fedgmullem2 - Mathbox for Thierry Arnoux

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Theorem fedgmullem2 33643

Description: Lemma for fedgmul 33644. (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 20626	. . . . . . . . . . . . . 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 33600	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
13	1, 2, 9, 12	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ LVec)
14		lveclmod 21128	. . . . . . . . . 10 ⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ LMod)
16		fedgmullem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))
17		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		eqid 2740	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
19	17, 18	lbsss 21099	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
20	16, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶))
21		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
22	21	subrgss 20600	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
23	3, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
24	5, 21	ressbas2 17296	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
29	28	subrgss 20600	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
30	4, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
31	27, 30	srabase 21200	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
32	25, 31	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
33	32, 23	eqsstrrd 4048	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
34	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝐸)‘𝑉))
35	21	subrgss 20600	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
36	9, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
37	34, 36	srabase 21200	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
38	33, 37	sseqtrd 4049	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐴))
39	20, 38	sstrd 4019	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐴))
40	34, 3, 36	srasubrg 33599	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐴))
41		subrgsubg 20605	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubRing‘𝐴) → 𝑈 ∈ (SubGrp‘𝐴))
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐴))
43	10, 1, 9	drgextvsca 33605	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
44	43	oveqdr 7476	. . . . . . . . . . . . 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 17308	. . . . . . . . . . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 ↾_s 𝑉) = ((𝐸 ↾_s 𝑈) ↾_s 𝑉)
52	50, 51, 11	3eqtr4g 2805	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = 𝐾)
53	27, 30	srasca 21206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = (Scalar‘𝐶))
54	52, 53	eqtr3d 2782	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
55	54	fveq2d 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
56	11, 21	ressbas2 17296	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
57	36, 56	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
58	34, 36	srasca 21206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑉) = (Scalar‘𝐴))
59	11, 58	eqtrid 2792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
60	52, 53, 59	3eqtr3rd 2789	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
61	60	fveq2d 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
62	55, 57, 61	3eqtr4d 2790	. . . . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑉)
65	47, 64	sseldd 4009	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐸) = (._r‘𝐸)
68	67	subrgmcl 20612	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑈)
69	45, 65, 66, 68	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑈)
70	44, 69	eqeltrrd 2845	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)
71	70	ralrimivva 3208	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ 𝑈 (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)
72		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
73		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
74		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
75		eqid 2740	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
76		eqid 2740	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝐴) = (LSubSp‘𝐴)
77	72, 73, 74, 75, 76	islss4 20983	. . . . . . . . . . . 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 4050	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑈)
81	18	lbslinds 21876	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) ⊆ (LIndS‘𝐶)
82	81, 16	sselid 4006	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶))
83	23, 37	sseqtrd 4049	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐴))
84		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ↾_s 𝑈) = (𝐴 ↾_s 𝑈)
85	84, 74	ressbas2 17296	. . . . . . . . . . . . . 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 2789	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(𝐴 ↾_s 𝑈)))
88	84, 72	resssca 17402	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Scalar‘𝐶) = (Scalar‘(𝐴 ↾_s 𝑈)))
91	90	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘(𝐴 ↾_s 𝑈))))
92	90	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘(𝐴 ↾_s 𝑈))))
93		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐸) = (+_g‘𝐸)
94	5, 93	ressplusg 17349	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (+_g‘𝐸) = (+_g‘𝐹))
95	3, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐹))
96	34, 36	sraaddg 21202	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐴))
97	27, 30	sraaddg 21202	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐹) = (+_g‘𝐶))
98	95, 96, 97	3eqtr3rd 2789	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘𝐴))
99		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝐴) = (+_g‘𝐴)
100	84, 99	ressplusg 17349	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 13 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘(𝐴 ↾_s 𝑈)))
103	102	oveqdr 7476	. . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) ∈ DivRing)
106		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾_s 𝑉) = (𝐹 ↾_s 𝑉)
107	26, 106	sralvec 33600	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾_s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
108	104, 105, 4, 107	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
109		lveclmod 21128	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
110	108, 109	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ LMod)
111		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
