fedgmullem2 - Mathbox for Thierry Arnoux

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Theorem fedgmullem2 33657

Description: Lemma for fedgmul 33658. (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 20614	. . . . . . . . . . . . . 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 33614	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
13	1, 2, 9, 12	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ LVec)
14		lveclmod 21122	. . . . . . . . . 10 ⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ LMod)
16		fedgmullem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))
17		eqid 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		eqid 2734	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
19	17, 18	lbsss 21093	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
20	16, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶))
21		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
22	21	subrgss 20588	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
23	3, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
24	5, 21	ressbas2 17282	. . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
29	28	subrgss 20588	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
30	4, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
31	27, 30	srabase 21194	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
32	25, 31	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
33	32, 23	eqsstrrd 4034	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
34	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝐸)‘𝑉))
35	21	subrgss 20588	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
36	9, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
37	34, 36	srabase 21194	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
38	33, 37	sseqtrd 4035	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐴))
39	20, 38	sstrd 4005	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐴))
40	34, 3, 36	srasubrg 33613	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐴))
41		subrgsubg 20593	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubRing‘𝐴) → 𝑈 ∈ (SubGrp‘𝐴))
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐴))
43	10, 1, 9	drgextvsca 33619	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐴))
44	43	oveqdr 7458	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
49		ressabs 17294	. . . . . . . . . . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 ↾_s 𝑉) = ((𝐸 ↾_s 𝑈) ↾_s 𝑉)
52	50, 51, 11	3eqtr4g 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = 𝐾)
53	27, 30	srasca 21200	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) = (Scalar‘𝐶))
54	52, 53	eqtr3d 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
55	54	fveq2d 6910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
56	11, 21	ressbas2 17282	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
57	36, 56	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
58	34, 36	srasca 21200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑉) = (Scalar‘𝐴))
59	11, 58	eqtrid 2786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
60	52, 53, 59	3eqtr3rd 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
61	60	fveq2d 6910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
62	55, 57, 61	3eqtr4d 2784	. . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑉)
65	47, 64	sseldd 3995	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
66		simprr 773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
67		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐸) = (._r‘𝐸)
68	67	subrgmcl 20600	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑈)
69	45, 65, 66, 68	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑈)
70	44, 69	eqeltrrd 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ 𝑈)) → (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)
71	70	ralrimivva 3199	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ 𝑈 (𝑥( ·_𝑠 ‘𝐴)𝑦) ∈ 𝑈)
72		eqid 2734	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
73		eqid 2734	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
74		eqid 2734	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
75		eqid 2734	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
76		eqid 2734	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝐴) = (LSubSp‘𝐴)
77	72, 73, 74, 75, 76	islss4 20977	. . . . . . . . . . . 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 837	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝐴))
80	20, 32	sseqtrrd 4036	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑈)
81	18	lbslinds 21870	. . . . . . . . . . . 12 ⊢ (LBasis‘𝐶) ⊆ (LIndS‘𝐶)
82	81, 16	sselid 3992	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶))
83	23, 37	sseqtrd 4035	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐴))
84		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ↾_s 𝑈) = (𝐴 ↾_s 𝑈)
85	84, 74	ressbas2 17282	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(𝐴 ↾_s 𝑈)))
88	84, 72	resssca 17388	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Scalar‘𝐶) = (Scalar‘(𝐴 ↾_s 𝑈)))
91	90	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘(𝐴 ↾_s 𝑈))))
92	90	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘(𝐴 ↾_s 𝑈))))
93		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐸) = (+_g‘𝐸)
94	5, 93	ressplusg 17335	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (+_g‘𝐸) = (+_g‘𝐹))
95	3, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐹))
96	34, 36	sraaddg 21196	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘𝐴))
97	27, 30	sraaddg 21196	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘𝐹) = (+_g‘𝐶))
98	95, 96, 97	3eqtr3rd 2783	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘𝐴))
99		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝐴) = (+_g‘𝐴)
100	84, 99	ressplusg 17335	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 13 ⊢ (𝜑 → (+_g‘𝐶) = (+_g‘(𝐴 ↾_s 𝑈)))
103	102	oveqdr 7458	. . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾_s 𝑉) ∈ DivRing)
106		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾_s 𝑉) = (𝐹 ↾_s 𝑉)
107	26, 106	sralvec 33614	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾_s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
108	104, 105, 4, 107	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
109		lveclmod 21122	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
110	108, 109	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ LMod)
