Theorem fedgmul 31614

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 19965	. . . . . . . . . . 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 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
8	7	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
9		ressabs 16885	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
10	2, 8, 9	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
11	4	oveq1i 7265	. . . . . . 7 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
12		fedgmul.k	. . . . . . 7 ⊢ 𝐾 = (𝐸 ↾s 𝑉)
13	10, 11, 12	3eqtr4g 2804	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = 𝐾)
14		fedgmul.3	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ DivRing)
15	13, 14	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing)
16		fedgmul.c	. . . . . 6 ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
17		eqid 2738	. . . . . 6 ⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉)
18	16, 17	sralvec 31577	. . . . 5 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
19	1, 15, 3, 18	syl3anc 1369	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
20		eqid 2738	. . . . 5 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
21	20	lbsex 20342	. . . 4 ⊢ (𝐶 ∈ LVec → (LBasis‘𝐶) ≠ ∅)
22	19, 21	syl 17	. . 3 ⊢ (𝜑 → (LBasis‘𝐶) ≠ ∅)
23		n0 4277	. . 3 ⊢ ((LBasis‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (LBasis‘𝐶))
24	22, 23	sylib 217	. 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 31577	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
28	25, 1, 2, 27	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ LVec)
29		eqid 2738	. . . . . . 7 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
30	29	lbsex 20342	. . . . . 6 ⊢ (𝐵 ∈ LVec → (LBasis‘𝐵) ≠ ∅)
31	28, 30	syl 17	. . . . 5 ⊢ (𝜑 → (LBasis‘𝐵) ≠ ∅)
32		n0 4277	. . . . 5 ⊢ ((LBasis‘𝐵) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
33	31, 32	sylib 217	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) → ∃𝑦 𝑦 ∈ (LBasis‘𝐵))
35		drngring 19913	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
36	25, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
37	36	ad4antr 728	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝐸 ∈ Ring)
38		simplr 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ∈ (LBasis‘𝐶))
39		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐶) = (Base‘𝐶)
40	39, 20	lbsss 20254	. . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
43	42	subrgss 19940	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
44	2, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
45	4, 42	ressbas2 16875	. . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
49	48	subrgss 19940	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
50	3, 49	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
51	47, 50	srabase 20356	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
52	46, 51	eqtrd 2778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
53	52, 44	eqsstrrd 3956	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
54	53	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐶) ⊆ (Base‘𝐸))
55	41, 54	sstrd 3927	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘𝐸))
56	55	ad2antrr 722	. . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐵) = (Base‘𝐵)
61	60, 29	lbsss 20254	. . . . . . . . . . . . . . . . 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 20356	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
65	64	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐵))
66	62, 65	sseqtrrd 3958	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ⊆ (Base‘𝐸))
67	66	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑦 ⊆ (Base‘𝐸))
68		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ 𝑦)
69	67, 68	sseldd 3918	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ (Base‘𝐸))
70		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (.r‘𝐸) = (.r‘𝐸)
71	42, 70	ringcl 19715	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
72	37, 58, 69, 71	syl3anc 1369	. . . . . . . . . . . 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 19940	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
77	75, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
78	74, 77	srabase 20356	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
79	78	ad4antr 728	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → (Base‘𝐸) = (Base‘𝐴))
80	72, 79	eleqtrd 2841	. . . . . . . . . . 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 3114	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
83		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → (𝑡(.r‘𝐸)𝑤) = (𝑡(.r‘𝐸)𝑗))
84		oveq1 7262	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑖 → (𝑡(.r‘𝐸)𝑗) = (𝑖(.r‘𝐸)𝑗))
85	83, 84	cbvmpov 7348	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ (𝑖(.r‘𝐸)𝑗))
86	85	fmpo 7881	. . . . . . . . 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 217	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)):(𝑦 × 𝑥)⟶(Base‘𝐴))
88		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