112		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘𝐶)
113		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
114	17, 111, 112, 113	lmodvscl 20898	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥( ·_𝑠 ‘𝐶)𝑦) ∈ (Base‘𝐶))
115	114	3expb 1120	. . . . . . . . . . . . 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 33605	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐵))
119	43, 118	eqtr3d 2782	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
120	84, 75	ressvsca 17403	. . . . . . . . . . . . . . 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 17366	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (._r‘𝐸) = (._r‘𝐹))
123	3, 122	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = (._r‘𝐹))
124	26, 104, 4	drgextvsca 33605	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐹) = ( ·_𝑠 ‘𝐶))
125	123, 118, 124	3eqtr3d 2788	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐶))
126	119, 121, 125	3eqtr3rd 2789	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘(𝐴 ↾_s 𝑈)))
127	126	oveqdr 7476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·_𝑠 ‘𝐶)𝑦) = (𝑥( ·_𝑠 ‘(𝐴 ↾_s 𝑈))𝑦))
128		ovexd 7483	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ↾_s 𝑈) ∈ V)
129	87, 91, 92, 103, 116, 127, 108, 128	lindspropd 33376	. . . . . . . . . . 11 ⊢ (𝜑 → (LIndS‘𝐶) = (LIndS‘(𝐴 ↾_s 𝑈)))
130	82, 129	eleqtrd 2846	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘(𝐴 ↾_s 𝑈)))
131	76, 84	lsslinds 21874	. . . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐴))
134		eqid 2740	. . . . . . . . . . 11 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
135		eqid 2740	. . . . . . . . . . 11 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
136	74, 73, 72, 75, 134, 135	islinds5 33360	. . . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
141		fvexd 6935	. . . . . . . . . 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 6935	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Scalar‘𝐴) ∈ V)
147	142, 16	xpexd 7786	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ V)
148		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)) = ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))
149		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) = (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))
150	148, 73, 135, 149	frlmelbas 21799	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 6935	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ∈ V)
155	154, 147	elmapd 8898	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑_m (𝑌 × 𝑋)) ↔ 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))))
156	153, 155	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
157	156	ffnd 6748	. . . . . . . . . . 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 20758	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
162	1, 161	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ Ring)
163		ringmnd 20270	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
164	162, 163	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Mnd)
165		subrgsubg 20605	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ∈ (SubGrp‘𝐸))
166	9, 165	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ (SubGrp‘𝐸))
167		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
168	167	subg0cl 19174	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (SubGrp‘𝐸) → (0_g‘𝐸) ∈ 𝑉)
169	166, 168	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑉)
170	46, 169	sseldd 4009	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑈)
171	5, 21, 167	ress0g 18800	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Mnd ∧ (0_g‘𝐸) ∈ 𝑈 ∧ 𝑈 ⊆ (Base‘𝐸)) → (0_g‘𝐸) = (0_g‘𝐹))
172	164, 170, 23, 171	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐹))
173	54	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐾) = (0_g‘(Scalar‘𝐶)))
174	11, 167	subrg0 20607	. . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐶)))
177	173, 175, 176	3eqtr4d 2790	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘(Scalar‘𝐴)))
178	172, 177	eqtr3d 2782	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘𝐹) = (0_g‘(Scalar‘𝐴)))
179	160, 178	breqtrrd 5194	. . . . . . . . . . 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 32739	. . . . . . . . 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 5192	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)))
184		eqidd 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
185	156	fovcdmda 7621	. . . . . . . . . . . . . . . 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 7042	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖) = (𝑗𝑊𝑖))
188	187	oveq1d 7463	. . . . . . . . . . . . 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 7465	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
191	188, 190	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
192	191	mpteq2dva 5266	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
193	192	oveq2d 7464	. . . . . . . . . 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 33609	. . . . . . . . . . 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 33609	. . . . . . . . . . 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 2782	. . . . . . . . . 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 2780	. . . . . . . . 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 7261	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ V)
204		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (0_g‘𝐵) = (0_g‘𝐵)
205	117, 5	sralvec 33600	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
206	1, 104, 3, 205	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ LVec)
207		lveclmod 21128	. . . . . . . . . . . . . . . . . . . . 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 20929	. . . . . . . . . . . . . . . . . . 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 33599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐵))
214		subrgsubg 20605	. . . . . . . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
223	17, 111, 112, 113	lmodvscl 20898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ LMod ∧ (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
224	217, 219, 222, 223	syl3anc 1371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
225	125	oveqd 7465	. . . . . . . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) ∈ 𝑈)
229	228	fmpttd 7149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)):𝑋⟶𝑈)
