fedgmullem2 - Mathbox for Thierry Arnoux

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Theorem fedgmullem2 33314

Description: Lemma for fedgmul 33315. (Contributed by Thierry Arnoux, 20-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))
fedgmullem.d	⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(._r‘𝐸)𝑗))
fedgmullem.h	⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))
fedgmullem.x	⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))
fedgmullem.y	⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵))
fedgmullem2.1	⊢ (𝜑 → 𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))
fedgmullem2.2	⊢ (𝜑 → (𝐴 Σ_g (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)) = (0_g‘𝐴))

**Assertion**
Ref	Expression
fedgmullem2	⊢ (𝜑 → 𝑊 = ((𝑌 × 𝑋) × {(0_g‘(Scalar‘𝐴))}))

Distinct variable groups: 𝐴,𝑖,𝑗 𝐵,𝑖,𝑗 𝐶,𝑖 𝐷,𝑖,𝑗 𝑖,𝐸,𝑗 𝑈,𝑖 𝑖,𝑊,𝑗 𝑖,𝑋,𝑗 𝑖,𝑌,𝑗 𝜑,𝑖,𝑗 Allowed substitution hints: 𝐶(𝑗) 𝑈(𝑗) 𝐹(𝑖,𝑗) 𝐺(𝑖,𝑗) 𝐻(𝑖,𝑗) 𝐾(𝑖,𝑗) 𝑉(𝑖,𝑗)

Proof of Theorem fedgmullem2 Dummy variables 𝑏 𝑤 𝑘 𝑥 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fedgmul.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ DivRing)
2		fedgmul.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ DivRing)
3		fedgmul.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))
4		fedgmul.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))
5		fedgmul.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝐸 ↾_s 𝑈)
6	5	subsubrg 20530	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈)))
7	6	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
8	3, 4, 7	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
9	8	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐸))
10		fedgmul.a	. . . . . . . . . . . 12 ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
11		fedgmul.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (𝐸 ↾_s 𝑉)
12	10, 11	sralvec 33271	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
13	1, 2, 9, 12	syl3anc 1369	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ LVec)
14		lveclmod 20984	. . . . . . . . . 10 ⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ LMod)
16		fedgmullem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))
17		eqid 2727	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		eqid 2727	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
19	17, 18	lbsss 20955	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
20	16, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶))
21		eqid 2727	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
22	21	subrgss 20504	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
23	3, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
24	5, 21	ressbas2 17211	. . . . . . . . . . . . . 14 ⊢ (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
25	23, 24	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 = (Base‘𝐹))
26		fedgmul.c	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 = ((subringAlg ‘𝐹)‘𝑉))
28		eqid 2727	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
29	28	subrgss 20504	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
30	4, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
31	27, 30	srabase 21056	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
32	25, 31	eqtrd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
33	32, 23	eqsstrrd 4017	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
34	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝐸)‘𝑉))
35	21	subrgss 20504	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
36	9, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
37	34, 36	srabase 21056	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
38	33, 37	sseqtrd 4018	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐴))
39	20, 38	sstrd 3988	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐴))
40	34, 3, 36	srasubrg 33270	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐴))
41		subrgsubg 20509	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubRing‘𝐴) → 𝑈 ∈ (SubGrp‘𝐴))
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐴))
43	10, 1, 9	drgextvsca 33276	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
44	43	oveqdr 7442	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → (𝑥(._r‘𝐸)𝑦) = (𝑥( ·_𝑠 ‘𝐴)𝑦))
45	3	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ (SubRing‘𝐸))
46	8	simprd 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
47	46	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑉 ⊆ 𝑈)
48		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
49		ressabs 17223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾_s 𝑈) ↾_s 𝑉) = (𝐸 ↾_s 𝑉))
50	3, 46, 49	syl2anc 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐸 ↾_s 𝑈) ↾_s 𝑉) = (𝐸 ↾_s 𝑉))
51	5	oveq1i 7424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 ↾_s 𝑉) = ((𝐸 ↾_s 𝑈) ↾_s 𝑉)
52	50, 51, 11	3eqtr4g 2792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = 𝐾)
53	27, 30	srasca 21062	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = (Scalar‘𝐶))
54	52, 53	eqtr3d 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
55	54	fveq2d 6895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
56	11, 21	ressbas2 17211	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
57	36, 56	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
58	34, 36	srasca 21062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑉) = (Scalar‘𝐴))
59	11, 58	eqtrid 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
60	52, 53, 59	3eqtr3rd 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
61	60	fveq2d 6895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
62	55, 57, 61	3eqtr4d 2777	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐴)))
63	62	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑉 = (Base‘(Scalar‘𝐴)))
64	48, 63	eleqtrrd 2831	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑉)
65	47, 64	sseldd 3979	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
66		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
67		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐸) = (._r‘𝐸)
68	67	subrgmcl 20516	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑈)
69	45, 65, 66, 68	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑈)
70	44, 69	eqeltrrd 2829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)
71	70	ralrimivva 3195	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ 𝑈 (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)
72		eqid 2727	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
73		eqid 2727	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
74		eqid 2727	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
75		eqid 2727	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
