fedgmullem2 - Mathbox for Thierry Arnoux

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Theorem fedgmullem2 33619

Description: Lemma for fedgmul 33620. (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 𝜑,𝑖,𝑗 𝑖,𝐸,𝑗 𝐷,𝑖,𝑗 𝐶,𝑖 𝑗,𝑊,𝑖 𝑖,𝑌,𝑗 𝑖,𝑋,𝑗 𝐵,𝑖,𝑗 𝑈,𝑖 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 𝑘 𝑙 𝑥 𝑦 𝑤 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 20518	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 33574	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
13	1, 2, 9, 12	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ LVec)
14		lveclmod 21045	. . . . . . . . . 10 ⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ LMod)
16		fedgmullem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))
17		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		eqid 2729	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
19	17, 18	lbsss 21016	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
20	16, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶))
21		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
22	21	subrgss 20492	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
23	3, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
24	5, 21	ressbas2 17184	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
29	28	subrgss 20492	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
30	4, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
31	27, 30	srabase 21116	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
32	25, 31	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
33	32, 23	eqsstrrd 3979	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
34	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝐸)‘𝑉))
35	21	subrgss 20492	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
36	9, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
37	34, 36	srabase 21116	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
38	33, 37	sseqtrd 3980	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐴))
39	20, 38	sstrd 3954	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐴))
40	34, 3, 36	srasubrg 33573	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐴))
41		subrgsubg 20497	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubRing‘𝐴) → 𝑈 ∈ (SubGrp‘𝐴))
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐴))
43	10, 1, 9	drgextvsca 33579	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
44	43	oveqdr 7397	. . . . . . . . . . . . 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 17194	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾_s 𝑈) ↾_s 𝑉) = (𝐸 ↾_s 𝑉))
50	3, 46, 49	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐸 ↾_s 𝑈) ↾_s 𝑉) = (𝐸 ↾_s 𝑉))
51	5	oveq1i 7379	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 ↾_s 𝑉) = ((𝐸 ↾_s 𝑈) ↾_s 𝑉)
52	50, 51, 11	3eqtr4g 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = 𝐾)
53	27, 30	srasca 21119	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = (Scalar‘𝐶))
54	52, 53	eqtr3d 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
55	54	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
56	11, 21	ressbas2 17184	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
57	36, 56	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
58	34, 36	srasca 21119	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑉) = (Scalar‘𝐴))
59	11, 58	eqtrid 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
60	52, 53, 59	3eqtr3rd 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
61	60	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
62	55, 57, 61	3eqtr4d 2774	. . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐸) = (._r‘𝐸)
68	67	subrgmcl 20504	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑈)
69	45, 65, 66, 68	syl3anc 1373	. . . . . . . . . . . . 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 3178	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ 𝑈 (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)
72		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
73		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
74		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
75		eqid 2729	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
76		eqid 2729	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝐴) = (LSubSp‘𝐴)
77	72, 73, 74, 75, 76	islss4 20900	. . . . . . . . . . . 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 3981	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑈)
81	18	lbslinds 21775	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) ⊆ (LIndS‘𝐶)
82	81, 16	sselid 3941	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶))
83	23, 37	sseqtrd 3980	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐴))
84		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ↾_s 𝑈) = (𝐴 ↾_s 𝑈)
85	84, 74	ressbas2 17184	. . . . . . . . . . . . . 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 2773	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(𝐴 ↾_s 𝑈)))
88	84, 72	resssca 17282	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Scalar‘𝐶) = (Scalar‘(𝐴 ↾_s 𝑈)))
91	90	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘(𝐴 ↾_s 𝑈))))
92	90	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘(𝐴 ↾_s 𝑈))))
93		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐸) = (+_g‘𝐸)
94	5, 93	ressplusg 17230	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (+_g‘𝐸) = (+_g‘𝐹))
95	3, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐹))
96	34, 36	sraaddg 21117	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐴))
97	27, 30	sraaddg 21117	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐹) = (+_g‘𝐶))
98	95, 96, 97	3eqtr3rd 2773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘𝐴))
99		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝐴) = (+_g‘𝐴)
100	84, 99	ressplusg 17230	. . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . 13 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘(𝐴 ↾_s 𝑈)))
103	102	oveqdr 7397	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾_s 𝑉) = (𝐹 ↾_s 𝑉)
107	26, 106	sralvec 33574	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾_s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
