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Theorem mdetrsca 22826

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 7433	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)))
3	2	adantr 486	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝐼 ∈ 𝑁)
6		snidg 4624	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ 𝑁 → 𝐼 ∈ {𝐼})
7	5, 6	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝐼 ∈ {𝐼})
8		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
9		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
10	8, 9	symgbasf1o 19503	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) → 𝑝:𝑁–1-1-onto→𝑁)
11	10	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁–1-1-onto→𝑁)
12		f1of 6821	. . . . . . . . . . . . . . 15 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝:𝑁⟶𝑁)
13	11, 12	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁⟶𝑁)
14	13, 5	ffvelcdmd 7081	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑝‘𝐼) ∈ 𝑁)
15		ovres 7582	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ {𝐼} ∧ (𝑝‘𝐼) ∈ 𝑁) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑋(𝑝‘𝐼)))
16	7, 14, 15	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑋(𝑝‘𝐼)))
17	7, 14	opelxpd 5698	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ⟨𝐼, (𝑝‘𝐼)⟩ ∈ ({𝐼} × 𝑁))
18		snfi 9053	. . . . . . . . . . . . . . . 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 22635	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
23	19, 22	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
24	23	simpld 500	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ Fin)
25	24	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 ∈ Fin)
26		xpfi 9292	. . . . . . . . . . . . . . . 16 ⊢ (({𝐼} ∈ Fin ∧ 𝑁 ∈ Fin) → ({𝐼} × 𝑁) ∈ Fin)
27	18, 25, 26	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} × 𝑁) ∈ Fin)
28		mdetrsca.y	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
29	28	adantr 486	. . . . . . . . . . . . . . 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 22645	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ 𝐵 → 𝑍 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
33		elmapi 8851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑍:(𝑁 × 𝑁)⟶𝐾)
34	30, 32, 33	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑍:(𝑁 × 𝑁)⟶𝐾)
35	34	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑍:(𝑁 × 𝑁)⟶𝐾)
36	35	ffnd 6707	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑍 Fn (𝑁 × 𝑁))
37	5	snssd 4750	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → {𝐼} ⊆ 𝑁)
38		xpss1 5678	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐼} ⊆ 𝑁 → ({𝐼} × 𝑁) ⊆ (𝑁 × 𝑁))
39	37, 38	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} × 𝑁) ⊆ (𝑁 × 𝑁))
40	36, 39	fnssresd 6660	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑍 ↾ ({𝐼} × 𝑁)) Fn ({𝐼} × 𝑁))
41		eqidd 2763	. . . . . . . . . . . . . . 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 7709	. . . . . . . . . . . . . 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 700	. . . . . . . . . . . . 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 7419	. . . . . . . . . . . . 13 ⊢ (𝐼((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)) = (((({𝐼} × 𝑁) × {𝑌}) ∘_f · (𝑍 ↾ ({𝐼} × 𝑁)))‘⟨𝐼, (𝑝‘𝐼)⟩)
45		df-ov 7419	. . . . . . . . . . . . . 14 ⊢ (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)
46	45	oveq2i 7427	. . . . . . . . . . . . 13 ⊢ (𝑌 · (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))) = (𝑌 · ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩))
47	43, 44, 46	3eqtr4g 2822	. . . . . . . . . . . 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 2805	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) = (𝑌 · (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))))
49		ovres 7582	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ {𝐼} ∧ (𝑝‘𝐼) ∈ 𝑁) → (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑍(𝑝‘𝐼)))
50	7, 14, 49	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑍(𝑝‘𝐼)))
51	50	oveq2d 7432	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑌 · (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))) = (𝑌 · (𝐼𝑍(𝑝‘𝐼))))
52	48, 51	eqtrd 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) = (𝑌 · (𝐼𝑍(𝑝‘𝐼))))
53	52	oveq1d 7431	. . . . . . . . 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 20386	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
56	55	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
57	35, 5, 14	fovcdmd 7589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑍(𝑝‘𝐼)) ∈ 𝐾)
58		eqid 2762	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