76		eqid 2727	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝐴) = (LSubSp‘𝐴)
77	72, 73, 74, 75, 76	islss4 20839	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ LMod → (𝑈 ∈ (LSubSp‘𝐴) ↔ (𝑈 ∈ (SubGrp‘𝐴) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ 𝑈 (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)))
78	77	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ LMod ∧ (𝑈 ∈ (SubGrp‘𝐴) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ 𝑈 (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)) → 𝑈 ∈ (LSubSp‘𝐴))
79	15, 42, 71, 78	syl12anc 836	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝐴))
80	20, 32	sseqtrrd 4019	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑈)
81	18	lbslinds 21760	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) ⊆ (LIndS‘𝐶)
82	81, 16	sselid 3976	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶))
83	23, 37	sseqtrd 4018	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐴))
84		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ↾_s 𝑈) = (𝐴 ↾_s 𝑈)
85	84, 74	ressbas2 17211	. . . . . . . . . . . . . 14 ⊢ (𝑈 ⊆ (Base‘𝐴) → 𝑈 = (Base‘(𝐴 ↾_s 𝑈)))
86	83, 85	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 = (Base‘(𝐴 ↾_s 𝑈)))
87	25, 86, 31	3eqtr3rd 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(𝐴 ↾_s 𝑈)))
88	84, 72	resssca 17317	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (Scalar‘𝐴) = (Scalar‘(𝐴 ↾_s 𝑈)))
89	3, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘(𝐴 ↾_s 𝑈)))
90	60, 89	eqtr3d 2769	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Scalar‘𝐶) = (Scalar‘(𝐴 ↾_s 𝑈)))
91	90	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘(𝐴 ↾_s 𝑈))))
92	90	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘(𝐴 ↾_s 𝑈))))
93		eqid 2727	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐸) = (+_g‘𝐸)
94	5, 93	ressplusg 17264	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (+_g‘𝐸) = (+_g‘𝐹))
95	3, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐹))
96	34, 36	sraaddg 21058	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐴))
97	27, 30	sraaddg 21058	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐹) = (+_g‘𝐶))
98	95, 96, 97	3eqtr3rd 2776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘𝐴))
99		eqid 2727	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝐴) = (+_g‘𝐴)
100	84, 99	ressplusg 17264	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (+_g‘𝐴) = (+_g‘(𝐴 ↾_s 𝑈)))
101	3, 100	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (+_g‘𝐴) = (+_g‘(𝐴 ↾_s 𝑈)))
102	98, 101	eqtrd 2767	. . . . . . . . . . . . 13 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘(𝐴 ↾_s 𝑈)))
103	102	oveqdr 7442	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(+_g‘𝐶)𝑦) = (𝑥(+_g‘(𝐴 ↾_s 𝑈))𝑦))
104		fedgmul.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ DivRing)
105	52, 2	eqeltrd 2828	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) ∈ DivRing)
106		eqid 2727	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾_s 𝑉) = (𝐹 ↾_s 𝑉)
107	26, 106	sralvec 33271	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾_s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
108	104, 105, 4, 107	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
109		lveclmod 20984	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
110	108, 109	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ LMod)
111		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
112		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘𝐶)
113		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
114	17, 111, 112, 113	lmodvscl 20754	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥( ·_𝑠 ‘𝐶)𝑦) ∈ (Base‘𝐶))
115	114	3expb 1118	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·_𝑠 ‘𝐶)𝑦) ∈ (Base‘𝐶))
116	110, 115	sylan 579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·_𝑠 ‘𝐶)𝑦) ∈ (Base‘𝐶))
117		fedgmul.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
118	117, 1, 3	drgextvsca 33276	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐵))
119	43, 118	eqtr3d 2769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
120	84, 75	ressvsca 17318	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘(𝐴 ↾_s 𝑈)))
121	3, 120	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘(𝐴 ↾_s 𝑈)))
122	5, 67	ressmulr 17281	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (._r‘𝐸) = (._r‘𝐹))
123	3, 122	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = (._r‘𝐹))
124	26, 104, 4	drgextvsca 33276	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐹) = ( ·_𝑠 ‘𝐶))
125	123, 118, 124	3eqtr3d 2775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐶))
126	119, 121, 125	3eqtr3rd 2776	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘(𝐴 ↾_s 𝑈)))
127	126	oveqdr 7442	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·_𝑠 ‘𝐶)𝑦) = (𝑥( ·_𝑠 ‘(𝐴 ↾_s 𝑈))𝑦))
128		ovexd 7449	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ↾_s 𝑈) ∈ V)
129	87, 91, 92, 103, 116, 127, 108, 128	lindspropd 33092	. . . . . . . . . . 11 ⊢ (𝜑 → (LIndS‘𝐶) = (LIndS‘(𝐴 ↾_s 𝑈)))
130	82, 129	eleqtrd 2830	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘(𝐴 ↾_s 𝑈)))
131	76, 84	lsslinds 21758	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝐴) ∧ 𝑋 ⊆ 𝑈) → (𝑋 ∈ (LIndS‘(𝐴 ↾_s 𝑈)) ↔ 𝑋 ∈ (LIndS‘𝐴)))
132	131	biimpa 476	. . . . . . . . . 10 ⊢ (((𝐴 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝐴) ∧ 𝑋 ⊆ 𝑈) ∧ 𝑋 ∈ (LIndS‘(𝐴 ↾_s 𝑈))) → 𝑋 ∈ (LIndS‘𝐴))
133	15, 79, 80, 130, 132	syl31anc 1371	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐴))
134		eqid 2727	. . . . . . . . . . 11 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
135		eqid 2727	. . . . . . . . . . 11 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
136	74, 73, 72, 75, 134, 135	islinds5 33073	. . . . . . . . . 10 ⊢ ((𝐴 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐴)) → (𝑋 ∈ (LIndS‘𝐴) ↔ ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋)((𝑤 finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → 𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))}))))
137	136	biimpa 476	. . . . . . . . 9 ⊢ (((𝐴 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐴)) ∧ 𝑋 ∈ (LIndS‘𝐴)) → ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋)((𝑤 finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → 𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))})))
138	15, 39, 133, 137	syl21anc 837	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋)((𝑤 finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → 𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))})))
139	138	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋)((𝑤 finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → 𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))})))