89		eqid 2738	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐵) = ( ·𝑠 ‘𝐵)
90		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐵) = (+g‘𝐵)
91		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (0g‘(Scalar‘𝐵)) = (0g‘(Scalar‘𝐵))
92	28	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LVec)
93	92	ad5antr 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)) → 𝐵 ∈ LVec)
94	29	lbslinds 20950	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
95	94, 59	sselid 3915	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑦 ∈ (LIndS‘𝐵))
96	95	ad5antr 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)) → 𝑦 ∈ (LIndS‘𝐵))
97	68	ad3antrrr 726	. . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . 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 20362	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
100	4, 99	syl5eq 2791	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
101	100	fveq2d 6760	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
102	101, 51	eqtr3d 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
103	102	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
104	41, 103	sseqtrrd 3958	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝑥 ⊆ (Base‘(Scalar‘𝐵)))
105	104	ad5antr 730	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LVec)
111		eqid 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
112	39, 20, 111	islbs4 20949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (LBasis‘𝐶) ↔ (𝑥 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑥) = (Base‘𝐶)))
113	38, 112	sylib 217	. . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘𝐶) = (0g‘𝐶)
116	115	0nellinds 31468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LIndS‘𝐶)) → ¬ (0g‘𝐶) ∈ 𝑥)
117	110, 114, 116	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ¬ (0g‘𝐶) ∈ 𝑥)
118	117	ad5antr 730	. . . . . . . . . . . . . . . 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 3041	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ 𝑥 ∧ ¬ (0g‘𝐶) ∈ 𝑥) → 𝑖 ≠ (0g‘𝐶))
120	106, 118, 119	syl2anc 583	. . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
122	16, 1, 3	drgext0g 31579	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
123	121, 122	eqtr3d 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
124	123	ad7antr 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)) → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
125	120, 124	neeqtrrd 3017	. . . . . . . . . . . . . 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 7290	. . . . . . . . . . . . . . . . 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 7398	. . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . 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 31580	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
131	130	oveqd 7272	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑖(.r‘𝐸)𝑗) = (𝑖( ·𝑠 ‘𝐵)𝑗))
132	131	ad7antr 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)) → (𝑖(.r‘𝐸)𝑗) = (𝑖( ·𝑠 ‘𝐵)𝑗))
133	129, 132	eqtrd 2778	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)) → 𝑖 = 𝑢)
136		simprl 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) ∧ (𝑗 = 𝑣 ∧ 𝑖 = 𝑢)) → 𝑗 = 𝑣)
137	135, 136	oveq12d 7273	. . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . 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 7290	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑣 ∈ 𝑦) ∧ 𝑢 ∈ 𝑥) → (𝑢(.r‘𝐸)𝑣) ∈ V)
141	134, 137, 138, 139, 140	ovmpod 7403	. . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 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 746	. . . . . . . . . . . . . . . . 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 7272	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑢(.r‘𝐸)𝑣) = (𝑢( ·𝑠 ‘𝐵)𝑣))
146	145	ad7antr 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)) → (𝑢(.r‘𝐸)𝑣) = (𝑢( ·𝑠 ‘𝐵)𝑣))
147	144, 146	eqtrd 2778	. . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . 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 31477	. . . . . . . . . . . . 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 3114	. . . . . . . . . 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 3114	. . . . . . . 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 7309	. . . . . . . 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 582	. . . . . . 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 7579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (𝑦 × 𝑥) ∈ V)
158		f1rnen 30865	. . . . . . 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 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ≈ (𝑦 × 𝑥))
160		hasheni 13990	. . . . . 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 31029	. . . . . 6 ⊢ ((𝑦 ∈ (LBasis‘𝐵) ∧ 𝑥 ∈ (LBasis‘𝐶)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
163	59, 38, 162	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘(𝑦 × 𝑥)) = ((♯‘𝑦) ·e (♯‘𝑥)))
164	161, 163	eqtrd 2778	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = ((♯‘𝑦) ·e (♯‘𝑥)))
165	73, 12	sralvec 31577	. . . . . . 7 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