111		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
112		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘𝐶)
113		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
114	17, 111, 112, 113	lmodvscl 20892	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥( ·_𝑠 ‘𝐶)𝑦) ∈ (Base‘𝐶))
115	114	3expb 1119	. . . . . . . . . . . . 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 33619	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = ( ·_𝑠 ‘𝐵))
119	43, 118	eqtr3d 2776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐵))
120	84, 75	ressvsca 17389	. . . . . . . . . . . . . . 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 17352	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (._r‘𝐸) = (._r‘𝐹))
123	3, 122	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐸) = (._r‘𝐹))
124	26, 104, 4	drgextvsca 33619	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (._r‘𝐹) = ( ·_𝑠 ‘𝐶))
125	123, 118, 124	3eqtr3d 2782	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐶))
126	119, 121, 125	3eqtr3rd 2783	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘(𝐴 ↾_s 𝑈)))
127	126	oveqdr 7458	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·_𝑠 ‘𝐶)𝑦) = (𝑥( ·_𝑠 ‘(𝐴 ↾_s 𝑈))𝑦))
128		ovexd 7465	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ↾_s 𝑈) ∈ V)
129	87, 91, 92, 103, 116, 127, 108, 128	lindspropd 33390	. . . . . . . . . . 11 ⊢ (𝜑 → (LIndS‘𝐶) = (LIndS‘(𝐴 ↾_s 𝑈)))
130	82, 129	eleqtrd 2840	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘(𝐴 ↾_s 𝑈)))
131	76, 84	lsslinds 21868	. . . . . . . . . . 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 1372	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐴))
134		eqid 2734	. . . . . . . . . . 11 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
135		eqid 2734	. . . . . . . . . . 11 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
136	74, 73, 72, 75, 134, 135	islinds5 33374	. . . . . . . . . 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 838	. . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
141		fvexd 6921	. . . . . . . . . 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 6921	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Scalar‘𝐴) ∈ V)
147	142, 16	xpexd 7769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ V)
148		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)) = ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))
149		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) = (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))
150	148, 73, 135, 149	frlmelbas 21793	. . . . . . . . . . . . . . . 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 6921	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ∈ V)
155	154, 147	elmapd 8878	. . . . . . . . . . . . 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 6737	. . . . . . . . . . 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 20752	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
162	1, 161	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ Ring)
163		ringmnd 20260	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
164	162, 163	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Mnd)
165		subrgsubg 20593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ∈ (SubGrp‘𝐸))
166	9, 165	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ (SubGrp‘𝐸))
167		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
168	167	subg0cl 19164	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (SubGrp‘𝐸) → (0_g‘𝐸) ∈ 𝑉)
169	166, 168	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑉)
170	46, 169	sseldd 3995	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝑈)
171	5, 21, 167	ress0g 18787	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Mnd ∧ (0_g‘𝐸) ∈ 𝑈 ∧ 𝑈 ⊆ (Base‘𝐸)) → (0_g‘𝐸) = (0_g‘𝐹))
172	164, 170, 23, 171	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐹))
173	54	fveq2d 6910	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝐾) = (0_g‘(Scalar‘𝐶)))
174	11, 167	subrg0 20595	. . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐶)))
177	173, 175, 176	3eqtr4d 2784	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘(Scalar‘𝐴)))
178	172, 177	eqtr3d 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘𝐹) = (0_g‘(Scalar‘𝐴)))
179	160, 178	breqtrrd 5175	. . . . . . . . . . 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 32742	. . . . . . . . 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 5173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴)))
184		eqidd 2735	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
185	156	fovcdmda 7603	. . . . . . . . . . . . . . . 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 7028	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖) = (𝑗𝑊𝑖))
188	187	oveq1d 7445	. . . . . . . . . . . . 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 7447	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
191	188, 190	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
192	191	mpteq2dva 5247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
193	192	oveq2d 7446	. . . . . . . . . 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 33623	. . . . . . . . . . 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 33623	. . . . . . . . . . 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 2776	. . . . . . . . . 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 2774	. . . . . . . . 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 7243	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ V)
204		eqid 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (0_g‘𝐵) = (0_g‘𝐵)
205	117, 5	sralvec 33614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
206	1, 104, 3, 205	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ LVec)
207		lveclmod 21122	. . . . . . . . . . . . . . . . . . . . 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 20923	. . . . . . . . . . . . . . . . . . 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 33613	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐵))
214		subrgsubg 20593	. . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
223	17, 111, 112, 113	lmodvscl 20892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ LMod ∧ (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
224	217, 219, 222, 223	syl3anc 1370	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