140		eqid 2727	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
141		fvexd 6906	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (0_g‘𝐹) ∈ V)
142		fedgmullem.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵))
143	142	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ∈ (LBasis‘𝐵))
144	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LBasis‘𝐶))
145		fedgmullem2.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))
146		fvexd 6906	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Scalar‘𝐴) ∈ V)
147	142, 16	xpexd 7747	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ V)
148		eqid 2727	. . . . . . . . . . . . . . . . 17 ⊢ ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)) = ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))
149		eqid 2727	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) = (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))
150	148, 73, 135, 149	frlmelbas 21683	. . . . . . . . . . . . . . . 16 ⊢ (((Scalar‘𝐴) ∈ V ∧ (𝑌 × 𝑋) ∈ V) → (𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) ↔ (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑_m (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0_g‘(Scalar‘𝐴)))))
151	146, 147, 150	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) ↔ (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑_m (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0_g‘(Scalar‘𝐴)))))
152	145, 151	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑_m (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0_g‘(Scalar‘𝐴))))
153	152	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑_m (𝑌 × 𝑋)))
154		fvexd 6906	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ∈ V)
155	154, 147	elmapd 8852	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑_m (𝑌 × 𝑋)) ↔ 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))))
156	153, 155	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
157	156	ffnd 6717	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 Fn (𝑌 × 𝑋))
158	157	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑊 Fn (𝑌 × 𝑋))
159		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑌)
160	152	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 finSupp (0_g‘(Scalar‘𝐴)))
161		drngring 20624	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
162	1, 161	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ Ring)
163		ringmnd 20176	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
164	162, 163	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Mnd)
165		subrgsubg 20509	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ∈ (SubGrp‘𝐸))
166	9, 165	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ (SubGrp‘𝐸))
167		eqid 2727	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
168	167	subg0cl 19082	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (SubGrp‘𝐸) → (0_g‘𝐸) ∈ 𝑉)
169	166, 168	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑉)
170	46, 169	sseldd 3979	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑈)
171	5, 21, 167	ress0g 18715	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Mnd ∧ (0_g‘𝐸) ∈ 𝑈 ∧ 𝑈 ⊆ (Base‘𝐸)) → (0_g‘𝐸) = (0_g‘𝐹))
172	164, 170, 23, 171	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐹))
173	54	fveq2d 6895	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐾) = (0_g‘(Scalar‘𝐶)))
174	11, 167	subrg0 20511	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → (0_g‘𝐸) = (0_g‘𝐾))
175	9, 174	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐾))
176	60	fveq2d 6895	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐶)))
177	173, 175, 176	3eqtr4d 2777	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘(Scalar‘𝐴)))
178	172, 177	eqtr3d 2769	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘𝐹) = (0_g‘(Scalar‘𝐴)))
179	160, 178	breqtrrd 5170	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 finSupp (0_g‘𝐹))
180	179	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑊 finSupp (0_g‘𝐹))
181	140, 141, 143, 144, 158, 159, 180	fsuppcurry1 32501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘𝐹))
182	178	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (0_g‘𝐹) = (0_g‘(Scalar‘𝐴)))
183	181, 182	breqtrd 5168	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)))
184		eqidd 2728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
185	156	fovcdmda 7586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐴)))
186	185	anassrs 467	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐴)))
187	184, 186	fvmpt2d 7012	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖) = (𝑗𝑊𝑖))
188	187	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))
189	119	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
190	189	oveqd 7431	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
191	188, 190	eqtrd 2767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
192	191	mpteq2dva 5242	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
193	192	oveq2d 7430	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
194	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐸 ∈ DivRing)
195	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑉 ∈ (SubRing‘𝐸))
196	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐾 ∈ DivRing)
197	10, 194, 195, 11, 196, 144	drgextgsum 33280	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
198	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubRing‘𝐸))
199	104	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐹 ∈ DivRing)
200	117, 194, 198, 5, 199, 144	drgextgsum 33280	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
201	197, 200	eqtr3d 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
202	193, 201	eqtrd 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
203	142	mptexd 7230	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ V)
204		eqid 2727	. . . . . . . . . . . . . . . . . 18 ⊢ (0_g‘𝐵) = (0_g‘𝐵)
205	117, 5	sralvec 33271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
206	1, 104, 3, 205	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ LVec)
207		lveclmod 20984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ LVec → 𝐵 ∈ LMod)
208	206, 207	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ LMod)
209	208	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐵 ∈ LMod)
210		lmodabl 20785	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ LMod → 𝐵 ∈ Abel)
211	209, 210	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐵 ∈ Abel)
212	117	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 = ((subringAlg ‘𝐸)‘𝑈))
213	212, 3, 23	srasubrg 33270	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐵))
214		subrgsubg 20509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 ∈ (SubRing‘𝐵) → 𝑈 ∈ (SubGrp‘𝐵))