108	104, 105, 4, 107	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
109		lveclmod 21045	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
110	108, 109	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ LMod)
111		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
112		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘𝐶)
113		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
114	17, 111, 112, 113	lmodvscl 20816	. . . . . . . . . . . . . 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 580	. . . . . . . . . . . 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 33579	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐵))
119	43, 118	eqtr3d 2766	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
120	84, 75	ressvsca 17283	. . . . . . . . . . . . . . 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 17246	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (._r‘𝐸) = (._r‘𝐹))
123	3, 122	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = (._r‘𝐹))
124	26, 104, 4	drgextvsca 33579	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐹) = ( ·_𝑠 ‘𝐶))
125	123, 118, 124	3eqtr3d 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐶))
126	119, 121, 125	3eqtr3rd 2773	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘(𝐴 ↾_s 𝑈)))
127	126	oveqdr 7397	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·_𝑠 ‘𝐶)𝑦) = (𝑥( ·_𝑠 ‘(𝐴 ↾_s 𝑈))𝑦))
128		ovexd 7404	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ↾_s 𝑈) ∈ V)
129	87, 91, 92, 103, 116, 127, 108, 128	lindspropd 33347	. . . . . . . . . . 11 ⊢ (𝜑 → (LIndS‘𝐶) = (LIndS‘(𝐴 ↾_s 𝑈)))
130	82, 129	eleqtrd 2830	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘(𝐴 ↾_s 𝑈)))
131	76, 84	lsslinds 21773	. . . . . . . . . . 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 1375	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐴))
134		eqid 2729	. . . . . . . . . . 11 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
135		eqid 2729	. . . . . . . . . . 11 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
136	74, 73, 72, 75, 134, 135	islinds5 33331	. . . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
141		fvexd 6855	. . . . . . . . . 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 6855	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Scalar‘𝐴) ∈ V)
147	142, 16	xpexd 7707	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ V)
148		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)) = ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))
149		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) = (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))
150	148, 73, 135, 149	frlmelbas 21698	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ∈ V)
155	154, 147	elmapd 8790	. . . . . . . . . . . . 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 6671	. . . . . . . . . . 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 20656	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
162	1, 161	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ Ring)
163		ringmnd 20163	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
164	162, 163	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Mnd)
165		subrgsubg 20497	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ∈ (SubGrp‘𝐸))
166	9, 165	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ (SubGrp‘𝐸))
167		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
168	167	subg0cl 19048	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (SubGrp‘𝐸) → (0_g‘𝐸) ∈ 𝑉)
169	166, 168	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑉)
170	46, 169	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑈)
171	5, 21, 167	ress0g 18671	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Mnd ∧ (0_g‘𝐸) ∈ 𝑈 ∧ 𝑈 ⊆ (Base‘𝐸)) → (0_g‘𝐸) = (0_g‘𝐹))
172	164, 170, 23, 171	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐹))
173	54	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐾) = (0_g‘(Scalar‘𝐶)))
174	11, 167	subrg0 20499	. . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐶)))
177	173, 175, 176	3eqtr4d 2774	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘(Scalar‘𝐴)))
178	172, 177	eqtr3d 2766	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘𝐹) = (0_g‘(Scalar‘𝐴)))
179	160, 178	breqtrrd 5130	. . . . . . . . . . 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 32698	. . . . . . . . 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 5128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)))
184		eqidd 2730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
185	156	fovcdmda 7540	. . . . . . . . . . . . . . . 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 6963	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖) = (𝑗𝑊𝑖))
188	187	oveq1d 7384	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))
189	119	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
190	189	oveqd 7386	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
191	188, 190	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
192	191	mpteq2dva 5195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
193	192	oveq2d 7385	. . . . . . . . . 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 33583	. . . . . . . . . . 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 33583	. . . . . . . . . . 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 2766	. . . . . . . . . 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 2764	. . . . . . . . 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 7180	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ V)
204		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (0_g‘𝐵) = (0_g‘𝐵)
205	117, 5	sralvec 33574	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
206	1, 104, 3, 205	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ LVec)
207		lveclmod 21045	. . . . . . . . . . . . . . . . . . . . 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 20847	. . . . . . . . . . . . . . . . . . 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 33573	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐵))