230	212, 23	srasca 21206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑈) = (Scalar‘𝐵))
231	5, 230	eqtrid 2792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
232	231	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐹 = (Scalar‘𝐵))
233		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐵) = (Base‘𝐵)
234		ovexd 7483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
235	20, 33	sstrd 4019	. . . . . . . . . . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . . . . . . . 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 21200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
241	240	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐵))
242	239, 241	eleqtrd 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐵))
243		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
244		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐵)
245	144, 209, 232, 233, 234, 242, 204, 243, 244, 181	mptscmfsupp0 20947	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)) finSupp (0_g‘𝐵))
246	204, 211, 144, 216, 229, 245	gsumsubgcl 19962	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ 𝑈)
247	231	fveq2d 6924	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
248	25, 247	eqtrd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘(Scalar‘𝐵)))
249	248	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 = (Base‘(Scalar‘𝐵)))
250	246, 249	eleqtrd 2846	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ (Base‘(Scalar‘𝐵)))
251	250	fmpttd 7149	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵)))
252	251	ffund 6751	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))))
253		fvexd 6935	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐵)) ∈ V)
254		fconstmpt 5762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 × {(0_g‘(Scalar‘𝐴))}) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
255	254	eqeq2i 2753	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
256		ovex 7481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘𝑊𝑖) ∈ V
257	256	rgenw 3071	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) ∈ V
258		mpteqb 7048	. . . . . . . . . . . . . . . . . . . . 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 2993	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ ¬ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
262		df-ov 7451	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘𝑊𝑖) = (𝑊‘⟨𝑘, 𝑖⟩)
263	262	eqcomi 2749	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖)
264	263	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖))
265	264	eqeq1d 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) = (0_g‘(Scalar‘𝐴)) ↔ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
266	265	necon3abid 2983	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
267	266	rexbidva 3183	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ∃𝑖 ∈ 𝑋 ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
268		rexnal 3106	. . . . . . . . . . . . . . . . . . 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 3450	. . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → (𝑊‘𝑧) = (𝑊‘⟨𝑘, 𝑖⟩))
273	272	neeq1d 3006	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → ((𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴)) ↔ (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))))
274	273	dmrab 32525	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} = dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
276		fvexd 6935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) ∈ V)
277		suppvalfn 8209	. . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 supp (0_g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
279	160	fsuppimpd 9439	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 supp (0_g‘(Scalar‘𝐴))) ∈ Fin)
280	278, 279	eqeltrrd 2845	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))} ∈ Fin)
281		dmfi 9403	. . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} ∈ Fin)
284		nfv 1913	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖𝜑
285		nfcv 2908	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖𝑌
286		nfmpt1 5274	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖))
287		nfcv 2908	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑋 × {(0_g‘(Scalar‘𝐴))})
288	286, 287	nfne 3049	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})
289	288, 285	nfrabw 3483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖{𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})}
290	285, 289	nfdif 4152	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑖(𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
291	290	nfcri 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
292	284, 291	nfan 1898	. . . . . . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → 𝑗 ∈ 𝑌)
295	293	eldifbd 3989	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑗 → (𝑘𝑊𝑖) = (𝑗𝑊𝑖))
297	296	mpteq2dv 5268	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑗 → (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
298	297	neeq1d 3006	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑗 → ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})))
299	298	elrab 3708	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2950	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}))
303	301, 302	sylib 218	. . . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . . 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 7481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗𝑊𝑖) ∈ V
306	305	rgenw 3071	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) ∈ V
307		mpteqb 7048	. . . . . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . . . 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 3257	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
311	310	oveq1d 7463	. . . . . . . . . . . . . . . . . . 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 33604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐵))
313	117, 1, 3	drgext0gsca 33606	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐵) = (0_g‘(Scalar‘𝐵)))
314	312, 177, 313	3eqtr3d 2788	. . . . . . . . . . . . . . . . . . . . 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 7463	. . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Scalar‘𝐵) = (Scalar‘𝐵)
320		eqid 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘(Scalar‘𝐵)) = (0_g‘(Scalar‘𝐵))