215	213, 214	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐵))
216	215	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝐵))
217	110	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐶 ∈ LMod)
218	61	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
219	186, 218	eleqtrd 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)))
220	20	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐶))
221		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
222	220, 221	sseldd 3979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
223	17, 111, 112, 113	lmodvscl 20754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ LMod ∧ (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
224	217, 219, 222, 223	syl3anc 1369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
225	125	oveqd 7431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖))
226	225	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖))
227	32	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑈 = (Base‘𝐶))
228	224, 226, 227	3eltr4d 2843	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) ∈ 𝑈)
229	228	fmpttd 7119	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)):𝑋⟶𝑈)
230	212, 23	srasca 21062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑈) = (Scalar‘𝐵))
231	5, 230	eqtrid 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
232	231	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐹 = (Scalar‘𝐵))
233		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐵) = (Base‘𝐵)
234		ovexd 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
235	20, 33	sstrd 3988	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐸))
236	235	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝐸))
237		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ 𝑋)
238	236, 237	sseldd 3979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ (Base‘𝐸))
239	238	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
240	212, 23	srabase 21056	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
241	240	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐵))
242	239, 241	eleqtrd 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐵))
243		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
244		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐵)
245	144, 209, 232, 233, 234, 242, 204, 243, 244, 181	mptscmfsupp0 20803	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)) finSupp (0_g‘𝐵))
246	204, 211, 144, 216, 229, 245	gsumsubgcl 19868	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ 𝑈)
247	231	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
248	25, 247	eqtrd 2767	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘(Scalar‘𝐵)))
249	248	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 = (Base‘(Scalar‘𝐵)))
250	246, 249	eleqtrd 2830	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ (Base‘(Scalar‘𝐵)))
251	250	fmpttd 7119	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵)))
252	251	ffund 6720	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))))
253		fvexd 6906	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐵)) ∈ V)
254		fconstmpt 5734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 × {(0_g‘(Scalar‘𝐴))}) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
255	254	eqeq2i 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
256		ovex 7447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘𝑊𝑖) ∈ V
257	256	rgenw 3060	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) ∈ V
258		mpteqb 7018	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) ∈ V → ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))) ↔ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
259	257, 258	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))) ↔ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
260	255, 259	bitri 275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
261	260	necon3abii 2982	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ ¬ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
262		df-ov 7417	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘𝑊𝑖) = (𝑊‘⟨𝑘, 𝑖⟩)
263	262	eqcomi 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖)
264	263	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖))
265	264	eqeq1d 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) = (0_g‘(Scalar‘𝐴)) ↔ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
266	265	necon3abid 2972	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
267	266	rexbidva 3171	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ∃𝑖 ∈ 𝑋 ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
268		rexnal 3095	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑖 ∈ 𝑋 ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)) ↔ ¬ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
269	267, 268	bitr2di 288	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (¬ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)) ↔ ∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))))
270	261, 269	bitrid 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ ∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))))
271	270	rabbidva 3434	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} = {𝑘 ∈ 𝑌 ∣ ∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))})
272		fveq2 6891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → (𝑊‘𝑧) = (𝑊‘⟨𝑘, 𝑖⟩))
273	272	neeq1d 2995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → ((𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴)) ↔ (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))))
274	273	dmrab 32288	. . . . . . . . . . . . . . . 16 ⊢ dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))} = {𝑘 ∈ 𝑌 ∣ ∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))}
275	271, 274	eqtr4di 2785	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} = dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
276		fvexd 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) ∈ V)
277		suppvalfn 8167	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 Fn (𝑌 × 𝑋) ∧ (𝑌 × 𝑋) ∈ V ∧ (0_g‘(Scalar‘𝐴)) ∈ V) → (𝑊 supp (0_g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
278	157, 147, 276, 277	syl3anc 1369	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 supp (0_g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
279	160	fsuppimpd 9387	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 supp (0_g‘(Scalar‘𝐴))) ∈ Fin)
280	278, 279	eqeltrrd 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))} ∈ Fin)
281		dmfi 9348	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))} ∈ Fin → dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))} ∈ Fin)
282	280, 281	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))} ∈ Fin)