225	125	oveqd 7447	. . . . . . . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖) ∈ 𝑈)
229	228	fmpttd 7134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)):𝑋⟶𝑈)
230	212, 23	srasca 21200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 ↾_s 𝑈) = (Scalar‘𝐵))
231	5, 230	eqtrid 2786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
232	231	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐹 = (Scalar‘𝐵))
233		eqid 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐵) = (Base‘𝐵)
234		ovexd 7465	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
235	20, 33	sstrd 4005	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐸))
236	235	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝐸))
237		simprr 773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ 𝑋)
238	236, 237	sseldd 3995	. . . . . . . . . . . . . . . . . . . . 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 21194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
241	240	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐵))
242	239, 241	eleqtrd 2840	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐵))
243		eqid 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
244		eqid 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ·_𝑠 ‘𝐵) = ( ·_𝑠 ‘𝐵)
245	144, 209, 232, 233, 234, 242, 204, 243, 244, 181	mptscmfsupp0 20941	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)) finSupp (0_g‘𝐵))
246	204, 211, 144, 216, 229, 245	gsumsubgcl 19952	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ 𝑈)
247	231	fveq2d 6910	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
248	25, 247	eqtrd 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘(Scalar‘𝐵)))
249	248	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 = (Base‘(Scalar‘𝐵)))
250	246, 249	eleqtrd 2840	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ (Base‘(Scalar‘𝐵)))
251	250	fmpttd 7134	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵)))
252	251	ffund 6740	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))))
253		fvexd 6921	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘(Scalar‘𝐵)) ∈ V)
254		fconstmpt 5750	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 × {(0_g‘(Scalar‘𝐴))}) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
255	254	eqeq2i 2747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴))))
256		ovex 7463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘𝑊𝑖) ∈ V
257	256	rgenw 3062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) ∈ V
258		mpteqb 7034	. . . . . . . . . . . . . . . . . . . . 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 2984	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ ¬ ∀𝑖 ∈ 𝑋 (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
262		df-ov 7433	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘𝑊𝑖) = (𝑊‘⟨𝑘, 𝑖⟩)
263	262	eqcomi 2743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖)
264	263	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖))
265	264	eqeq1d 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) = (0_g‘(Scalar‘𝐴)) ↔ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
266	265	necon3abid 2974	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
267	266	rexbidva 3174	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (∃𝑖 ∈ 𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴)) ↔ ∃𝑖 ∈ 𝑋 ¬ (𝑘𝑊𝑖) = (0_g‘(Scalar‘𝐴))))
268		rexnal 3097	. . . . . . . . . . . . . . . . . . 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 3439	. . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → (𝑊‘𝑧) = (𝑊‘⟨𝑘, 𝑖⟩))
273	272	neeq1d 2997	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑘, 𝑖⟩ → ((𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴)) ↔ (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0_g‘(Scalar‘𝐴))))
274	273	dmrab 32524	. . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} = dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
276		fvexd 6921	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0_g‘(Scalar‘𝐴)) ∈ V)
277		suppvalfn 8191	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 supp (0_g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))})
279	160	fsuppimpd 9406	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 supp (0_g‘(Scalar‘𝐴))) ∈ Fin)
280	278, 279	eqeltrrd 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊‘𝑧) ≠ (0_g‘(Scalar‘𝐴))} ∈ Fin)
281		dmfi 9372	. . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})} ∈ Fin)
284		nfv 1911	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖𝜑
285		nfcv 2902	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖𝑌
286		nfmpt1 5255	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖))
287		nfcv 2902	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖(𝑋 × {(0_g‘(Scalar‘𝐴))})
288	286, 287	nfne 3040	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})
289	288, 285	nfrabw 3472	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖{𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})}
290	285, 289	nfdif 4138	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑖(𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
291	290	nfcri 2894	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})
292	284, 291	nfan 1896	. . . . . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) → 𝑗 ∈ 𝑌)
295	293	eldifbd 3975	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑗 → (𝑘𝑊𝑖) = (𝑗𝑊𝑖))
297	296	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑗 → (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
298	297	neeq1d 2997	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑗 → ((𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))}) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})))
299	298	elrab 3694	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2941	. . . . . . . . . . . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . . . . . . . . . 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 7463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗𝑊𝑖) ∈ V
306	305	rgenw 3062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) ∈ V
307		mpteqb 7034	. . . . . . . . . . . . . . . . . . . . . . 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 3248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ {𝑘 ∈ 𝑌 ∣ (𝑖 ∈ 𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0_g‘(Scalar‘𝐴))})})) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