214		subrgsubg 20497	. . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐶 ∈ LMod)
218	61	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐶))
221		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
222	220, 221	sseldd 3944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
223	17, 111, 112, 113	lmodvscl 20816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ LMod ∧ (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
224	217, 219, 222, 223	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
225	125	oveqd 7386	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖))
226	225	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖))
227	32	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 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 7069	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)):𝑋⟶𝑈)
230	212, 23	srasca 21119	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑈) = (Scalar‘𝐵))
231	5, 230	eqtrid 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
232	231	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐹 = (Scalar‘𝐵))
233		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐵) = (Base‘𝐵)
234		ovexd 7404	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
235	20, 33	sstrd 3954	. . . . . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . . . 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 21116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
241	240	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
244		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐵)
245	144, 209, 232, 233, 234, 242, 204, 243, 244, 181	mptscmfsupp0 20865	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)) finSupp (0_g‘𝐵))
246	204, 211, 144, 216, 229, 245	gsumsubgcl 19834	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ 𝑈)
247	231	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
248	25, 247	eqtrd 2764	. . . . . . . . . . . . . . . . . 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 7069	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵)))
252	251	ffund 6674	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))))
253		fvexd 6855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐵)) ∈ V)
254		fconstmpt 5693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 × {(0_g‘(Scalar‘𝐴))}) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
255	254	eqeq2i 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
256		ovex 7402	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘𝑊𝑖) ∈ V
257	256	rgenw 3048	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) ∈ V
258		mpteqb 6969	. . . . . . . . . . . . . . . . . . . . 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 2971	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ ¬ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
262		df-ov 7372	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘𝑊𝑖) = (𝑊‘⟨𝑘, 𝑖⟩)
263	262	eqcomi 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖)
264	263	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖))
265	264	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) = (0_g‘(Scalar‘𝐴)) ↔ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
266	265	necon3abid 2961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
267	266	rexbidva 3155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ∃𝑖 ∈ 𝑋 ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
268		rexnal 3082	. . . . . . . . . . . . . . . . . . 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 3409	. . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → (𝑊‘𝑧) = (𝑊‘⟨𝑘, 𝑖⟩))
273	272	neeq1d 2984	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → ((𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴)) ↔ (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))))
274	273	dmrab 32476	. . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} = dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
276		fvexd 6855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) ∈ V)
277		suppvalfn 8124	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 supp (0_g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
279	160	fsuppimpd 9296	. . . . . . . . . . . . . . . . 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 9262	. . . . . . . . . . . . . . . 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 1914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖𝜑
285		nfcv 2891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖𝑌
286		nfmpt1 5201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖))
287		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑋 × {(0_g‘(Scalar‘𝐴))})
288	286, 287	nfne 3026	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})
289	288, 285	nfrabw 3440	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖{𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})}
290	285, 289	nfdif 4088	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑖(𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
291	290	nfcri 2883	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
292	284, 291	nfan 1899	. . . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → 𝑗 ∈ 𝑌)
295	293	eldifbd 3924	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑗 → (𝑘𝑊𝑖) = (𝑗𝑊𝑖))
297	296	mpteq2dv 5196	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑗 → (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
298	297	neeq1d 2984	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑗 → ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})))
299	298	elrab 3656	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2929	. . . . . . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗𝑊𝑖) ∈ V
306	305	rgenw 3048	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) ∈ V
307		mpteqb 6969	. . . . . . . . . . . . . . . . . . . . . . 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 3227	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
311	310	oveq1d 7384	. . . . . . . . . . . . . . . . . . 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 33578	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐵))