283	275, 282	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} ∈ Fin)
284		nfv 1910	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖𝜑
285		nfcv 2898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖𝑌
286		nfmpt1 5250	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖))
287		nfcv 2898	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑋 × {(0_g‘(Scalar‘𝐴))})
288	286, 287	nfne 3038	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})
289	288, 285	nfrabw 3463	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖{𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})}
290	285, 289	nfdif 4121	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑖(𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
291	290	nfcri 2885	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
292	284, 291	nfan 1895	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})}))
293		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})}))
294	293	eldifad 3956	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → 𝑗 ∈ 𝑌)
295	293	eldifbd 3957	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → ¬ 𝑗 ∈ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
296		oveq1 7421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑗 → (𝑘𝑊𝑖) = (𝑗𝑊𝑖))
297	296	mpteq2dv 5244	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑗 → (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
298	297	neeq1d 2995	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑗 → ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})))
299	298	elrab 3680	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} ↔ (𝑗 ∈ 𝑌 ∧ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})))
300	295, 299	sylnib 328	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → ¬ (𝑗 ∈ 𝑌 ∧ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})))
301	294, 300	mpnanrd 409	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → ¬ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}))
302		nne 2939	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}))
303	301, 302	sylib 217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}))
304	303, 254	eqtrdi 2783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
305		ovex 7447	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗𝑊𝑖) ∈ V
306	305	rgenw 3060	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) ∈ V
307		mpteqb 7018	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) ∈ V → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))) ↔ ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
308	306, 307	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))) ↔ ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
309	304, 308	sylib 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
310	309	r19.21bi 3243	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
311	310	oveq1d 7429	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) = ((0_g‘(Scalar‘𝐴))( ·_𝑠 ‘𝐵)𝑖))
312	117, 1, 3	drgext0g 33275	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐵))
313	117, 1, 3	drgext0gsca 33277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐵) = (0_g‘(Scalar‘𝐵)))
314	312, 177, 313	3eqtr3d 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐵)))
315	314	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐵)))
316	315	oveq1d 7429	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → ((0_g‘(Scalar‘𝐴))( ·_𝑠 ‘𝐵)𝑖) = ((0_g‘(Scalar‘𝐵))( ·_𝑠 ‘𝐵)𝑖))
317	208	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → 𝐵 ∈ LMod)
318	294, 242	syldanl 601	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐵))
319		eqid 2727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Scalar‘𝐵) = (Scalar‘𝐵)
320		eqid 2727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘(Scalar‘𝐵)) = (0_g‘(Scalar‘𝐵))
321	233, 319, 244, 320, 204	lmod0vs 20771	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ LMod ∧ 𝑖 ∈ (Base‘𝐵)) → ((0_g‘(Scalar‘𝐵))( ·_𝑠 ‘𝐵)𝑖) = (0_g‘𝐵))
322	317, 318, 321	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → ((0_g‘(Scalar‘𝐵))( ·_𝑠 ‘𝐵)𝑖) = (0_g‘𝐵))
323	311, 316, 322	3eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) = (0_g‘𝐵))
324	292, 323	mpteq2da 5240	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵)))
325	324	oveq2d 7430	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵))))
326	208, 210	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ Abel)
327		ablgrp 19733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Abel → 𝐵 ∈ Grp)
328		grpmnd 18890	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Grp → 𝐵 ∈ Mnd)
329	326, 327, 328	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ Mnd)
330	204	gsumz 18781	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Mnd ∧ 𝑋 ∈ (LBasis‘𝐶)) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵))) = (0_g‘𝐵))
331	329, 16, 330	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵))) = (0_g‘𝐵))
332	331	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵))) = (0_g‘𝐵))
333	313	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (0_g‘𝐵) = (0_g‘(Scalar‘𝐵)))
334	325, 332, 333	3eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
335	334, 142	suppss2 8199	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) supp (0_g‘(Scalar‘𝐵))) ⊆ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
336		suppssfifsupp 9397	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ V ∧ Fun (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∧ (0_g‘(Scalar‘𝐵)) ∈ V) ∧ ({𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} ∈ Fin ∧ ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) supp (0_g‘(Scalar‘𝐵))) ⊆ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)))
337	203, 252, 253, 283, 335, 336	syl32anc 1376	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)))
338		eqidd 2728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))))
339		ovexd 7449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V)
340	338, 339	fvmpt2d 7012	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗) = (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
341	340	oveq1d 7429	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗) = ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))
342	341	mpteq2dva 5242	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗)) = (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗)))
343	342	oveq2d 7430	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))))
344	119	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
345	43	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
346	345	oveqd 7431	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))
347	346	mpteq2dva 5242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))
348	118	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (._r‘𝐸) = ( ·_𝑠 ‘𝐵))
