Theorem fedgmul 33600

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 20483	. . . . . . . . . . 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 17194	. . . . . . . 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 7379	. . . . . . 7 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
12		fedgmul.k	. . . . . . 7 ⊢ 𝐾 = (𝐸 ↾s 𝑉)
13	10, 11, 12	3eqtr4g 2789	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = 𝐾)
14		fedgmul.3	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ DivRing)
15	13, 14	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing)
16		fedgmul.c	. . . . . 6 ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
17		eqid 2729	. . . . . 6 ⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉)
18	16, 17	sralvec 33554	. . . . 5 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
19	1, 15, 3, 18	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
20		eqid 2729	. . . . 5 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
21	20	lbsex 21051	. . . 4 ⊢ (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
22	19, 21	syl 17	. . 3 ⊢ (𝜑 → (LBasis‘𝐶) ≠ ∅)
23		n0 4312	. . 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 33554	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
28	25, 1, 2, 27	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ LVec)
29		eqid 2729	. . . . . . 7 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
30	29	lbsex 21051	. . . . . 6 ⊢ (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
31	28, 30	syl 17	. . . . 5 ⊢ (𝜑 → (LBasis‘𝐵) ≠ ∅)
32		n0 4312	. . . . 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 20621	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐶) = (Base‘𝐶)
40	39, 20	lbsss 20960	. . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
43	42	subrgss 20457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
44	2, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
45	4, 42	ressbas2 17184	. . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
49	48	subrgss 20457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
50	3, 49	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
51	47, 50	srabase 21060	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
52	46, 51	eqtrd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
53	52, 44	eqsstrrd 3979	. . . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐵) = (Base‘𝐵)
61	60, 29	lbsss 20960	. . . . . . . . . . . . . . . . 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 21060	. . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ (Base‘𝐸))
70		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (.r‘𝐸) = (.r‘𝐸)
71	42, 70	ringcl 20135	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
72	37, 58, 69, 71	syl3anc 1373	. . . . . . . . . . . 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 20457	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
77	75, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
78	74, 77	srabase 21060	. . . . . . . . . . . . 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 2830	. . . . . . . . . . 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 3178	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
83		oveq2 7377	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → (𝑡(.r‘𝐸)𝑤) = (𝑡(.r‘𝐸)𝑗))
84		oveq1 7376	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑖 → (𝑡(.r‘𝐸)𝑗) = (𝑖(.r‘𝐸)𝑗))
85	83, 84	cbvmpov 7464	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ (𝑖(.r‘𝐸)𝑗))
86	85	fmpo 8026	. . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
89		eqid 2729	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐵) = ( ·𝑠 ‘𝐵)
90		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐵) = (+g‘𝐵)
91		eqid 2729	. . . . . . . . . . . . . 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 21718	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
95	94, 59	sselid 3941	. . . . . . . . . . . . . . 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 21063	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
100	4, 99	eqtrid 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
101	100	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102	101, 51	eqtr3d 2766	. . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
112	39, 20, 111	islbs4 21717	. . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘𝐶) = (0g‘𝐶)
116	115	0nellinds 33314	. . . . . . . . . . . . . . . . . 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 3023	. . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
122	16, 1, 3	drgext0g 33558	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
123	121, 122	eqtr3d 2766	. . . . . . . . . . . . . . . 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 2999	. . . . . . . . . . . . . 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 7404	. . . . . . . . . . . . . . . . 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 7516	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 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 33559	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
131	130	oveqd 7386	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 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 7387	. . . . . . . . . . . . . . . . . . . 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 7404	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑢(.r‘𝐸)𝑣) ∈ V)
141	134, 137, 138, 139, 140	ovmpod 7521	. . . . . . . . . . . . . . . . . . 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 7386	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 33325	. . . . . . . . . . . . 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 3178	. . . . . . . . . 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 3178	. . . . . . . 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 7425	. . . . . . . 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 7707	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158		f1rnen 32526	. . . . . . 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 14289	. . . . . 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 32705	. . . . . 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 2764	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = ((♯‘𝑦) ·e (♯‘𝑥)))
165	73, 12	sralvec 33554	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
166	25, 14, 75, 165	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ LVec)
167	166	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168		lveclmod 20989	. . . . . . . . 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 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 7464	. . . . . . . . . 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 33599	. . . . . . . . 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 3125	. . . . . . 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 2729	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
188		eqid 2729	. . . . . . . . 9 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
189		eqid 2729	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
190		eqid 2729	. . . . . . . . 9 ⊢ (0g‘𝐴) = (0g‘𝐴)
191		eqid 2729	. . . . . . . . 9 ⊢ (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
192		eqid 2729	. . . . . . . . 9 ⊢ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥))) = (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))
193	187, 188, 189, 190, 191, 192	islindf4 21723	. . . . . . . 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 1375	. . . . . 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 3178	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
198	85	rnmposs 32571	. . . . . . . . . 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 3980	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ⊆ (Base‘𝐴))
202		eqid 2729	. . . . . . . . 9 ⊢ (LSpan‘𝐴) = (LSpan‘𝐴)