321	233, 319, 244, 320, 204	lmod0vs 20915	. . . . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . . 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 5264	. . . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . 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		ablgrp 19827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Abel → 𝐵 ∈ Grp)
327		grpmnd 18980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Grp → 𝐵 ∈ Mnd)
328	208, 210, 326, 327	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ Mnd)
329	204	gsumz 18871	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Mnd ∧ 𝑋 ∈ (LBasis‘𝐶)) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵))) = (0_g‘𝐵))
330	328, 16, 329	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵))) = (0_g‘𝐵))
331	330	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))
332	313	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → (0_g‘𝐵) = (0_g‘(Scalar‘𝐵)))
333	325, 331, 332	3eqtrd 2784	. . . . . . . . . . . . . . 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)))
334	333, 142	suppss2 8241	. . . . . . . . . . . . . 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))})})
335		suppssfifsupp 9449	. . . . . . . . . . . . . 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)))
336	203, 252, 253, 283, 334, 335	syl32anc 1378	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)))
337		eqidd 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))))
338		ovexd 7483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V)
339	337, 338	fvmpt2d 7042	. . . . . . . . . . . . . . . . 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))))
340	339	oveq1d 7463	. . . . . . . . . . . . . . . 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))
341	340	mpteq2dva 5266	. . . . . . . . . . . . . . 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)))
342	341	oveq2d 7464	. . . . . . . . . . . . . 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))))
343	119	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
344	43	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
345	344	oveqd 7465	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))
346	345	mpteq2dva 5266	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))
347	118	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (._r‘𝐸) = ( ·_𝑠 ‘𝐵))
348	347	oveqd 7465	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
349	348	mpteq2dv 5268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
350	346, 349	eqtr3d 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
351	350	oveq2d 7464	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
352		eqidd 2741	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 = 𝑗)
353	343, 351, 352	oveq123d 7469	. . . . . . . . . . . . . . . . . 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))
354	201	oveq1d 7463	. . . . . . . . . . . . . . . . . 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))
355	353, 354	eqtrd 2780	. . . . . . . . . . . . . . . . 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))
356	355	mpteq2dva 5266	. . . . . . . . . . . . . . . 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)))
357	356	oveq2d 7464	. . . . . . . . . . . . . . 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))))
358	10, 21	sraring 21216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ Ring ∧ 𝑉 ⊆ (Base‘𝐸)) → 𝐴 ∈ Ring)
359	162, 36, 358	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ Ring)
360		ringcmn 20305	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
361	359, 360	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ CMnd)
362	162	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝐸 ∈ Ring)
363		eqid 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
364	233, 363	lbsss 21099	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
365	142, 364	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
366	365, 240	sseqtrrd 4050	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸))
367	366	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑌 ⊆ (Base‘𝐸))
368		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑗 ∈ 𝑌)
369	367, 368	sseldd 4009	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑗 ∈ (Base‘𝐸))
370	21, 67	ringcl 20277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐸))
371	362, 238, 369, 370	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐸))
372	37	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (Base‘𝐸) = (Base‘𝐴))
373	371, 372	eleqtrd 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴))
374	373	ralrimivva 3208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴))
375		fedgmullem.d	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(._r‘𝐸)𝑗))
376	375	fmpo 8109	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
377	374, 376	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
378	72, 73, 75, 74, 15, 156, 377, 147	lcomf 20921	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐴))
379	72, 73, 75, 74, 15, 156, 377, 147, 134, 135, 160	lcomfsupp 20922	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) finSupp (0_g‘𝐴))
380	74, 134, 361, 142, 16, 378, 379	gsumxp 20018	. . . . . . . . . . . . . . . . 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))))))
381		fedgmullem2.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 Σ_g (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)) = (0_g‘𝐴))
382	162	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
383	156	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
384	57, 55	eqtrd 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
385	384, 36	eqsstrrd 4048	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
386	61, 385	eqsstrd 4047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
387	386, 37	sseqtrd 4049	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
388	387	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
389	383, 388	fssd 6764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘𝐴))
390		simp2 1137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
391		simp3 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
392	389, 390, 391	fovcdmd 7622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐴))
393	37	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴))
394	392, 393	eleqtrrd 2847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
395	238	3impb 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
396	369	3impb 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸))
397	21, 67	ringass 20280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