311	310	oveq1d 7445	. . . . . . . . . . . . . . . . . . 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 33618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐵))
313	117, 1, 3	drgext0gsca 33620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0_g‘𝐵) = (0_g‘(Scalar‘𝐵)))
314	312, 177, 313	3eqtr3d 2782	. . . . . . . . . . . . . . . . . . . . 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 7445	. . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Scalar‘𝐵) = (Scalar‘𝐵)
320		eqid 2734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘(Scalar‘𝐵)) = (0_g‘(Scalar‘𝐵))
321	233, 319, 244, 320, 204	lmod0vs 20909	. . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . 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 5245	. . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . 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 19817	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Abel → 𝐵 ∈ Grp)
327		grpmnd 18970	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Grp → 𝐵 ∈ Mnd)
328	208, 210, 326, 327	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ Mnd)
329	204	gsumz 18861	. . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 8223	. . . . . . . . . . . . . 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 9417	. . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) finSupp (0_g‘(Scalar‘𝐵)))
337		eqidd 2735	. . . . . . . . . . . . . . . . . 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 7465	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V)
339	337, 338	fvmpt2d 7028	. . . . . . . . . . . . . . . . 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 7445	. . . . . . . . . . . . . . . 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 5247	. . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))
346	345	mpteq2dva 5247	. . . . . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) = ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))
349	348	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
350	346, 349	eqtr3d 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖)) = (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))
351	350	oveq2d 7446	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐴)𝑖))) = (𝐴 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
352		eqidd 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 = 𝑗)
353	343, 351, 352	oveq123d 7451	. . . . . . . . . . . . . . . . . 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 7445	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 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 5247	. . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . 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 21210	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ Ring ∧ 𝑉 ⊆ (Base‘𝐸)) → 𝐴 ∈ Ring)
359	162, 36, 358	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ Ring)
360		ringcmn 20295	. . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
364	233, 363	lbsss 21093	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
365	142, 364	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
366	365, 240	sseqtrrd 4036	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸))
367	366	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑌 ⊆ (Base‘𝐸))
368		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑗 ∈ 𝑌)
369	367, 368	sseldd 3995	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑗 ∈ (Base‘𝐸))
370	21, 67	ringcl 20267	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐸))
371	362, 238, 369, 370	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(._r‘𝐸)𝑗) ∈ (Base‘𝐴))
374	373	ralrimivva 3199	. . . . . . . . . . . . . . . . . . . 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 8091	. . . . . . . . . . . . . . . . . . . 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 20915	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐴))
379	72, 73, 75, 74, 15, 156, 377, 147, 134, 135, 160	lcomfsupp 20916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷) finSupp (0_g‘𝐴))
380	74, 134, 361, 142, 16, 378, 379	gsumxp 20008	. . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
383	156	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
384	57, 55	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
385	384, 36	eqsstrrd 4034	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
386	61, 385	eqsstrd 4033	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
387	386, 37	sseqtrd 4035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
388	387	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
389	383, 388	fssd 6753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘𝐴))
390		simp2 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
391		simp3 1137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
392	389, 390, 391	fovcdmd 7604	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐴))
393	37	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴))
394	392, 393	eleqtrrd 2841	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) = ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗)))
399	398	mpoeq3dva 7509	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ V)
401		ovexd 7465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑖(._r‘𝐸)𝑗) ∈ V)
402		fnov 7563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8111	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)(𝑖(._r‘𝐸)𝑗))))
406	43	ofeqd 7698	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ∘_f (._r‘𝐸) = ∘_f ( ·_𝑠 ‘𝐴))
407	406	oveqd 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑊 ∘_f (._r‘𝐸)𝐷) = (𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷))
408	399, 405, 407	3eqtr2rd 2781	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
412		ovexd 7465	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗) ∈ V)
413		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7579	. . . . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑊 ∘_f ( ·_𝑠 ‘𝐴)𝐷)𝑖) = (((𝑗𝑊𝑖)(._r‘𝐸)𝑖)(._r‘𝐸)𝑗))
417	416	mpteq2dva 5247	. . . . . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
424	21, 67	ringcl 20267	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 ∈ Ring ∧ (𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → ((𝑗𝑊𝑖)(._r‘𝐸)𝑖) ∈ (Base‘𝐸))
425	421, 423, 239, 424	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) finSupp (0_g‘𝐸))