59	58, 31	mgpbas 20279	. . . . . . . . . . 11 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
60	58	crngmgp 20381	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
61	54, 60	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
62	61	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
63		difssd 4087	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑁 ∖ {𝐼}) ⊆ 𝑁)
64	25, 63	ssfid 9242	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑁 ∖ {𝐼}) ∈ Fin)
65		eldifi 4081	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (𝑁 ∖ {𝐼}) → 𝑟 ∈ 𝑁)
66	34	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑍:(𝑁 × 𝑁)⟶𝐾)
67		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑟 ∈ 𝑁)
68	13	ffvelcdmda 7080	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑝‘𝑟) ∈ 𝑁)
69	66, 67, 68	fovcdmd 7589	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
70	65, 69	sylan2 605	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
71	70	ralrimiva 3156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑟 ∈ (𝑁 ∖ {𝐼})(𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
72	59, 62, 64, 71	gsummptcl 20095	. . . . . . . . . 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 20393	. . . . . . . . . 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 1399	. . . . . . . . 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 2797	. . . . . . . 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 20278	. . . . . . . . . 10 ⊢ · = (+_g‘(mulGrp‘𝑅))
78	20, 31, 21	matbas2i 22645	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
79		elmapi 8851	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐾)
80	19, 78, 79	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐾)
81	80	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶𝐾)
82	81, 67, 68	fovcdmd 7589	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑟𝑋(𝑝‘𝑟)) ∈ 𝐾)
83		disjdif 4429	. . . . . . . . . . 11 ⊢ ({𝐼} ∩ (𝑁 ∖ {𝐼})) = ∅
84	83	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} ∩ (𝑁 ∖ {𝐼})) = ∅)
85		undif 4441	. . . . . . . . . . . 12 ⊢ ({𝐼} ⊆ 𝑁 ↔ ({𝐼} ∪ (𝑁 ∖ {𝐼})) = 𝑁)
86	37, 85	sylib 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} ∪ (𝑁 ∖ {𝐼})) = 𝑁)
87	86	eqcomd 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 = ({𝐼} ∪ (𝑁 ∖ {𝐼})))
88	59, 77, 62, 25, 82, 84, 87	gsummptfidmsplit 20058	. . . . . . . . 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 19932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ Mnd)
90	80	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑋:(𝑁 × 𝑁)⟶𝐾)
91	90, 5, 14	fovcdmd 7589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) ∈ 𝐾)
92		id 23	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → 𝑟 = 𝐼)
93		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → (𝑝‘𝑟) = (𝑝‘𝐼))
94	92, 93	oveq12d 7434	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝐼 → (𝑟𝑋(𝑝‘𝑟)) = (𝐼𝑋(𝑝‘𝐼)))
95	59, 94	gsumsn 20082	. . . . . . . . . . 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 1398	. . . . . . . . . 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 7433	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)))
99	98	ad2antrr 739	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → 𝑟 ∈ (𝑁 ∖ {𝐼}))
101	65, 68	sylan2 605	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑝‘𝑟) ∈ 𝑁)
102		ovres 7582	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ (𝑁 ∖ {𝐼}) ∧ (𝑝‘𝑟) ∈ 𝑁) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
103	100, 101, 102	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
104		ovres 7582	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ (𝑁 ∖ {𝐼}) ∧ (𝑝‘𝑟) ∈ 𝑁) → (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
105	100, 101, 104	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
106	99, 103, 105	3eqtr3d 2805	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟𝑋(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
107	106	mpteq2dva 5202	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))) = (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))
108	107	oveq2d 7432	. . . . . . . . . 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 7434	. . . . . . . . 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 2797	. . . . . . . 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 20058	. . . . . . . . . 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 7434	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → (𝑟𝑍(𝑝‘𝑟)) = (𝐼𝑍(𝑝‘𝐼)))
113	59, 112	gsumsn 20082	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) = (𝐼𝑍(𝑝‘𝐼)))
