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Theorem mdetrsca 22590

Description: The determinant function is homogeneous for each row: If the matrices 𝑋 and 𝑍 are identical except for the 𝐼-th row, and the 𝐼-th row of the matrix 𝑋 is the componentwise product of the 𝐼-th row of the matrix 𝑍 and the scalar 𝑌, then the determinant of 𝑋 is the determinant of 𝑍 multiplied by 𝑌. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)

**Hypotheses**
Ref	Expression
mdetrsca.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetrsca.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetrsca.b	⊢ 𝐵 = (Base‘𝐴)
mdetrsca.k	⊢ 𝐾 = (Base‘𝑅)
mdetrsca.t	⊢ · = (._r‘𝑅)
mdetrsca.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mdetrsca.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mdetrsca.y	⊢ (𝜑 → 𝑌 ∈ 𝐾)
mdetrsca.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
mdetrsca.i	⊢ (𝜑 → 𝐼 ∈ 𝑁)
mdetrsca.eq	⊢ (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁))))
mdetrsca.ne	⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))

**Assertion**
Ref	Expression
mdetrsca	⊢ (𝜑 → (𝐷‘𝑋) = (𝑌 · (𝐷‘𝑍)))

Proof of Theorem mdetrsca Dummy variables 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetrsca.eq	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁))))
2	1	oveqd 7377	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)))
3	2	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)))
4		mdetrsca.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ 𝑁)
5	4	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝐼 ∈ 𝑁)
6		snidg 4595	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ 𝑁 → 𝐼 ∈ {𝐼})
7	5, 6	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝐼 ∈ {𝐼})
8		eqid 2741	. . . . . . . . . . . . . . . . 17 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
9		eqid 2741	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
10	8, 9	symgbasf1o 19345	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) → 𝑝:𝑁–1-1-onto→𝑁)
11	10	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁–1-1-onto→𝑁)
12		f1of 6771	. . . . . . . . . . . . . . 15 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝:𝑁⟶𝑁)
13	11, 12	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁⟶𝑁)
14	13, 5	ffvelcdmd 7030	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑝‘𝐼) ∈ 𝑁)
15		ovres 7526	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ {𝐼} ∧ (𝑝‘𝐼) ∈ 𝑁) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑋(𝑝‘𝐼)))
16	7, 14, 15	syl2anc 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑋(𝑝‘𝐼)))
17	7, 14	opelxpd 5660	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ⟨𝐼, (𝑝‘𝐼)⟩ ∈ ({𝐼} × 𝑁))
18		snfi 8984	. . . . . . . . . . . . . . . 16 ⊢ {𝐼} ∈ Fin
19		mdetrsca.x	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
20		mdetrsca.a	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 = (𝑁 Mat 𝑅)
21		mdetrsca.b	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = (Base‘𝐴)
22	20, 21	matrcl 22399	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
23	19, 22	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
24	23	simpld 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ Fin)
25	24	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 ∈ Fin)
26		xpfi 9224	. . . . . . . . . . . . . . . 16 ⊢ (({𝐼} ∈ Fin ∧ 𝑁 ∈ Fin) → ({𝐼} × 𝑁) ∈ Fin)
27	18, 25, 26	sylancr 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} × 𝑁) ∈ Fin)
28		mdetrsca.y	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
29	28	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑌 ∈ 𝐾)
30		mdetrsca.z	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
31		mdetrsca.k	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐾 = (Base‘𝑅)
32	20, 31, 21	matbas2i 22409	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ 𝐵 → 𝑍 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
33		elmapi 8790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑍:(𝑁 × 𝑁)⟶𝐾)
34	30, 32, 33	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑍:(𝑁 × 𝑁)⟶𝐾)
35	34	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑍:(𝑁 × 𝑁)⟶𝐾)
36	35	ffnd 6660	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑍 Fn (𝑁 × 𝑁))
37	5	snssd 4721	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → {𝐼} ⊆ 𝑁)
38		xpss1 5640	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐼} ⊆ 𝑁 → ({𝐼} × 𝑁) ⊆ (𝑁 × 𝑁))
39	37, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} × 𝑁) ⊆ (𝑁 × 𝑁))
40	36, 39	fnssresd 6613	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑍 ↾ ({𝐼} × 𝑁)) Fn ({𝐼} × 𝑁))
41		eqidd 2742	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ ⟨𝐼, (𝑝‘𝐼)⟩ ∈ ({𝐼} × 𝑁)) → ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩) = ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩))
42	27, 29, 40, 41	ofc1 7652	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ ⟨𝐼, (𝑝‘𝐼)⟩ ∈ ({𝐼} × 𝑁)) → (((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))‘⟨𝐼, (𝑝‘𝐼)⟩) = (𝑌 · ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)))
