Theorem fedgmul 33788

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 20531	. . . . . . . . . . 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 17175	. . . . . . . 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 7368	. . . . . . 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 33741	. . . . 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 2736	. . . . 5 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
21	20	lbsex 21120	. . . 4 ⊢ (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
22	19, 21	syl 17	. . 3 ⊢ (𝜑 → (LBasis‘𝐶) ≠ ∅)
23		n0 4305	. . 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 33741	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
28	25, 1, 2, 27	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ LVec)
29		eqid 2736	. . . . . . 7 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
30	29	lbsex 21120	. . . . . 6 ⊢ (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
31	28, 30	syl 17	. . . . 5 ⊢ (𝜑 → (LBasis‘𝐵) ≠ ∅)
32		n0 4305	. . . . 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 20669	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
36	25, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
37	36	ad4antr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝐸 ∈ Ring)
38		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐶) = (Base‘𝐶)
40	39, 20	lbsss 21029	. . . . . . . . . . . . . . . . 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 20505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
44	2, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
45	4, 42	ressbas2 17165	. . . . . . . . . . . . . . . . . . . 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 20505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
50	3, 49	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
51	47, 50	srabase 21129	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
52	46, 51	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
53	52, 44	eqsstrrd 3969	. . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . 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 21029	. . . . . . . . . . . . . . . . 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 21129	. . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . 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 20185	. . . . . . . . . . . . 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 20505	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
77	75, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
78	74, 77	srabase 21129	. . . . . . . . . . . . 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 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 3179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
83		oveq2 7366	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → (𝑡(.r‘𝐸)𝑤) = (𝑡(.r‘𝐸)𝑗))
84		oveq1 7365	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑖 → (𝑡(.r‘𝐸)𝑗) = (𝑖(.r‘𝐸)𝑗))
85	83, 84	cbvmpov 7453	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ (𝑖(.r‘𝐸)𝑗))
86	85	fmpo 8012	. . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
87	82, 86	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
88		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
89		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐵) = ( ·𝑠 ‘𝐵)
90		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐵) = (+g‘𝐵)
91		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0g‘(Scalar‘𝐵)) = (0g‘(Scalar‘𝐵))
92	28	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LVec)
93	92	ad5antr 734	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝐵 ∈ LVec)
94	29	lbslinds 21788	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
95	94, 59	sselid 3931	. . . . . . . . . . . . . . 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 21132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
100	4, 99	eqtrid 2783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
101	100	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102	101, 51	eqtr3d 2773	. . . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑖) = (𝑣(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))𝑢)) → 𝑢 ∈ (Base‘(Scalar‘𝐵)))
110	19	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LVec)
111		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
112	39, 20, 111	islbs4 21787	. . . . . . . . . . . . . . . . . . . 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 33451	. . . . . . . . . . . . . . . . . 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 3030	. . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
122	16, 1, 3	drgext0g 33746	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
123	121, 122	eqtr3d 2773	. . . . . . . . . . . . . . . 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 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 7393	. . . . . . . . . . . . . . . . 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 7505	. . . . . . . . . . . . . . . . 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 33747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
131	130	oveqd 7375	. . . . . . . . . . . . . . . . 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 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 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 7376	. . . . . . . . . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑢(.r‘𝐸)𝑣) ∈ V)
141	134, 137, 138, 139, 140	ovmpod 7510	. . . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . 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 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 33462	. . . . . . . . . . . . 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 3179	. . . . . . . . . 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 3179	. . . . . . . 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 7414	. . . . . . . 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 7696	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158		f1rnen 32706	. . . . . . 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 14271	. . . . . 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 32887	. . . . . 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 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 33741	. . . . . . 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 21058	. . . . . . . . 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 6834	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑗 → (𝑓‘𝑤) = (𝑓‘𝑗))
177	176	fveq1d 6836	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑗)‘𝑣))
178		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑗)‘𝑣) = ((𝑓‘𝑗)‘𝑖))
179	177, 178	cbvmpov 7453	. . . . . . . . . 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 33787	. . . . . . . . 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 3128	. . . . . . 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 21793	. . . . . . . 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 3179	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
198	85	rnmposs 32752	. . . . . . . . . 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 3970	. . . . . . . 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 20934	. . . . . . . 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 8786	. . . . . . . . . . . . . . . . . 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 7030	. . . . . . . . . . . . . . . 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 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 21058	. . . . . . . . . . . . . . . . . . 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 33449	. . . . . . . . . . . . . . . 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 3128	. . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → (𝑎‘𝑤) = (𝑎‘𝑗))
228		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → (𝑏‘𝑣) = (𝑏‘𝑖))
229		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → 𝑣 = 𝑖)
230	228, 229	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑖 → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
231	230	cbvmptv 5202	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
232	231	oveq2i 7369	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
233	232	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
234	227, 233	eqeq12d 2752	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) ↔ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
235	234	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑗 → ((𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ (𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))))))
236	235	rexbidv 3160	. . . . . . . . . . . . . . 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 3214	. . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
239		breq1 5101	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶))))
