mdetpmtr1 - Mathbox for Thierry Arnoux

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Theorem mdetpmtr1 32791

Description: The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)

**Hypotheses**
Ref	Expression
mdetpmtr.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetpmtr.b	⊢ 𝐵 = (Base‘𝐴)
mdetpmtr.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetpmtr.g	⊢ 𝐺 = (Base‘(SymGrp‘𝑁))
mdetpmtr.s	⊢ 𝑆 = (pmSgn‘𝑁)
mdetpmtr.z	⊢ 𝑍 = (ℤRHom‘𝑅)
mdetpmtr.t	⊢ · = (._r‘𝑅)
mdetpmtr1.e	⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗))

**Assertion**
Ref	Expression
mdetpmtr1	⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))

Distinct variable groups: 𝐵,𝑖,𝑗 𝑖,𝐺,𝑗 𝑖,𝑀,𝑗 𝑖,𝑁,𝑗 𝑃,𝑖,𝑗 𝑅,𝑖,𝑗 Allowed substitution hints: 𝐴(𝑖,𝑗) 𝐷(𝑖,𝑗) 𝑆(𝑖,𝑗) · (𝑖,𝑗) 𝐸(𝑖,𝑗) 𝑍(𝑖,𝑗)

Proof of Theorem mdetpmtr1 Dummy variables 𝑝 𝑞 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2732	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
3		mdetpmtr.t	. . 3 ⊢ · = (._r‘𝑅)
4		crngring 20061	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	ad2antrr 724	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ Ring)
6		mdetpmtr.g	. . . . 5 ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))
7	6	fvexi 6902	. . . 4 ⊢ 𝐺 ∈ V
8	7	a1i 11	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐺 ∈ V)
9		simplr 767	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑁 ∈ Fin)
10		mdetpmtr.s	. . . . . . 7 ⊢ 𝑆 = (pmSgn‘𝑁)
11	10, 6	psgndmfi 32244	. . . . . 6 ⊢ (𝑁 ∈ Fin → 𝑆 Fn 𝐺)
12		fnfun 6646	. . . . . 6 ⊢ (𝑆 Fn 𝐺 → Fun 𝑆)
13	9, 11, 12	3syl 18	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → Fun 𝑆)
14		simprr 771	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑃 ∈ 𝐺)
15		fndm 6649	. . . . . . 7 ⊢ (𝑆 Fn 𝐺 → dom 𝑆 = 𝐺)
16	9, 11, 15	3syl 18	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → dom 𝑆 = 𝐺)
17	14, 16	eleqtrrd 2836	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑃 ∈ dom 𝑆)
18		fvco 6986	. . . . 5 ⊢ ((Fun 𝑆 ∧ 𝑃 ∈ dom 𝑆) → ((𝑍 ∘ 𝑆)‘𝑃) = (𝑍‘(𝑆‘𝑃)))
19	13, 17, 18	syl2anc 584	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → ((𝑍 ∘ 𝑆)‘𝑃) = (𝑍‘(𝑆‘𝑃)))
20		mdetpmtr.z	. . . . . 6 ⊢ 𝑍 = (ℤRHom‘𝑅)
21	6, 10, 20	zrhpsgnelbas 21138	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺) → (𝑍‘(𝑆‘𝑃)) ∈ (Base‘𝑅))
22	5, 9, 14, 21	syl3anc 1371	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑍‘(𝑆‘𝑃)) ∈ (Base‘𝑅))
23	19, 22	eqeltrd 2833	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → ((𝑍 ∘ 𝑆)‘𝑃) ∈ (Base‘𝑅))
24	5	adantr 481	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑅 ∈ Ring)
25	6, 10	cofipsgn 21137	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑝) = (𝑍‘(𝑆‘𝑝)))
26	9, 25	sylan 580	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑝) = (𝑍‘(𝑆‘𝑝)))
27		simpllr 774	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑁 ∈ Fin)
28		simpr 485	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑝 ∈ 𝐺)
29	6, 10, 20	zrhpsgnelbas 21138	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑝 ∈ 𝐺) → (𝑍‘(𝑆‘𝑝)) ∈ (Base‘𝑅))
30	24, 27, 28, 29	syl3anc 1371	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑍‘(𝑆‘𝑝)) ∈ (Base‘𝑅))
31	26, 30	eqeltrd 2833	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑝) ∈ (Base‘𝑅))
32		eqid 2732	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
33	32, 1	mgpbas 19987	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
34	32	crngmgp 20057	. . . . . 6 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
35	34	ad3antrrr 728	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (mulGrp‘𝑅) ∈ CMnd)
36		mdetpmtr.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
37		mdetpmtr.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
38		eqid 2732	. . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
39	38, 6	symgfv 19241	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝐺 ∧ 𝑥 ∈ 𝑁) → (𝑝‘𝑥) ∈ 𝑁)
40	39	adantll 712	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → (𝑝‘𝑥) ∈ 𝑁)
41		simpr 485	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁)
42		mdetpmtr1.e	. . . . . . . . 9 ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗))
43		simpll 765	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ CRing)
44		simp1rr 1239	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ 𝐺)
45		simp2 1137	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
46	38, 6	symgfv 19241	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑖 ∈ 𝑁) → (𝑃‘𝑖) ∈ 𝑁)
47	44, 45, 46	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑖) ∈ 𝑁)