428	21, 167, 67, 419, 144, 420, 425, 427	gsummulc1 20329	. . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . 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 7243	. . . . . . . . . . . . . . . . . . . . 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 33032	. . . . . . . . . . . . . . . . . . . 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 7243	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)(._r‘𝐸)𝑖)) ∈ V)
435	10, 434, 194, 431, 432	gsumsra 33032	. . . . . . . . . . . . . . . . . . . . . 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 7445	. . . . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . . . 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 7451	. . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . . . 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 5247	. . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . 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 33618	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘𝐴))
446	444, 445, 312	3eqtr2d 2780	. . . . . . . . . . . . . . 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 33623	. . . . . . . . . . . . . . . 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 33623	. . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 5150	. . . . . . . . . . . . . . . 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 5255	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
454	453	nfeq2 2920	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑏 = (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))))
455		fveq1 6905	. . . . . . . . . . . . . . . . . . . . 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 7445	. . . . . . . . . . . . . . . . . . . 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 5245	. . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . 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 21870	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
465	464, 142	sselid 3992	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (LIndS‘𝐵))
466		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
467	233, 466, 319, 244, 204, 320	islinds5 33374	. . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 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 6921	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘(Scalar‘𝐵)) ∈ V)
471		elmapg 8877	. . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑_m 𝑌))
474	463, 469, 473	rspcdva 3622	. . . . . . . . . . . . 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 5750	. . . . . . . . . . . 12 ⊢ (𝑌 × {(0_g‘(Scalar‘𝐵))}) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵)))
477	475, 476	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖)))) = (𝑗 ∈ 𝑌 ↦ (0_g‘(Scalar‘𝐵))))
478		ovex 7463	. . . . . . . . . . . . 13 ⊢ (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
479	478	rgenw 3062	. . . . . . . . . . . 12 ⊢ ∀𝑗 ∈ 𝑌 (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) ∈ V
480		mpteqb 7034	. . . . . . . . . . . 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 3248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐵 Σ_g (𝑖 ∈ 𝑋 ↦ ((𝑗𝑊𝑖)( ·_𝑠 ‘𝐵)𝑖))) = (0_g‘(Scalar‘𝐵)))
484	312, 445, 313	3eqtr3rd 2783	. . . . . . . . . 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 2778	. . . . . . . 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 7134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴)))
489		fvexd 6921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐴)) ∈ V)
490	489, 144	elmapd 8878	. . . . . . . . 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 5157	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 finSupp (0_g‘(Scalar‘𝐴)) ↔ (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0_g‘(Scalar‘𝐴))))
494		nfv 1911	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝑌)
495		nfmpt1 5255	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
496	495	nfeq2 2920	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))
497	494, 496	nfan 1896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
498		simplr 769	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖)))
499	498	fveq1d 6908	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑤 = (𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑤‘𝑖) = ((𝑖 ∈ 𝑋 ↦ (𝑗𝑊𝑖))‘𝑖))
500	499	oveq1d 7445	. . . . . . . . . . . . 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 5245	. . . . . . . . . . . 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 7446	. . . . . . . . . . 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 2736	. . . . . . . . . 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 2736	. . . . . . . . 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 3613	. . . . . . 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 2790	. . . . 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 3143	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑗𝑊𝑖) = (0_g‘(Scalar‘𝐴)))
512		eqidd 2735	. . . 4 ⊢ ((𝑗 = 𝑘 ∧ 𝑖 = 𝑙) → (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴)))
513		fvexd 6921	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
514		fvexd 6921	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌 ∧ 𝑙 ∈ 𝑋) → (0_g‘(Scalar‘𝐴)) ∈ V)
515	157, 512, 513, 514	fnmpoovd 8110	. . 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 7549	. 2 ⊢ ((𝑌 × 𝑋) × {(0_g‘(Scalar‘𝐴))}) = (𝑘 ∈ 𝑌, 𝑙 ∈ 𝑋 ↦ (0_g‘(Scalar‘𝐴)))
518	516, 517	eqtr4di 2792	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 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3432 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 ⟨cop 4636 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 dom cdm 5688 Fun wfun 6556 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ∘_f cof 7694 supp csupp 8183 ↑_m cmap 8864 Fincfn 8983 finSupp cfsupp 9398 Basecbs 17244 ↾_s cress 17273 +_gcplusg 17297 ._rcmulr 17298 Scalarcsca 17300 ·_𝑠 cvsca 17301 0_gc0g 17485 Σ_g cgsu 17486 Mndcmnd 18759 Grpcgrp 18963 SubGrpcsubg 19150 CMndccmn 19812 Abelcabl 19813 Ringcrg 20250 SubRingcsubrg 20585 DivRingcdr 20745 LModclmod 20874 LSubSpclss 20946 LBasisclbs 21090 LVecclvec 21118 subringAlg csra 21187 freeLMod cfrlm 21783 LIndSclinds 21842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-nzr 20529 df-subrng 20562 df-subrg 20586 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lmhm 21038 df-lbs 21091 df-lvec 21119 df-sra 21189 df-rgmod 21190 df-dsmm 21769 df-frlm 21784 df-uvc 21820 df-lindf 21843 df-linds 21844

This theorem is referenced by: fedgmul 33658

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