43	17, 42	mpdan 694	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))‘⟨𝐼, (𝑝‘𝐼)⟩) = (𝑌 · ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)))
44		df-ov 7363	. . . . . . . . . . . . 13 ⊢ (𝐼((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)) = (((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))‘⟨𝐼, (𝑝‘𝐼)⟩)
45		df-ov 7363	. . . . . . . . . . . . . 14 ⊢ (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)
46	45	oveq2i 7371	. . . . . . . . . . . . 13 ⊢ (𝑌 · (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))) = (𝑌 · ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩))
47	43, 44, 46	3eqtr4g 2801	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)) = (𝑌 · (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))))
48	3, 16, 47	3eqtr3d 2784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) = (𝑌 · (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))))
49		ovres 7526	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ {𝐼} ∧ (𝑝‘𝐼) ∈ 𝑁) → (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑍(𝑝‘𝐼)))
50	7, 14, 49	syl2anc 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑍(𝑝‘𝐼)))
51	50	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑌 · (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))) = (𝑌 · (𝐼𝑍(𝑝‘𝐼))))
52	48, 51	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) = (𝑌 · (𝐼𝑍(𝑝‘𝐼))))
53	52	oveq1d 7375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((𝐼𝑋(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((𝑌 · (𝐼𝑍(𝑝‘𝐼))) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))))
54		mdetrsca.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ CRing)
55	54	crngringd 20222	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
56	55	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
57	35, 5, 14	fovcdmd 7532	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑍(𝑝‘𝐼)) ∈ 𝐾)
58		eqid 2741	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
59	58, 31	mgpbas 20121	. . . . . . . . . . 11 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
60	58	crngmgp 20217	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
61	54, 60	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
62	61	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
63		difssd 4070	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑁 ∖ {𝐼}) ⊆ 𝑁)
64	25, 63	ssfid 9173	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑁 ∖ {𝐼}) ∈ Fin)
65		eldifi 4064	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (𝑁 ∖ {𝐼}) → 𝑟 ∈ 𝑁)
66	34	ad2antrr 733	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑍:(𝑁 × 𝑁)⟶𝐾)
67		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑟 ∈ 𝑁)
68	13	ffvelcdmda 7029	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑝‘𝑟) ∈ 𝑁)
69	66, 67, 68	fovcdmd 7532	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
70	65, 69	sylan2 600	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
71	70	ralrimiva 3133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑟 ∈ (𝑁 ∖ {𝐼})(𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
72	59, 62, 64, 71	gsummptcl 19937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ 𝐾)
73		mdetrsca.t	. . . . . . . . . . 11 ⊢ · = (._r‘𝑅)
74	31, 73	ringass 20229	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝑌 ∈ 𝐾 ∧ (𝐼𝑍(𝑝‘𝐼)) ∈ 𝐾 ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ 𝐾)) → ((𝑌 · (𝐼𝑍(𝑝‘𝐼))) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))) = (𝑌 · ((𝐼𝑍(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))))))
75	56, 29, 57, 72, 74	syl13anc 1381	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((𝑌 · (𝐼𝑍(𝑝‘𝐼))) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))) = (𝑌 · ((𝐼𝑍(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))))))
76	53, 75	eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((𝐼𝑋(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))) = (𝑌 · ((𝐼𝑍(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))))))
77	58, 73	mgpplusg 20120	. . . . . . . . . 10 ⊢ · = (+_g‘(mulGrp‘𝑅))
78	20, 31, 21	matbas2i 22409	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
79		elmapi 8790	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐾)
80	19, 78, 79	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐾)
81	80	ad2antrr 733	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶𝐾)
82	81, 67, 68	fovcdmd 7532	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑟𝑋(𝑝‘𝑟)) ∈ 𝐾)
83		disjdif 4403	. . . . . . . . . . 11 ⊢ ({𝐼} ∩ (𝑁 ∖ {𝐼})) = ∅
84	83	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} ∩ (𝑁 ∖ {𝐼})) = ∅)
85		undif 4413	. . . . . . . . . . . 12 ⊢ ({𝐼} ⊆ 𝑁 ↔ ({𝐼} ∪ (𝑁 ∖ {𝐼})) = 𝑁)
86	37, 85	sylib 220	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} ∪ (𝑁 ∖ {𝐼})) = 𝑁)
87	86	eqcomd 2747	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 = ({𝐼} ∪ (𝑁 ∖ {𝐼})))
88	59, 77, 62, 25, 82, 84, 87	gsummptfidmsplit 19900	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟)))) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
