑗))
85	83, 84	cbvmpov 7462	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ (𝑖(.r‘𝐸)𝑗))
86	85	fmpo 8021	. . . . . . . . 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 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LVec)
93	92	ad5antr 735	. . . . . . . . . . . . . 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 21813	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
95	94, 59	sselid 3919	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LIndS‘𝐵))
96	95	ad5antr 735	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 21175	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
100	4, 99	eqtrid 2783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
101	100	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102	101, 51	eqtr3d 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
103	102	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
104	41, 103	sseqtrrd 3959	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
105	104	ad5antr 735	. . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LVec)
111		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
112	39, 20, 111	islbs4 21812	. . . . . . . . . . . . . . . . . . . 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 33430	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LIndS‘𝐶)) → ¬ (0g‘𝐶) ∈ 𝑥)
117	110, 114, 116	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ¬ (0g‘𝐶) ∈ 𝑥)
118	117	ad5antr 735	. . . . . . . . . . . . . . . 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 3030	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ 𝑥 ∧ ¬ (0g‘𝐶) ∈ 𝑥) → 𝑖 ≠ (0g‘𝐶))
120	106, 118, 119	syl2anc 585	. . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
122	16, 1, 3	drgext0g 33734	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
123	121, 122	eqtr3d 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
124	123	ad7antr 739	. . . . . . . . . . . . . . 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 3006	. . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . 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 7514	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 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 33735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
131	130	oveqd 7384	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑖(.r‘𝐸)𝑗) = (𝑖( ·𝑠 ‘𝐵)𝑗))
132	131	ad7antr 739	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)) → 𝑖 = 𝑢)
136		simprl 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)) → 𝑗 = 𝑣)
137	135, 136	oveq12d 7385	. . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑢(.r‘𝐸)𝑣) ∈ V)
141	134, 137, 138, 139, 140	ovmpod 7519	. . . . . . . . . . . . . . . . . . 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 720	. . . . . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑢(.r‘𝐸)𝑣) = (𝑢( ·𝑠 ‘𝐵)𝑣))
146	145	ad7antr 739	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 33441	. . . . . . . . . . . . 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 3180	. . . . . . . . . 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 3180	. . . . . . . 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 7423	. . . . . . . 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 584	. . . . . . 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 7705	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158		f1rnen 32701	. . . . . . 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 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ≈ (𝑦 × 𝑥))
160		hasheni 14310	. . . . . 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 32880	. . . . . 6 ⊢ ((𝑦 ∈ (LBasis‘𝐵) ∧ 𝑥 ∈ (LBasis‘𝐶)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
163	59, 38, 162	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
164	161, 163	eqtrd 2771	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = ((♯‘𝑦) ·e (♯‘𝑥)))
165	73, 12	sralvec 33729	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
166	25, 14, 75, 165	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ LVec)
167	166	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168		lveclmod 21101	. . . . . . . . 9 ⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
169	166, 168	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ LMod)
170	169	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LMod)
171	25	ad4antr 733	. . . . . . . . . 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 733	. . . . . . . . . 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 733	. . . . . . . . . 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 733	. . . . . . . . . 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 733	. . . . . . . . . 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 6840	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑗 → (𝑓‘𝑤) = (𝑓‘𝑗))
177	176	fveq1d 6842	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑗)‘𝑣))
178		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑗)‘𝑣) = ((𝑓‘𝑗)‘𝑖))
179	177, 178	cbvmpov 7462	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ ((𝑓‘𝑗)‘𝑖))
180		simp-4r 784	. . . . . . . . . 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 776	. . . . . . . . . 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 769	. . . . . . . . . 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 33774	. . . . . . . . 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 3129	. . . . . . 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 21818	. . . . . . . 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 1376	. . . . . 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 3180	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
198	85	rnmposs 32746	. . . . . . . . . 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 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐴))
201	199, 200	sseqtrd 3958	. . . . . . . 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 20978	. . . . . . . 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 585	. . . . . . 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 733	. . . . . . . . . . . . . . 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 8796	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
208	207	ad4antlr 734	. . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . 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 21101	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
221	19, 220	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ LMod)
222	221	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LMod)
223	111, 39, 216, 217, 218, 219, 222, 41	ellspds 33428	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → (𝑎‘𝑤) = (𝑎‘𝑗))
228		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → (𝑏‘𝑣) = (𝑏‘𝑖))
229		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → 𝑣 = 𝑖)
230	228, 229	oveq12d 7385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑖 → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
231	230	cbvmptv 5189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
232	231	oveq2i 7378	. . . . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . . . 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 3215	. . . . . . . . . . . . . 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 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
239		breq1 5088	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶))))
240		fveq1 6839	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏‘𝑣) = ((𝑓‘𝑤)‘𝑣))
241	240	oveq1d 7382	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))