313	117, 1, 3	drgext0gsca 33580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐵) = (0_g‘(Scalar‘𝐵)))
314	312, 177, 313	3eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐵)))
315	314	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → 𝐵 ∈ LMod)
318	294, 242	syldanl 602	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐵))
319		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Scalar‘𝐵) = (Scalar‘𝐵)
320		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘(Scalar‘𝐵)) = (0_g‘(Scalar‘𝐵))
321	233, 319, 244, 320, 204	lmod0vs 20833	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ LMod ∧ 𝑖 ∈ (Base‘𝐵)) → ((0_g‘(Scalar‘𝐵))( ·_𝑠 ‘𝐵)𝑖) = (0_g‘𝐵))
322	317, 318, 321	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . 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 5194	. . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . 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 19699	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Abel → 𝐵 ∈ Grp)
327		grpmnd 18854	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Grp → 𝐵 ∈ Mnd)
328	208, 210, 326, 327	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ Mnd)
329	204	gsumz 18745	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Mnd ∧ 𝑋 ∈ (LBasis‘𝐶)) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ (0_g‘𝐵))) = (0_g‘𝐵))
330	328, 16, 329	syl2anc 584	. . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . 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 8156	. . . . . . . . . . . . . 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 9307	. . . . . . . . . . . . . 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 1380	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)))
337		eqidd 2730	. . . . . . . . . . . . . . . . . 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 7404	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V)
339	337, 338	fvmpt2d 6963	. . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . 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 5195	. . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
345	344	oveqd 7386	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))
346	345	mpteq2dva 5195	. . . . . . . . . . . . . . . . . . . . 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 7386	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
349	348	mpteq2dv 5196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
350	346, 349	eqtr3d 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
351	350	oveq2d 7385	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
352		eqidd 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 = 𝑗)
353	343, 351, 352	oveq123d 7390	. . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 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 5195	. . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . 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 21125	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ Ring ∧ 𝑉 ⊆ (Base‘𝐸)) → 𝐴 ∈ Ring)
359	162, 36, 358	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ Ring)
360		ringcmn 20202	. . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
364	233, 363	lbsss 21016	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
365	142, 364	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
366	365, 240	sseqtrrd 3981	. . . . . . . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑗 ∈ (Base‘𝐸))
370	21, 67	ringcl 20170	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐸))
371	362, 238, 369, 370	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴))
374	373	ralrimivva 3178	. . . . . . . . . . . . . . . . . . . 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 8026	. . . . . . . . . . . . . . . . . . . 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 20839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐴))
379	72, 73, 75, 74, 15, 156, 377, 147, 134, 135, 160	lcomfsupp 20840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) finSupp (0_g‘𝐴))
380	74, 134, 361, 142, 16, 378, 379	gsumxp 19890	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
385	384, 36	eqsstrrd 3979	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
386	61, 385	eqsstrd 3978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
387	386, 37	sseqtrd 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7541	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
395	238	3impb 1114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
396	369	3impb 1114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸))
397	21, 67	ringass 20173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) = ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗)))
399	398	mpoeq3dva 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7404	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
401		ovexd 7404	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑖(._r‘𝐸)𝑗) ∈ V)
402		fnov 7500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8044	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗))))
406	43	ofeqd 7635	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ∘_f (._r‘𝐸) = ∘_f ( ·_𝑠 ‘𝐴))
407	406	oveqd 7386	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷))
408	399, 405, 407	3eqtr2rd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗)))
409	408	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 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 7386	. . . . . . . . . . . . . . . . . . . . . . . 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 7404	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) ∈ V)
413		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7516	. . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖) = (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
417	416	mpteq2dva 5195	. . . . . . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸))
421	162	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
422	385	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
423	422, 219	sseldd 3944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