166	25, 14, 75, 165	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ LVec)
167	166	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LVec)
168		lveclmod 20283	. . . . . . . . 9 ⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
169	166, 168	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ LMod)
170	169	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐴 ∈ LMod)
171	25	ad4antr 728	. . . . . . . . . 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 728	. . . . . . . . . 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 728	. . . . . . . . . 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 728	. . . . . . . . . 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 728	. . . . . . . . . 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 6756	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑗 → (𝑓‘𝑤) = (𝑓‘𝑗))
177	176	fveq1d 6758	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑗)‘𝑣))
178		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑗)‘𝑣) = ((𝑓‘𝑗)‘𝑖))
179	177, 178	cbvmpov 7348	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦, 𝑣 ∈ 𝑥 ↦ ((𝑓‘𝑤)‘𝑣)) = (𝑗 ∈ 𝑦, 𝑖 ∈ 𝑥 ↦ ((𝑓‘𝑗)‘𝑖))
180		simp-4r 780	. . . . . . . . . 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 772	. . . . . . . . . 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 765	. . . . . . . . . 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 31613	. . . . . . . . 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 3107	. . . . . . 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 2738	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
188		eqid 2738	. . . . . . . . 9 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
189		eqid 2738	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
190		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝐴) = (0g‘𝐴)
191		eqid 2738	. . . . . . . . 9 ⊢ (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
192		eqid 2738	. . . . . . . . 9 ⊢ (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥))) = (Base‘((Scalar‘𝐴) freeLMod (𝑦 × 𝑥)))
193	187, 188, 189, 190, 191, 192	islindf4 20955	. . . . . . . 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 1371	. . . . . 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 3114	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ∀𝑗 ∈ 𝑦 ∀𝑖 ∈ 𝑥 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
198	85	rnmposs 30913	. . . . . . . . . 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 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Base‘𝐸) = (Base‘𝐴))
201	199, 200	sseqtrd 3957	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ⊆ (Base‘𝐴))
202		eqid 2738	. . . . . . . . 9 ⊢ (LSpan‘𝐴) = (LSpan‘𝐴)
203	187, 202	lspssv 20160	. . . . . . . 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 583	. . . . . . 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 728	. . . . . . . . . . . . . . 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 8595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦) → 𝑎:𝑦⟶(Base‘(Scalar‘𝐵)))
208	207	ad4antlr 729	. . . . . . . . . . . . . . . . 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	ffvelrnd 6944	. . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑦)) ∧ 𝑎 finSupp (0g‘(Scalar‘𝐵))) ∧ 𝑧 = (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)))) ∧ 𝑗 ∈ 𝑦) → (Base‘(Scalar‘𝐵)) = (Base‘𝐶))
214	212, 213	eqtr4d 2781	. . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
217		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
218		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
219		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘𝐶)
220		lveclmod 20283	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
221	19, 220	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ LMod)
222	221	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐶 ∈ LMod)
223	111, 39, 216, 217, 218, 219, 222, 41	ellspds 31466	. . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 3107	. . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → (𝑎‘𝑤) = (𝑎‘𝑗))
228		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → (𝑏‘𝑣) = (𝑏‘𝑖))
229		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑖 → 𝑣 = 𝑖)
230	228, 229	oveq12d 7273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑖 → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
231	230	cbvmptv 5183	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ ((𝑏‘𝑖)( ·𝑠 ‘𝐶)𝑖))
232	231	oveq2i 7266	. . . . . . . . . . . . . . . . . . 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 2754	. . . . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . . . 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 3372	. . . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
239		breq1 5073	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶))))
240		fveq1 6755	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑏‘𝑣) = ((𝑓‘𝑤)‘𝑣))
241	240	oveq1d 7270	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = (𝑓‘𝑤) → ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣))
242	241	mpteq2dv 5172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘𝑤) → (𝑣 ∈ 𝑥 ↦ ((𝑏‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)))