115	114	oveq1d 7431	. . . . . . . . . 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 2797	. . . . . . . . 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 7432	. . . . . . . 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 2807	. . . . . . 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 7432	. . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ CRing)
121		zrhpsgnmhm 21798	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
122	55, 24, 121	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
123	9, 59	mhmf 18898	. . . . . . . . . . 11 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
124	122, 123	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
125	124	ffvelcdmda 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾)
126	31, 73	crngcom 20391	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · 𝑌) = (𝑌 · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)))
128	127	oveq1d 7431	. . . . . . 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 3156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑟 ∈ 𝑁 (𝑟𝑍(𝑝‘𝑟)) ∈ 𝐾)
130	59, 62, 25, 129	gsummptcl 20095	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ 𝐾)
131	31, 73	ringass 20393	. . . . . . . 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 1399	. . . . . . 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 20393	. . . . . . . 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 1399	. . . . . . 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 2805	. . . . . 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 2797	. . . . 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 5202	. . . 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 7432	. . 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 2762	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
140	8, 9	symgbasfi 19507	. . . . 5 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
141	24, 140	syl 18	. . . 4 ⊢ (𝜑 → (Base‘(SymGrp‘𝑁)) ∈ Fin)
142	31, 73, 56, 125, 130	ringcld 20397	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) ∈ 𝐾)
143		eqid 2762	. . . . 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 7451	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) ∈ V)
145		fvexd 6897	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
146	143, 141, 144, 145	fsuppmptdm 9349	. . . 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 20458	. . 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 2797	. 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 2762	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
151		eqid 2762	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
152	149, 20, 21, 9, 150, 151, 73, 58	mdetleib2 22811	. . 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 596	. 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 22811	. . . 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 596	. . 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 7432	. 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 2807	1 ⊢ (𝜑 → (𝐷‘𝑋) = (𝑌 · (𝐷‘𝑍)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 {csn 4587 ⟨cop 4593 ↦ cmpt 5190 × cxp 5657 ↾ cres 5661 ∘ ccom 5663 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7416 ∘_f cof 7679 ↑_m cmap 8829 Fincfn 8955 Basecbs 17305 ._rcmulr 17347 0_gc0g 17528 Σ_g cgsu 17529 Mndcmnd 18838 MndHom cmhm 18890 SymGrpcsymg 19497 pmSgncpsgn 19617 CMndccmn 19908 mulGrpcmgp 20274 Ringcrg 20373 CRingccrg 20374 ℤRHomczrh 21713 Mat cmat 22630 maDet cmdat 22807

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-addf 11206 ax-mulf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-xnn0 12605 df-z 12619 df-dec 12740 df-uz 12891 df-rp 13045 df-fz 13564 df-fzo 13712 df-seq 14068 df-exp 14128 df-hash 14397 df-word 14581 df-lsw 14630 df-concat 14638 df-s1 14665 df-substr 14711 df-pfx 14743 df-splice 14821 df-reverse 14830 df-s2 14921 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-0g 17530 df-gsum 17531 df-prds 17536 df-pws 17538 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-mhm 18892 df-submnd 18893 df-efmnd 18979 df-grp 19061 df-minusg 19062 df-mulg 19192 df-subg 19247 df-ghm 19342 df-gim 19387 df-cntz 19445 df-oppg 19474 df-symg 19498 df-pmtr 19570 df-psgn 19619 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-cring 20376 df-oppr 20479 df-dvdsr 20499 df-unit 20500 df-invr 20530 df-dvr 20543 df-rhm 20614 df-subrng 20709 df-subrg 20733 df-drng 20893 df-sra 21358 df-rgmod 21359 df-cnfld 21587 df-zring 21661 df-zrh 21717 df-dsmm 21946 df-frlm 21961 df-mat 22631 df-mdet 22808

This theorem is used by: mdetrsca2 22827 mdetuni0 22844 mdetmul 22846 smadiadetg 22896

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