349	348	oveqd 7431	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
350	349	mpteq2dv 5244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
351	347, 350	eqtr3d 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
352	351	oveq2d 7430	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
353		eqidd 2728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 = 𝑗)
354	344, 352, 353	oveq123d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗) = ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))
355	201	oveq1d 7429	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗) = ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))
356	354, 355	eqtrd 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗) = ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))
357	356	mpteq2dva 5242	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗)) = (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗)))
358	357	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))) = (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))))
359	10, 21	sraring 21072	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ Ring ∧ 𝑉 ⊆ (Base‘𝐸)) → 𝐴 ∈ Ring)
360	162, 36, 359	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ Ring)
361		ringcmn 20211	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
362	360, 361	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ CMnd)
363	162	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝐸 ∈ Ring)
364		eqid 2727	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
365	233, 364	lbsss 20955	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
366	142, 365	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
367	366, 240	sseqtrrd 4019	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸))
368	367	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑌 ⊆ (Base‘𝐸))
369		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑗 ∈ 𝑌)
370	368, 369	sseldd 3979	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑗 ∈ (Base‘𝐸))
371	21, 67	ringcl 20183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐸))
372	363, 238, 370, 371	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐸))
373	37	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (Base‘𝐸) = (Base‘𝐴))
374	372, 373	eleqtrd 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴))
375	374	ralrimivva 3195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴))
376		fedgmullem.d	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(._r‘𝐸)𝑗))
377	376	fmpo 8066	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
378	375, 377	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
379	72, 73, 75, 74, 15, 156, 378, 147	lcomf 20777	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐴))
380	72, 73, 75, 74, 15, 156, 378, 147, 134, 135, 160	lcomfsupp 20778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) finSupp (0_g‘𝐴))
381	74, 134, 362, 142, 16, 379, 380	gsumxp 19924	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 Σ_g (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)) = (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖))))))
382		fedgmullem2.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 Σ_g (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)) = (0_g‘𝐴))
383	162	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
384	156	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
385	57, 55	eqtrd 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
386	385, 36	eqsstrrd 4017	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
387	61, 386	eqsstrd 4016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
388	387, 37	sseqtrd 4018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
389	388	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
390	384, 389	fssd 6734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘𝐴))
391		simp2 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
392		simp3 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
393	390, 391, 392	fovcdmd 7587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐴))
394	37	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴))
395	393, 394	eleqtrrd 2831	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
396	238	3impb 1113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
397	370	3impb 1113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸))
398	21, 67	ringass 20186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐸 ∈ Ring ∧ ((𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸))) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) = ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗)))
399	383, 395, 396, 397, 398	syl13anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) = ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗)))
400	399	mpoeq3dva 7491	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗))))
401		ovexd 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
402		ovexd 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑖(._r‘𝐸)𝑗) ∈ V)
403		fnov 7546	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑊 Fn (𝑌 × 𝑋) ↔ 𝑊 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
404	157, 403	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑊 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
405	376	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(._r‘𝐸)𝑗)))
406	142, 16, 401, 402, 404, 405	offval22 8087	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗))))
407	43	ofeqd 7681	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ∘_f (._r‘𝐸) = ∘_f ( ·_𝑠 ‘𝐴))
408	407	oveqd 7431	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷))
409	400, 406, 408	3eqtr2rd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)))
410	409	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)))
411	410	oveqd 7431	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖) = (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))𝑖))
412		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
413		ovexd 7449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) ∈ V)
414		eqid 2727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
415	414	ovmpt4g 7562	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ∧ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) ∈ V) → (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))𝑖) = (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
416	412, 221, 413, 415	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))𝑖) = (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
417	411, 416	eqtrd 2767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖) = (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
418	417	mpteq2dva 5242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)))