89	62	cmnmndd 19774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ Mnd)
90	80	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑋:(𝑁 × 𝑁)⟶𝐾)
91	90, 5, 14	fovcdmd 7532	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) ∈ 𝐾)
92		id 22	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → 𝑟 = 𝐼)
93		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → (𝑝‘𝑟) = (𝑝‘𝐼))
94	92, 93	oveq12d 7378	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝐼 → (𝑟𝑋(𝑝‘𝑟)) = (𝐼𝑋(𝑝‘𝐼)))
95	59, 94	gsumsn 19924	. . . . . . . . . . 11 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑁 ∧ (𝐼𝑋(𝑝‘𝐼)) ∈ 𝐾) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟)))) = (𝐼𝑋(𝑝‘𝐼)))
96	89, 5, 91, 95	syl3anc 1380	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟)))) = (𝐼𝑋(𝑝‘𝐼)))
97		mdetrsca.ne	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
98	97	oveqd 7377	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)))
99	98	ad2antrr 733	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)))
100		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → 𝑟 ∈ (𝑁 ∖ {𝐼}))
101	65, 68	sylan2 600	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑝‘𝑟) ∈ 𝑁)
102		ovres 7526	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ (𝑁 ∖ {𝐼}) ∧ (𝑝‘𝑟) ∈ 𝑁) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
103	100, 101, 102	syl2anc 591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
104		ovres 7526	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ (𝑁 ∖ {𝐼}) ∧ (𝑝‘𝑟) ∈ 𝑁) → (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
105	100, 101, 104	syl2anc 591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
106	99, 103, 105	3eqtr3d 2784	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟𝑋(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
107	106	mpteq2dva 5168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))) = (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))
108	107	oveq2d 7376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))) = ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))))
109	96, 108	oveq12d 7378	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟)))) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))) = ((𝐼𝑋(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))))
110	88, 109	eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))) = ((𝐼𝑋(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))))
111	59, 77, 62, 25, 69, 84, 87	gsummptfidmsplit 19900	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))))
112	92, 93	oveq12d 7378	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → (𝑟𝑍(𝑝‘𝑟)) = (𝐼𝑍(𝑝‘𝐼)))
113	59, 112	gsumsn 19924	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑁 ∧ (𝐼𝑍(𝑝‘𝐼)) ∈ 𝐾) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) = (𝐼𝑍(𝑝‘𝐼)))
114	89, 5, 57, 113	syl3anc 1380	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) = (𝐼𝑍(𝑝‘𝐼)))
115	114	oveq1d 7375	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((𝐼𝑍(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))))
116	111, 115	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) = ((𝐼𝑍(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))))
117	116	oveq2d 7376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑌 · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = (𝑌 · ((𝐼𝑍(𝑝‘𝐼)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))))))
118	76, 110, 117	3eqtr4d 2786	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))) = (𝑌 · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))
119	118	oveq2d 7376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · (𝑌 · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
120	54	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ CRing)
121		zrhpsgnmhm 21563	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
122	55, 24, 121	syl2anc 591	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
123	9, 59	mhmf 18752	. . . . . . . . . . 11 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
124	122, 123	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
125	124	ffvelcdmda 7029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾)
126	31, 73	crngcom 20227	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · 𝑌) = (𝑌 · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)))
127	120, 125, 29, 126	syl3anc 1380	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · 𝑌) = (𝑌 · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)))
128	127	oveq1d 7375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · 𝑌) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((𝑌 · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))
129	69	ralrimiva 3133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑟 ∈ 𝑁 (𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
130	59, 62, 25, 129	gsummptcl 19937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ 𝐾)