398	382, 394, 395, 396, 397	syl13anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) = ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗)))
399	398	mpoeq3dva 7527	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗))))
400		ovexd 7483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
401		ovexd 7483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑖(._r‘𝐸)𝑗) ∈ V)
402		fnov 7581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑊 Fn (𝑌 × 𝑋) ↔ 𝑊 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
403	157, 402	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑊 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
404	375	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(._r‘𝐸)𝑗)))
405	142, 16, 400, 401, 403, 404	offval22 8129	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗))))
406	43	ofeqd 7716	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ∘_f (._r‘𝐸) = ∘_f ( ·_𝑠 ‘𝐴))
407	406	oveqd 7465	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷))
408	399, 405, 407	3eqtr2rd 2787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)))
409	408	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)))
410	409	oveqd 7465	. . . . . . . . . . . . . . . . . . . . . . . 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))
411		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
412		ovexd 7483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) ∈ V)
413		eqid 2740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
414	413	ovmpt4g 7597	. . . . . . . . . . . . . . . . . . . . . . . . 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))
415	411, 221, 412, 414	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . 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))
416	410, 415	eqtrd 2780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖) = (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
417	416	mpteq2dva 5266	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)))
418	417	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . 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))))
419	162	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐸 ∈ Ring)
420	366	sselda 4008	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸))
421	162	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
422	385	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
423	422, 219	sseldd 4009	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
424	21, 67	ringcl 20277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ (𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) ∈ (Base‘𝐸))
425	421, 423, 239, 424	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) ∈ (Base‘𝐸))
426	312	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (0_g‘𝐸) = (0_g‘𝐵))
427	245, 349, 426	3brtr4d 5198	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) finSupp (0_g‘𝐸))
428	21, 167, 67, 419, 144, 420, 425, 427	gsummulc1 20339	. . . . . . . . . . . . . . . . . . . . 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))
429	418, 428	eqtrd 2780	. . . . . . . . . . . . . . . . . . . 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))
430	144	mptexd 7261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖)) ∈ V)
431	15	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐴 ∈ LMod)
432	36	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑉 ⊆ (Base‘𝐸))
433	10, 430, 194, 431, 432	gsumsra 33030	. . . . . . . . . . . . . . . . . . . 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))))
434	144	mptexd 7261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) ∈ V)
435	10, 434, 194, 431, 432	gsumsra 33030	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖))))
436	435	oveq1d 7463	. . . . . . . . . . . . . . . . . . . . 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))
437	43	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
438	346	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))))
439	437, 438, 352	oveq123d 7469	. . . . . . . . . . . . . . . . . . . . 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))
440	436, 439	eqtrd 2780	. . . . . . . . . . . . . . . . . . . 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))
441	429, 433, 440	3eqtr3d 2788	. . . . . . . . . . . . . . . . . . 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))
442	441	mpteq2dva 5266	. . . . . . . . . . . . . . . . . 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)))
443	442	oveq2d 7464	. . . . . . . . . . . . . . . . 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))))
444	380, 381, 443	3eqtr3rd 2789	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))) = (0_g‘𝐴))
445	10, 1, 9	drgext0g 33604	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐴))
446	444, 445, 312	3eqtr2d 2786	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)))( ·_𝑠 ‘𝐴)𝑗))) = (0_g‘𝐵))
447	10, 1, 9, 11, 2, 142	drgextgsum 33609	. . . . . . . . . . . . . . . 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))))
448	117, 1, 3, 5, 104, 142	drgextgsum 33609	. . . . . . . . . . . . . . . 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))))
449	447, 448	eqtr3d 2782	. . . . . . . . . . . . . . 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))))
450	357, 446, 449	3eqtr3rd 2789	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 Σ_g (𝑗 ∈ 𝑌 ↦ ((𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))( ·_𝑠 ‘𝐵)𝑗))) = (0_g‘𝐵))
451	342, 450	eqtrd 2780	. . . . . . . . . . . . 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))
452		breq1 5169	. . . . . . . . . . . . . . . 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))))
453		nfmpt1 5274	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
454	453	nfeq2 2926	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
455		fveq1 6919	. . . . . . . . . . . . . . . . . . . . 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))
456	455	oveq1d 7463	. . . . . . . . . . . . . . . . . . . 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))
457	456	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))
458	454, 457	mpteq2da 5264	. . . . . . . . . . . . . . . . . 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)))
459	458	oveq2d 7464	. . . . . . . . . . . . . . . . 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))))
460	459	eqeq1d 2742	. . . . . . . . . . . . . . . 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)))