424	21, 67	ringcl 20170	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ (𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) ∈ (Base‘𝐸))
425	421, 423, 239, 424	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 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 5134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) finSupp (0_g‘𝐸))
428	21, 167, 67, 419, 144, 420, 425, 427	gsummulc1 20236	. . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . 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 7180	. . . . . . . . . . . . . . . . . . . . 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 7180	. . . . . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . 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 5195	. . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . 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 33578	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐴))
446	444, 445, 312	3eqtr2d 2770	. . . . . . . . . . . . . . 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 33583	. . . . . . . . . . . . . . . 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 33583	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
454	453	nfeq2 2909	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
455		fveq1 6839	. . . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . 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 5194	. . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 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 21775	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
465	464, 142	sselid 3941	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (LIndS‘𝐵))
466		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
467	233, 466, 319, 244, 204, 320	islinds5 33331	. . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘(Scalar‘𝐵)) ∈ V)
471		elmapg 8789	. . . . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . 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 699	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑌 × {(0_g‘(Scalar‘𝐵))}))
476		fconstmpt 5693	. . . . . . . . . . . 12 ⊢ (𝑌 × {(0_g‘(Scalar‘𝐵))}) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵)))
477	475, 476	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵))))
478		ovex 7402	. . . . . . . . . . . . 13 ⊢ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
479	478	rgenw 3048	. . . . . . . . . . . 12 ⊢ ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
480		mpteqb 6969	. . . . . . . . . . . 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 3227	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
484	312, 445, 313	3eqtr3rd 2773	. . . . . . . . . 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 2768	. . . . . . . 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 7069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴)))
489		fvexd 6855	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐴)) ∈ V)
490	489, 144	elmapd 8790	. . . . . . . . 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 5112	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 finSupp (0_g‘(Scalar‘𝐴)) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴))))
494		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝑌)
495		nfmpt1 5201	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
496	495	nfeq2 2909	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
497	494, 496	nfan 1899	. . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑤‘𝑖) = ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖))
500	499	oveq1d 7384	. . . . . . . . . . . . 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 5194	. . . . . . . . . . . 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 7385	. . . . . . . . . . 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 2731	. . . . . . . . . 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 632	. . . . . . . . 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 2731	. . . . . . . . 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 3577	. . . . . . 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 2780	. . . . 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 3125	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
512		eqidd 2730	. . . 4 ⊢ ((𝑗 = 𝑘 ∧ 𝑖 = 𝑙) → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴)))
513		fvexd 6855	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
514		fvexd 6855	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌 ∧ 𝑙 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
515	157, 512, 513, 514	fnmpoovd 8043	. . 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 7486	. 2 ⊢ ((𝑌 × 𝑋) × {(0_g‘(Scalar‘𝐴))}) = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
518	516, 517	eqtr4di 2782	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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3402 Vcvv 3444 ∖ cdif 3908 ⊆ wss 3911 {csn 4585 ⟨cop 4591 class class class wbr 5102 ↦ cmpt 5183 × cxp 5629 dom cdm 5631 Fun wfun 6493 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 ∘_f cof 7631 supp csupp 8116 ↑_m cmap 8776 Fincfn 8895 finSupp cfsupp 9288 Basecbs 17155 ↾_s cress 17176 +_gcplusg 17196 ._rcmulr 17197 Scalarcsca 17199 ·_𝑠 cvsca 17200 0_gc0g 17378 Σ_g cgsu 17379 Mndcmnd 18643 Grpcgrp 18847 SubGrpcsubg 19034 CMndccmn 19694 Abelcabl 19695 Ringcrg 20153 SubRingcsubrg 20489 DivRingcdr 20649 LModclmod 20798 LSubSpclss 20869 LBasisclbs 21013 LVecclvec 21041 subringAlg csra 21110 freeLMod cfrlm 21688 LIndSclinds 21747

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

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-fzo 13592 df-seq 13943 df-hash 14272 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17380 df-gsum 17381 df-prds 17386 df-pws 17388 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-nzr 20433 df-subrng 20466 df-subrg 20490 df-drng 20651 df-lmod 20800 df-lss 20870 df-lsp 20910 df-lmhm 20961 df-lbs 21014 df-lvec 21042 df-sra 21112 df-rgmod 21113 df-dsmm 21674 df-frlm 21689 df-uvc 21725 df-lindf 21748 df-linds 21749

This theorem is referenced by: fedgmul 33620

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