131	31, 73	ringass 20229	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ 𝐾)) → (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · 𝑌) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · (𝑌 · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
132	56, 125, 29, 130, 131	syl13anc 1381	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · 𝑌) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · (𝑌 · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
133	31, 73	ringass 20229	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑌 ∈ 𝐾 ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾 ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ 𝐾)) → ((𝑌 · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = (𝑌 · ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
134	56, 29, 125, 130, 133	syl13anc 1381	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((𝑌 · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = (𝑌 · ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
135	128, 132, 134	3eqtr3d 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · (𝑌 · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) = (𝑌 · ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
136	119, 135	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))) = (𝑌 · ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
137	136	mpteq2dva 5168	. . . 4 ⊢ (𝜑 → (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ (𝑌 · ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))))
138	137	oveq2d 7376	. . 3 ⊢ (𝜑 → (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ (𝑌 · ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))))
139		eqid 2741	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
140	8, 9	symgbasfi 19349	. . . . 5 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
141	24, 140	syl 17	. . . 4 ⊢ (𝜑 → (Base‘(SymGrp‘𝑁)) ∈ Fin)
142	31, 73, 56, 125, 130	ringcld 20236	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) ∈ 𝐾)
143		eqid 2741	. . . . 5 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))
144		ovexd 7395	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) ∈ V)
145		fvexd 6846	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
146	143, 141, 144, 145	fsuppmptdm 9283	. . . 4 ⊢ (𝜑 → (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) finSupp (0_g‘𝑅))
147	31, 139, 73, 55, 141, 28, 142, 146	gsummulc2 20291	. . 3 ⊢ (𝜑 → (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ (𝑌 · ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))) = (𝑌 · (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))))
148	138, 147	eqtrd 2776	. 2 ⊢ (𝜑 → (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))) = (𝑌 · (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))))
149		mdetrsca.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
150		eqid 2741	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
151		eqid 2741	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
152	149, 20, 21, 9, 150, 151, 73, 58	mdetleib2 22575	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐷‘𝑋) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))))
153	54, 19, 152	syl2anc 591	. 2 ⊢ (𝜑 → (𝐷‘𝑋) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))))
154	149, 20, 21, 9, 150, 151, 73, 58	mdetleib2 22575	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑍 ∈ 𝐵) → (𝐷‘𝑍) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))))
155	54, 30, 154	syl2anc 591	. . 3 ⊢ (𝜑 → (𝐷‘𝑍) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))))
156	155	oveq2d 7376	. 2 ⊢ (𝜑 → (𝑌 · (𝐷‘𝑍)) = (𝑌 · (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))))
157	148, 153, 156	3eqtr4d 2786	1 ⊢ (𝜑 → (𝐷‘𝑋) = (𝑌 · (𝐷‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 Vcvv 3433 ∖ cdif 3882 ∪ cun 3883 ∩ cin 3884 ⊆ wss 3885 ∅c0 4264 {csn 4558 ⟨cop 4564 ↦ cmpt 5156 × cxp 5619 ↾ cres 5623 ∘ ccom 5625 ⟶wf 6485 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7360 ∘_f cof 7622 ↑_m cmap 8767 Fincfn 8887 Basecbs 17174 ._rcmulr 17216 0_gc0g 17397 Σ_g cgsu 17398 Mndcmnd 18697 MndHom cmhm 18744 SymGrpcsymg 19339 pmSgncpsgn 19459 CMndccmn 19750 mulGrpcmgp 20116 Ringcrg 20209 CRingccrg 20210 ℤRHomczrh 21478 Mat cmat 22394 maDet cmdat 22571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-xor 1520 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-word 14471 df-lsw 14520 df-concat 14528 df-s1 14554 df-substr 14599 df-pfx 14629 df-splice 14707 df-reverse 14716 df-s2 14805 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-efmnd 18832 df-grp 18907 df-minusg 18908 df-mulg 19039 df-subg 19094 df-ghm 19183 df-gim 19229 df-cntz 19287 df-oppg 19316 df-symg 19340 df-pmtr 19412 df-psgn 19461 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-subrng 20522 df-subrg 20546 df-drng 20707 df-sra 21167 df-rgmod 21168 df-cnfld 21352 df-zring 21426 df-zrh 21482 df-dsmm 21711 df-frlm 21726 df-mat 22395 df-mdet 22572

This theorem is referenced by: mdetrsca2 22591 mdetuni0 22608 mdetmul 22610 smadiadetg 22660

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