461	452, 460	anbi12d 631	. . . . . . . . . . . . . . 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))))
462		eqeq1 2744	. . . . . . . . . . . . . . 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))})))
463	461, 462	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))}))))
464	363	lbslinds 21876	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
465	464, 142	sselid 4006	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (LIndS‘𝐵))
466		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
467	233, 466, 319, 244, 204, 320	islinds5 33360	. . . . . . . . . . . . . . . 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))}))))
468	467	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))})))
469	208, 365, 465, 468	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))})))
470		fvexd 6935	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘(Scalar‘𝐵)) ∈ V)
471		elmapg 8897	. . . . . . . . . . . . . . . 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))))
472	471	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))
473	470, 142, 251, 472	syl21anc 837	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌))
474	463, 469, 473	rspcdva 3636	. . . . . . . . . . . . 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))})))
475	336, 451, 474	mp2and 698	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑌 × {(0_g‘(Scalar‘𝐵))}))
476		fconstmpt 5762	. . . . . . . . . . . 12 ⊢ (𝑌 × {(0_g‘(Scalar‘𝐵))}) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵)))
477	475, 476	eqtrdi 2796	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵))))
478		ovex 7481	. . . . . . . . . . . . 13 ⊢ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
479	478	rgenw 3071	. . . . . . . . . . . 12 ⊢ ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
480		mpteqb 7048	. . . . . . . . . . . 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))))
481	479, 480	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)))
482	477, 481	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
483	482	r19.21bi 3257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
484	312, 445, 313	3eqtr3rd 2789	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘(Scalar‘𝐵)) = (0_g‘𝐴))
485	484	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (0_g‘(Scalar‘𝐵)) = (0_g‘𝐴))
486	202, 483, 485	3eqtrd 2784	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (0_g‘𝐴))
487	183, 486	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)))
488	186	fmpttd 7149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴)))
489		fvexd 6935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐴)) ∈ V)
490	489, 144	elmapd 8898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴))))
491	488, 490	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑_m 𝑋))
492		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
493	492	breq1d 5176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 finSupp (0_g‘(Scalar‘𝐴)) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴))))
494		nfv 1913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝑌)
495		nfmpt1 5274	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
496	495	nfeq2 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
497	494, 496	nfan 1898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
498		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
499	498	fveq1d 6922	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑤‘𝑖) = ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖))
500	499	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖))
501	497, 500	mpteq2da 5264	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑖 ∈ 𝑋 ↦ ((𝑤‘𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖)))
502	501	oveq2d 7464	. . . . . . . . . . 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))))
503	502	eqeq1d 2742	. . . . . . . . . 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)))
504	493, 503	anbi12d 631	. . . . . . . . 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))))
505	492	eqeq1d 2742	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 = (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))})))
506	504, 505	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))}))))
507	491, 506	rspcdv 3627	. . . . . . 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))}))))
508	139, 487, 507	mp2d 49	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}))
509	508, 254	eqtrdi 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
510	509, 308	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
511	510	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
512		eqidd 2741	. . . 4 ⊢ ((𝑗 = 𝑘 ∧ 𝑖 = 𝑙) → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴)))
513		fvexd 6935	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
514		fvexd 6935	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌 ∧ 𝑙 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
515	157, 512, 513, 514	fnmpoovd 8128	. . 3 ⊢ (𝜑 → (𝑊 = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))) ↔ ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
516	511, 515	mpbird 257	. 2 ⊢ (𝜑 → 𝑊 = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
517		fconstmpo 7567	. 2 ⊢ ((𝑌 × 𝑋) × {(0_g‘(Scalar‘𝐴))}) = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
518	516, 517	eqtr4di 2798	1 ⊢ (𝜑 → 𝑊 = ((𝑌 × 𝑋) × {(0_g‘(Scalar‘𝐴))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 {csn 4648 ⟨cop 4654 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 dom cdm 5700 Fun wfun 6567 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 ∘_f cof 7712 supp csupp 8201 ↑_m cmap 8884 Fincfn 9003 finSupp cfsupp 9431 Basecbs 17258 ↾_s cress 17287 +_gcplusg 17311 ._rcmulr 17312 Scalarcsca 17314 ·_𝑠 cvsca 17315 0_gc0g 17499 Σ_g cgsu 17500 Mndcmnd 18772 Grpcgrp 18973 SubGrpcsubg 19160 CMndccmn 19822 Abelcabl 19823 Ringcrg 20260 SubRingcsubrg 20595 DivRingcdr 20751 LModclmod 20880 LSubSpclss 20952 LBasisclbs 21096 LVecclvec 21124 subringAlg csra 21193 freeLMod cfrlm 21789 LIndSclinds 21848

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-nzr 20539 df-subrng 20572 df-subrg 20597 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lmhm 21044 df-lbs 21097 df-lvec 21125 df-sra 21195 df-rgmod 21196 df-dsmm 21775 df-frlm 21790 df-uvc 21826 df-lindf 21849 df-linds 21850

This theorem is referenced by: fedgmul 33644

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