240		fveq1 6833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏‘𝑣) = ((𝑓‘𝑤)‘𝑣))
241	240	oveq1d 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))
242	241	mpteq2dv 5192	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))
243	242	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑓‘𝑤) → (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))
244	243	eqeq2d 2747	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣))) ↔ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))))
245	239, 244	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑏 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))))
246	238, 245	ac6s 10394	. . . . . . . . . . . . . 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 7030	. . . . . . . . . . . . . . . . . . . . . . . 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 8786	. . . . . . . . . . . . . . . . . . . . . . . 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 7030	. . . . . . . . . . . . . . . . . . . . 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 21132	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
258	12, 257	eqtrid 2783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
259	47, 50	srasca 21132	. . . . . . . . . . . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . . . . . . 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 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 3179	. . . . . . . . . . . . . . . . . . 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 8012	. . . . . . . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . . . . . . . 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 8777	. . . . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) = (𝐴 Σg ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))))
279	278	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → (𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))) ↔ 𝑧 = (𝐴 Σg ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))))
280	276, 279	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) ∧ 𝑐 = (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣))) → ((𝑐 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg (𝑐 ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))) ↔ ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑧 = (𝐴 Σg ((𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) ∘f ( ·𝑠 ‘𝐴)(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))))))
281	25	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝐸 ∈ DivRing)
282	1	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝐹 ∈ DivRing)
283	14	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝐾 ∈ DivRing)
284	2	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑈 ∈ (SubRing‘𝐸))
285	3	ad8antr 740	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑉 ∈ (SubRing‘𝐹))
286	38	ad6antr 736	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑥 ∈ (LBasis‘𝐶))
287	59	ad6antr 736	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑦 ∈ (LBasis‘𝐵))
288		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ (Base‘𝐴))
289	288	ad5antr 734	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑧 ∈ (Base‘𝐴))
290	207	ad5antlr 735	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
291		simp-4r 783	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑎 finSupp (0g‘(Scalar‘𝐵)))
292		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ ∀𝑤 ∈ 𝑦 ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))))) → 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))))
293		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑗 → 𝑤 = 𝑗)
294	227, 293	oveq12d 7376	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤) = ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
295	294	cbvmptv 5202	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)) = (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
296	295	oveq2i 7369	. . . . . . . . . . . . . . . 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 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 5108	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
301		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑤)‘𝑖))
302	301, 229	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑖 → (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
303	302	cbvmptv 5202	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
304	176	fveq1d 6836	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑖) = ((𝑓‘𝑗)‘𝑖))
305	304	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑗 → (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
306	305	mpteq2dv 5192	. . . . . . . . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑗 → (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
309	227, 308	eqeq12d 2752	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))) ↔ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))))
310	300, 309	anbi12d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑗 → (((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑤) = (𝐶 Σg (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))) ↔ ((𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝑎‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))))
311	310	cbvralvw 3214	. . . . . . . . . . . . . . . . . 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 3228	. . . . . . . . . . . . . . . 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 33786	. . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . 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 1936	. . . . . . . . . . 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 21787	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 21058	. . . . . . . . . . . . . . 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 33449	. . . . . . . . . . . 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 3144	. . . . . . . . 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 21757	. . . . . . . . . 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 6663	. . . . . . . . . . 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 6622	. . . . . . . . . 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 6838	. . . . . . . 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 3955	. . . . . 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 20681	. . . . . . . . . 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 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 33460	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ∈ (LBasis‘𝐴) ↔ ((𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) LIndF 𝐴 ∧ ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘𝐴))))
355	195, 348, 354	mpbir2and 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ∈ (LBasis‘𝐴))
356		eqid 2736	. . . . . 6 ⊢ (LBasis‘𝐴) = (LBasis‘𝐴)
357	356	dimval 33757	. . . . 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 33757	. . . . . 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 33757	. . . . . 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 7376	. . . 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 1936	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
366	24, 365	exlimddv 1936	1 ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  {csn 4580   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  ran crn 5625   “ cima 5627   Fn wfn 6487  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   ∘_f cof 7620   ↑_m cmap 8763   ≈ cen 8880   finSupp cfsupp 9264   ·_e cxmu 13025  ♯chash 14253  Basecbs 17136   ↾_s cress 17157  +_gcplusg 17177  ._rcmulr 17178  Scalarcsca 17180   ·_𝑠 cvsca 17181  0_gc0g 17359   Σ_g cgsu 17360  Ringcrg 20168  NzRingcnzr 20445  SubRingcsubrg 20502  DivRingcdr 20662  LModclmod 20811  LSpanclspn 20922  LBasisclbs 21026  LVecclvec 21054  subringAlg csra 21123   freeLMod cfrlm 21701   LIndF clindf 21759  LIndSclinds 21760  dimcldim 33755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550  ax-ac2 10373  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-rpss 7668  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-r1 9676  df-rank 9677  df-dju 9813  df-card 9851  df-acn 9854  df-ac 10026  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-xmul 13028  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ocomp 17198  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-mri 17507  df-acs 17508  df-proset 18217  df-drs 18218  df-poset 18236  df-ipo 18451  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-ring 20170  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-invr 20324  df-nzr 20446  df-subrng 20479  df-subrg 20503  df-drng 20664  df-lmod 20813  df-lss 20883  df-lsp 20923  df-lmhm 20974  df-lbs 21027  df-lvec 21055  df-sra 21125  df-rgmod 21126  df-dsmm 21687  df-frlm 21702  df-uvc 21738  df-lindf 21761  df-linds 21762  df-dim 33756

This theorem is referenced by:  extdgmul  33820