419	418	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖))) = (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))))
420	162	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐸 ∈ Ring)
421	367	sselda 3978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸))
422	162	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
423	386	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
424	423, 219	sseldd 3979	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
425	21, 67	ringcl 20183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ (𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) ∈ (Base‘𝐸))
426	422, 424, 239, 425	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) ∈ (Base‘𝐸))
427	312	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (0_g‘𝐸) = (0_g‘𝐵))
428	245, 350, 427	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) finSupp (0_g‘𝐸))
429	21, 167, 67, 420, 144, 421, 426, 428	gsummulc1 20245	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))) = ((𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)))(._r‘𝐸)𝑗))
430	419, 429	eqtrd 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖))) = ((𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)))(._r‘𝐸)𝑗))
431	144	mptexd 7230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖)) ∈ V)
432	15	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐴 ∈ LMod)
433	36	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑉 ⊆ (Base‘𝐸))
434	10, 431, 194, 432, 433	gsumsra 32755	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖))))
435	144	mptexd 7230	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) ∈ V)
436	10, 435, 194, 432, 433	gsumsra 32755	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖))))
437	436	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)))(._r‘𝐸)𝑗) = ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)))(._r‘𝐸)𝑗))
438	43	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
439	347	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))))
440	438, 439, 353	oveq123d 7435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)))(._r‘𝐸)𝑗) = ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))
441	437, 440	eqtrd 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)))(._r‘𝐸)𝑗) = ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))
442	430, 434, 441	3eqtr3d 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖))) = ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))
443	442	mpteq2dva 5242	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖)))) = (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗)))
444	443	oveq2d 7430	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖))))) = (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))))
445	381, 382, 444	3eqtr3rd 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))) = (0_g‘𝐴))
446	10, 1, 9	drgext0g 33275	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐴))
447	445, 446, 312	3eqtr2d 2773	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))) = (0_g‘𝐵))
448	10, 1, 9, 11, 2, 142	drgextgsum 33280	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))) = (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))))
449	117, 1, 3, 5, 104, 142	drgextgsum 33280	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))) = (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))))
450	448, 449	eqtr3d 2769	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))) = (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))))
451	358, 447, 450	3eqtr3rd 2776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵))
452	343, 451	eqtrd 2767	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵))
453		breq1 5145	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → (𝑏 finSupp (0_g‘(Scalar‘𝐵)) ↔ (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵))))
454		nfmpt1 5250	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
455	454	nfeq2 2915	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
456		fveq1 6890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → (𝑏‘𝑗) = ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗))
457	456	oveq1d 7429	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗) = (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))
458	457	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∧ 𝑗 ∈ 𝑌) → ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗) = (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))
459	455, 458	mpteq2da 5240	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗)) = (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗)))
460	459	oveq2d 7430	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))))
461	460	eqeq1d 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → ((𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵) ↔ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)))
462	453, 461	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → ((𝑏 finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)) ↔ ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵))))
463		eqeq1 2731	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → (𝑏 = (𝑌 × {(0_g‘(Scalar‘𝐵))}) ↔ (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑌 × {(0_g‘(Scalar‘𝐵))})))
464	462, 463	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) → (((𝑏 finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)) → 𝑏 = (𝑌 × {(0_g‘(Scalar‘𝐵))})) ↔ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)) → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑌 × {(0_g‘(Scalar‘𝐵))}))))
465	364	lbslinds 21760	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
466	465, 142	sselid 3976	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (LIndS‘𝐵))
467		eqid 2727	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
468	233, 467, 319, 244, 204, 320	islinds5 33073	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ LMod ∧ 𝑌 ⊆ (Base‘𝐵)) → (𝑌 ∈ (LIndS‘𝐵) ↔ ∀𝑏 ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌)((𝑏 finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)) → 𝑏 = (𝑌 × {(0_g‘(Scalar‘𝐵))}))))
469	468	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ LMod ∧ 𝑌 ⊆ (Base‘𝐵)) ∧ 𝑌 ∈ (LIndS‘𝐵)) → ∀𝑏 ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌)((𝑏 finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)) → 𝑏 = (𝑌 × {(0_g‘(Scalar‘𝐵))})))
470	208, 366, 466, 469	syl21anc 837	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑏 ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌)((𝑏 finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝑏‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)) → 𝑏 = (𝑌 × {(0_g‘(Scalar‘𝐵))})))
471		fvexd 6906	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘(Scalar‘𝐵)) ∈ V)