242	241	mpteq2dv 5179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))
243	242	oveq2d 7383	. . . . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . . 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 10406	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . . . . . . 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 8796	. . . . . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . . . 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 21175	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
258	12, 257	eqtrid 2783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
259	47, 50	srasca 21175	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
260	13, 259	eqtr3d 2773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
261	258, 260	eqtr3d 2773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262	261	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
263	262	ad4antr 733	. . . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . . . . . . . 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 8021	. . . . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . . 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 8787	. . . . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . 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 5095	. . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . 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 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))))) → 𝑧 ∈ (Base‘𝐴))
290	207	ad5antlr 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))))) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
291		simp-4r 784	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤) = ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
295	294	cbvmptv 5189	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)) = (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
296	295	oveq2i 7378	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))) = (𝐵 Σg (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗)))
297	292, 296	eqtrdi 2787	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 5095	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
301		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑤)‘𝑖))
302	301, 229	oveq12d 7385	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑖 → (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
303	302	cbvmptv 5189	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
304	176	fveq1d 6842	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑖) = ((𝑓‘𝑗)‘𝑖))
305	304	oveq1d 7382	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑗 → (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
306	305	mpteq2dv 5179	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑗 → (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
307	303, 306	eqtrid 2783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑗 → (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
308	307	oveq2d 7383	. . . . . . . . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . . . . . . 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 3215	. . . . . . . . . . . . . . . . . 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 3229	. . . . . . . . . . . . . . . 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 33773	. . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . 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 1937	. . . . . . . . . . 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 21812	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
327	326	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
328	325, 327	eqtr4d 2774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐴))
329	288, 328	eleqtrrd 2839	. . . . . . . . . . 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 21101	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ LVec → 𝐵 ∈ LMod)
332	28, 331	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ LMod)
333	332	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LMod)
334	321, 60, 88, 330, 91, 89, 333, 62	ellspds 33428	. . . . . . . . . . . 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 585	. . . . . . . . . 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 3145	. . . . . . . . 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 21782	. . . . . . . . . 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 6669	. . . . . . . . . . 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 6628	. . . . . . . . . 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 6844	. . . . . . . 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 2838	. . . . . . 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 3943	. . . . . 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 20725	. . . . . . . . . 10 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
351	14, 350	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ NzRing)
352	258, 351	eqeltrrd 2837	. . . . . . . 8 ⊢ (𝜑 → (Scalar‘𝐴) ∈ NzRing)
353	352	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Scalar‘𝐴) ∈ NzRing)
354	187, 349, 188, 189, 190, 191, 202, 170, 353, 157, 156	lindflbs 33439	. . . . . 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 714	. . . . 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 33745	. . . . 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 585	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))
359	29	dimval 33745	. . . . . 6 ⊢ ((𝐵 ∈ LVec ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
360	92, 59, 359	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
361	20	dimval 33745	. . . . . 6 ⊢ ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐶) = (♯‘𝑥))
362	110, 38, 361	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐶) = (♯‘𝑥))
363	360, 362	oveq12d 7385	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((dim‘𝐵) ·e (dim‘𝐶)) = ((♯‘𝑦) ·e (♯‘𝑥)))
364	164, 358, 363	3eqtr4d 2781	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
365	34, 364	exlimddv 1937	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
366	24, 365	exlimddv 1937	1 ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  {csn 4567   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629  ran crn 5632   “ cima 5634   Fn wfn 6493  ⟶wf 6494  –1-1→wf1 6495  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ∘_f cof 7629   ↑_m cmap 8773   ≈ cen 8890   finSupp cfsupp 9274   ·_e cxmu 13062  ♯chash 14292  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   Σ_g cgsu 17403  Ringcrg 20214  NzRingcnzr 20489  SubRingcsubrg 20546  DivRingcdr 20706  LModclmod 20855  LSpanclspn 20966  LBasisclbs 21069  LVecclvec 21097  subringAlg csra 21166   freeLMod cfrlm 21726   LIndF clindf 21784  LIndSclinds 21785  dimcldim 33743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-rpss 7677  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-r1 9688  df-rank 9689  df-dju 9825  df-card 9863  df-acn 9866  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-xmul 13065  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ocomp 17241  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-mri 17550  df-acs 17551  df-proset 18260  df-drs 18261  df-poset 18279  df-ipo 18494  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-nzr 20490  df-subrng 20523  df-subrg 20547  df-drng 20708  df-lmod 20857  df-lss 20927  df-lsp 20967  df-lmhm 21017  df-lbs 21070  df-lvec 21098  df-sra 21168  df-rgmod 21169  df-dsmm 21712  df-frlm 21727  df-uvc 21763  df-lindf 21786  df-linds 21787  df-dim 33744

This theorem is referenced by:  extdgmul  33807