203	187, 202	lspssv 20865	. . . . . . . 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 8799	. . . . . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
217		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
218		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
219		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘𝐶)
220		lveclmod 20989	. . . . . . . . . . . . . . . . . . 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 33312	. . . . . . . . . . . . . . . 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 3125	. . . . . . . . . . . . 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 7387	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑖 → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
231	230	cbvmptv 5206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
232	231	oveq2i 7380	. . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . 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 3448	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
239		breq1 5105	. . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))
242	241	mpteq2dv 5196	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))
243	242	oveq2d 7385	. . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 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 10413	. . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . . . . . . . . . . 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 8799	. . . . . . . . . . . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . . . . . . . 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 21063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
258	12, 257	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
259	47, 50	srasca 21063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
260	13, 259	eqtr3d 2766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
261	258, 260	eqtr3d 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262	261	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . . 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 8026	. . . . . . . . . . . . . . . . . . 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 8790	. . . . . . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 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 7387	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤) = ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
295	294	cbvmptv 5206	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)) = (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
296	295	oveq2i 7380	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))) = (𝐵 Σg (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗)))
297	292, 296	eqtrdi 2780	. . . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . . . . . . 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 7387	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑖 → (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
303	302	cbvmptv 5206	. . . . . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑗 → (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
306	305	mpteq2dv 5196	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑗 → (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
307	303, 306	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑗 → (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
308	307	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . . . . . 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 3227	. . . . . . . . . . . . . . . 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 33598	. . . . . . . . . . . . . 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 3587	. . . . . . . . . . . . 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 1935	. . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . 17 ⊢ (LSpan‘𝐵) = (LSpan‘𝐵)
322	60, 29, 321	islbs4 21717	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐴))
329	288, 328	eleqtrrd 2831	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐵)‘𝑦))
330		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐵) = (Scalar‘𝐵)
331		lveclmod 20989	. . . . . . . . . . . . . . 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 33312	. . . . . . . . . . . 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 3141	. . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
339	202, 187, 338, 188, 191, 189, 87, 170, 157	ellspd 21687	. . . . . . . . . 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 6671	. . . . . . . . . . 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 6630	. . . . . . . . . 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 2830	. . . . . . 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 3965	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘𝐴))
349		eqid 2729	. . . . . . 7 ⊢ (Base‘(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))
350		drngnzr 20633	. . . . . . . . . 10 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
351	14, 350	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ NzRing)
352	258, 351	eqeltrrd 2829	. . . . . . . 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 33323	. . . . . 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 2729	. . . . . 6 ⊢ (LBasis‘𝐴) = (LBasis‘𝐴)
357	356	dimval 33569	. . . . 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 33569	. . . . . 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 33569	. . . . . 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 7387	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((dim‘𝐵) ·e (dim‘𝐶)) = ((♯‘𝑦) ·e (♯‘𝑥)))
364	164, 358, 363	3eqtr4d 2774	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
365	34, 364	exlimddv 1935	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
366	24, 365	exlimddv 1935	1 ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ⊆ wss 3911  ∅c0 4292  {csn 4585   class class class wbr 5102   ↦ cmpt 5183   × cxp 5629  ran crn 5632   “ cima 5634   Fn wfn 6494  ⟶wf 6495  –1-1→wf1 6496  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371   ∘_f cof 7631   ↑_m cmap 8776   ≈ cen 8892   finSupp cfsupp 9288   ·_e cxmu 13047  ♯chash 14271  Basecbs 17155   ↾_s cress 17176  +_gcplusg 17196  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  0_gc0g 17378   Σ_g cgsu 17379  Ringcrg 20118  NzRingcnzr 20397  SubRingcsubrg 20454  DivRingcdr 20614  LModclmod 20742  LSpanclspn 20853  LBasisclbs 20957  LVecclvec 20985  subringAlg csra 21054   freeLMod cfrlm 21631   LIndF clindf 21689  LIndSclinds 21690  dimcldim 33567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570  ax-ac2 10392  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  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 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-rpss 7679  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-oi 9439  df-r1 9693  df-rank 9694  df-dju 9830  df-card 9868  df-acn 9871  df-ac 10045  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-xmul 13050  df-fz 13445  df-fzo 13592  df-seq 13943  df-hash 14272  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ocomp 17217  df-ds 17218  df-hom 17220  df-cco 17221  df-0g 17380  df-gsum 17381  df-prds 17386  df-pws 17388  df-mre 17523  df-mrc 17524  df-mri 17525  df-acs 17526  df-proset 18231  df-drs 18232  df-poset 18250  df-ipo 18463  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19121  df-cntz 19225  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-nzr 20398  df-subrng 20431  df-subrg 20455  df-drng 20616  df-lmod 20744  df-lss 20814  df-lsp 20854  df-lmhm 20905  df-lbs 20958  df-lvec 20986  df-sra 21056  df-rgmod 21057  df-dsmm 21617  df-frlm 21632  df-uvc 21668  df-lindf 21691  df-linds 21692  df-dim 33568

This theorem is referenced by:  extdgmul  33632