472		elmapg 8851	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘(Scalar‘𝐵)) ∈ V ∧ 𝑌 ∈ (LBasis‘𝐵)) → ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌) ↔ (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵))))
473	472	biimpar 477	. . . . . . . . . . . . . . 15 ⊢ ((((Base‘(Scalar‘𝐵)) ∈ V ∧ 𝑌 ∈ (LBasis‘𝐵)) ∧ (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵))) → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌))
474	471, 142, 251, 473	syl21anc 837	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌))
475	464, 470, 474	rspcdva 3608	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)) ∧ (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ (((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))‘𝑗)( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵)) → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑌 × {(0_g‘(Scalar‘𝐵))})))
476	337, 452, 475	mp2and 698	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑌 × {(0_g‘(Scalar‘𝐵))}))
477		fconstmpt 5734	. . . . . . . . . . . 12 ⊢ (𝑌 × {(0_g‘(Scalar‘𝐵))}) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵)))
478	476, 477	eqtrdi 2783	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵))))
479		ovex 7447	. . . . . . . . . . . . 13 ⊢ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
480	479	rgenw 3060	. . . . . . . . . . . 12 ⊢ ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
481		mpteqb 7018	. . . . . . . . . . . 12 ⊢ (∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V → ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵))) ↔ ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵))))
482	480, 481	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵))) ↔ ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
483	478, 482	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
484	483	r19.21bi 3243	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
485	312, 446, 313	3eqtr3rd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘(Scalar‘𝐵)) = (0_g‘𝐴))
486	485	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (0_g‘(Scalar‘𝐵)) = (0_g‘𝐴))
487	202, 484, 486	3eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴))
488	183, 487	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)))
489	186	fmpttd 7119	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴)))
490		fvexd 6906	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐴)) ∈ V)
491	490, 144	elmapd 8852	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴))))
492	489, 491	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋))
493		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
494	493	breq1d 5152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 finSupp (0_g‘(Scalar‘𝐴)) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴))))
495		nfv 1910	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝑌)
496		nfmpt1 5250	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
497	496	nfeq2 2915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
498	495, 497	nfan 1895	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
499		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
500	499	fveq1d 6893	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑤‘𝑖) = ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖))
501	500	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))
502	498, 501	mpteq2da 5240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖)))
503	502	oveq2d 7430	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))))
504	503	eqeq1d 2729	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴) ↔ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)))
505	494, 504	anbi12d 630	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → ((𝑤 finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) ↔ ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴))))
506	493	eqeq1d 2729	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))})))
507	505, 506	imbi12d 344	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (((𝑤 finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → 𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))})) ↔ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}))))
508	492, 507	rspcdv 3599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋)((𝑤 finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → 𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))})) → (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)) ∧ (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴)) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}))))
509	139, 488, 508	mp2d 49	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}))
510	509, 254	eqtrdi 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
511	510, 308	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
512	511	ralrimiva 3141	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
513		eqidd 2728	. . . 4 ⊢ ((𝑗 = 𝑘 ∧ 𝑖 = 𝑙) → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴)))
514		fvexd 6906	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
515		fvexd 6906	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌 ∧ 𝑙 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
516	157, 513, 514, 515	fnmpoovd 8086	. . 3 ⊢ (𝜑 → (𝑊 = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))) ↔ ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
517	512, 516	mpbird 257	. 2 ⊢ (𝜑 → 𝑊 = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
518		fconstmpo 7531	. 2 ⊢ ((𝑌 × 𝑋) × {(0_g‘(Scalar‘𝐴))}) = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
519	517, 518	eqtr4di 2785	1 ⊢ (𝜑 → 𝑊 = ((𝑌 × 𝑋) × {(0_g‘(Scalar‘𝐴))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 {crab 3427 Vcvv 3469 ∖ cdif 3941 ⊆ wss 3944 {csn 4624 ⟨cop 4630 class class class wbr 5142 ↦ cmpt 5225 × cxp 5670 dom cdm 5672 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ∘_f cof 7677 supp csupp 8159 ↑_m cmap 8838 Fincfn 8957 finSupp cfsupp 9379 Basecbs 17173 ↾_s cress 17202 +_gcplusg 17226 ._rcmulr 17227 Scalarcsca 17229 ·_𝑠 cvsca 17230 0_gc0g 17414 Σ_g cgsu 17415 Mndcmnd 18687 Grpcgrp 18883 SubGrpcsubg 19068 CMndccmn 19728 Abelcabl 19729 Ringcrg 20166 SubRingcsubrg 20499 DivRingcdr 20617 LModclmod 20736 LSubSpclss 20808 LBasisclbs 20952 LVecclvec 20980 subringAlg csra 21049 freeLMod cfrlm 21673 LIndSclinds 21732

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-nzr 20445 df-subrng 20476 df-subrg 20501 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lmhm 20900 df-lbs 20953 df-lvec 20981 df-sra 21051 df-rgmod 21052 df-dsmm 21659 df-frlm 21674 df-uvc 21710 df-lindf 21733 df-linds 21734

This theorem is referenced by: fedgmul 33315

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