243	242	oveq2d 7271	. . . . . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . 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 10171	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ 𝑗 ∈ 𝑦) ∧ 𝑖 ∈ 𝑥) → 𝑗 ∈ 𝑦)
251	249, 250	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . . . . . . 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 8595	. . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑓:𝑦⟶((Base‘(Scalar‘𝐶)) ↑m 𝑥)) ∧ (𝑗 ∈ 𝑦 ∧ 𝑖 ∈ 𝑥)) → 𝑖 ∈ 𝑥)
256	254, 255	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . . . 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 20362	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
258	12, 257	syl5eq 2791	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
259	47, 50	srasca 20362	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
260	13, 259	eqtr3d 2780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
261	258, 260	eqtr3d 2780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
262	261	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
263	262	ad4antr 728	. . . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . . . 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 3114	. . . . . . . . . . . . . . . . . . 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 7881	. . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . 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 6771	. . . . . . . . . . . . . . . . . . 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 8587	. . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . 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 755	. . . . . . . . . . . . . . . 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 746	. . . . . . . . . . . . . 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 5080	. . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . 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 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))))) → 𝐸 ∈ DivRing)
282	1	ad8antr 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))))) → 𝐹 ∈ DivRing)
283	14	ad8antr 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))))) → 𝐾 ∈ DivRing)
284	2	ad8antr 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))))) → 𝑈 ∈ (SubRing‘𝐸))
285	3	ad8antr 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))))) → 𝑉 ∈ (SubRing‘𝐹))
286	38	ad6antr 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)))) ∧ 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 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)))) ∧ 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 730	. . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 7273	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑗 → ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤) = ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
295	294	cbvmptv 5183	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤)) = (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗))
296	295	oveq2i 7266	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 Σg (𝑤 ∈ 𝑦 ↦ ((𝑎‘𝑤)( ·𝑠 ‘𝐵)𝑤))) = (𝐵 Σg (𝑗 ∈ 𝑦 ↦ ((𝑎‘𝑗)( ·𝑠 ‘𝐵)𝑗)))
297	292, 296	eqtrdi 2795	. . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . 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 5080	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝑓‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
301		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = 𝑖 → ((𝑓‘𝑤)‘𝑣) = ((𝑓‘𝑤)‘𝑖))
302	301, 229	oveq12d 7273	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑖 → (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣) = (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
303	302	cbvmptv 5183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
304	176	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑗 → ((𝑓‘𝑤)‘𝑖) = ((𝑓‘𝑗)‘𝑖))
305	304	oveq1d 7270	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑗 → (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
306	305	mpteq2dv 5172	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑗 → (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
307	303, 306	syl5eq 2791	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑗 → (𝑣 ∈ 𝑥 ↦ (((𝑓‘𝑤)‘𝑣)( ·𝑠 ‘𝐶)𝑣)) = (𝑖 ∈ 𝑥 ↦ (((𝑓‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
308	307	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . 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 2754	. . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . 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 3372	. . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . . . . . 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 31612	. . . . . . . . . . . . . 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 3555	. . . . . . . . . . . . 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 1939	. . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ (LSpan‘𝐵) = (LSpan‘𝐵)
322	60, 29, 321	islbs4 20949	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (LBasis‘𝐵) ↔ (𝑦 ∈ (LIndS‘𝐵) ∧ ((LSpan‘𝐵)‘𝑦) = (Base‘𝐵)))
323	59, 322	sylib 217	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
327	326	ad3antrrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
328	325, 327	eqtr4d 2781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → ((LSpan‘𝐵)‘𝑦) = (Base‘𝐴))
329	288, 328	eleqtrrd 2842	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) ∧ 𝑧 ∈ (Base‘𝐴)) → 𝑧 ∈ ((LSpan‘𝐵)‘𝑦))
330		eqid 2738	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐵) = (Scalar‘𝐵)
331		lveclmod 20283	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ LVec → 𝐵 ∈ LMod)
332	28, 331	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ LMod)
333	332	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → 𝐵 ∈ LMod)
334	321, 60, 88, 330, 91, 89, 333, 62	ellspds 31466	. . . . . . . . . . . 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 583	. . . . . . . . . 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 3217	. . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
339	202, 187, 338, 188, 191, 189, 87, 170, 157	ellspd 20919	. . . . . . . . . 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 256	. . . . . . . 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 6585	. . . . . . . . . . 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 6547	. . . . . . . . . 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 6760	. . . . . . . 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 2841	. . . . . . 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 3938	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((LSpan‘𝐴)‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘𝐴))
349		eqid 2738	. . . . . . 7 ⊢ (Base‘(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))) = (Base‘(𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)))
350		drngnzr 20446	. . . . . . . . . 10 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
351	14, 350	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ NzRing)
352	258, 351	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → (Scalar‘𝐴) ∈ NzRing)
353	352	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (Scalar‘𝐴) ∈ NzRing)
354	187, 349, 188, 189, 190, 191, 202, 170, 353, 157, 156	lindflbs 31476	. . . . . 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 709	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤)) ∈ (LBasis‘𝐴))
356		eqid 2738	. . . . . 6 ⊢ (LBasis‘𝐴) = (LBasis‘𝐴)
357	356	dimval 31588	. . . . 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 583	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = (♯‘ran (𝑤 ∈ 𝑦, 𝑡 ∈ 𝑥 ↦ (𝑡(.r‘𝐸)𝑤))))
359	29	dimval 31588	. . . . . 6 ⊢ ((𝐵 ∈ LVec ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
360	92, 59, 359	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐵) = (♯‘𝑦))
361	20	dimval 31588	. . . . . 6 ⊢ ((𝐶 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐶) = (♯‘𝑥))
362	110, 38, 361	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐶) = (♯‘𝑥))
363	360, 362	oveq12d 7273	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → ((dim‘𝐵) ·e (dim‘𝐶)) = ((♯‘𝑦) ·e (♯‘𝑥)))
364	164, 358, 363	3eqtr4d 2788	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) ∧ 𝑦 ∈ (LBasis‘𝐵)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
365	34, 364	exlimddv 1939	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐶)) → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
366	24, 365	exlimddv 1939	1 ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ran crn 5581   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ∘_f cof 7509   ↑_m cmap 8573   ≈ cen 8688   finSupp cfsupp 9058   ·_e cxmu 12776  ♯chash 13972  Basecbs 16840   ↾_s cress 16867  +_gcplusg 16888  ._rcmulr 16889  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067   Σ_g cgsu 17068  Ringcrg 19698  DivRingcdr 19906  SubRingcsubrg 19935  LModclmod 20038  LSpanclspn 20148  LBasisclbs 20251  LVecclvec 20279  subringAlg csra 20345  NzRingcnzr 20441   freeLMod cfrlm 20863   LIndF clindf 20921  LIndSclinds 20922  dimcldim 31586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329  ax-ac2 10150  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-rpss 7554  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-oi 9199  df-r1 9453  df-rank 9454  df-dju 9590  df-card 9628  df-acn 9631  df-ac 9803  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-dec 12367  df-uz 12512  df-xmul 12779  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ocomp 16909  df-ds 16910  df-hom 16912  df-cco 16913  df-0g 17069  df-gsum 17070  df-prds 17075  df-pws 17077  df-mre 17212  df-mrc 17213  df-mri 17214  df-acs 17215  df-proset 17928  df-drs 17929  df-poset 17946  df-ipo 18161  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-drng 19908  df-subrg 19937  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lmhm 20199  df-lbs 20252  df-lvec 20280  df-sra 20349  df-rgmod 20350  df-nzr 20442  df-dsmm 20849  df-frlm 20864  df-uvc 20900  df-lindf 20923  df-linds 20924  df-dim 31587

This theorem is referenced by:  extdgmul  31638