48		simp3 1138	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
49		simp1rl 1238	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
50	36, 1, 37, 47, 48, 49	matecld 21919	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑃‘𝑖)𝑀𝑗) ∈ (Base‘𝑅))
51	36, 1, 37, 9, 43, 50	matbas2d 21916	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗)) ∈ 𝐵)
52	42, 51	eqeltrid 2837	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐸 ∈ 𝐵)
53	52	ad2antrr 724	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝐸 ∈ 𝐵)
54	36, 1, 37, 40, 41, 53	matecld 21919	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → ((𝑝‘𝑥)𝐸𝑥) ∈ (Base‘𝑅))
55	54	ralrimiva 3146	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ∀𝑥 ∈ 𝑁 ((𝑝‘𝑥)𝐸𝑥) ∈ (Base‘𝑅))
56	33, 35, 27, 55	gsummptcl 19829	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))) ∈ (Base‘𝑅))
57	1, 3	ringcl 20066	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((𝑍 ∘ 𝑆)‘𝑝) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))) ∈ (Base‘𝑅)) → (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))) ∈ (Base‘𝑅))
58	24, 31, 56, 57	syl3anc 1371	. . 3 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))) ∈ (Base‘𝑅))
59		eqid 2732	. . . 4 ⊢ (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))) = (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))))
60	38, 6	symgbasfi 19240	. . . . 5 ⊢ (𝑁 ∈ Fin → 𝐺 ∈ Fin)
61	9, 60	syl 17	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐺 ∈ Fin)
62		ovexd 7440	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))) ∈ V)
63		fvexd 6903	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (0_g‘𝑅) ∈ V)
64	59, 61, 62, 63	fsuppmptdm 9370	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))) finSupp (0_g‘𝑅))
65	1, 2, 3, 5, 8, 23, 58, 64	gsummulc2 20122	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑃) · (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))))) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))))))
66		nfcv 2903	. . . 4 ⊢ Ⅎ𝑞(((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))))
67		fveq2 6888	. . . . 5 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → ((𝑍 ∘ 𝑆)‘𝑞) = ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)))
68		fveq1 6887	. . . . . . . 8 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → (𝑞‘𝑥) = ((𝑃 ∘ 𝑝)‘𝑥))
69	68	oveq1d 7420	. . . . . . 7 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → ((𝑞‘𝑥)𝑀𝑥) = (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))
70	69	mpteq2dv 5249	. . . . . 6 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)) = (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))
71	70	oveq2d 7421	. . . . 5 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥))) = ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))))
72	67, 71	oveq12d 7423	. . . 4 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → (((𝑍 ∘ 𝑆)‘𝑞) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)))) = (((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))))
73		ringcmn 20092	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
74	5, 73	syl 17	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ CMnd)
75		ssidd 4004	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (Base‘𝑅) ⊆ (Base‘𝑅))
76	5	adantr 481	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑅 ∈ Ring)
77	6, 10	cofipsgn 21137	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑞 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑞) = (𝑍‘(𝑆‘𝑞)))
78	9, 77	sylan 580	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑞) = (𝑍‘(𝑆‘𝑞)))
79		simpllr 774	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑁 ∈ Fin)
80		simpr 485	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑞 ∈ 𝐺)
81	6, 10, 20	zrhpsgnelbas 21138	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑞 ∈ 𝐺) → (𝑍‘(𝑆‘𝑞)) ∈ (Base‘𝑅))
82	76, 79, 80, 81	syl3anc 1371	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → (𝑍‘(𝑆‘𝑞)) ∈ (Base‘𝑅))
83	78, 82	eqeltrd 2833	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑞) ∈ (Base‘𝑅))
84	34	ad3antrrr 728	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → (mulGrp‘𝑅) ∈ CMnd)
85	38, 6	symgfv 19241	. . . . . . . . 9 ⊢ ((𝑞 ∈ 𝐺 ∧ 𝑥 ∈ 𝑁) → (𝑞‘𝑥) ∈ 𝑁)
86	85	adantll 712	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → (𝑞‘𝑥) ∈ 𝑁)
87		simpr 485	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁)
88		simprl 769	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑀 ∈ 𝐵)
89	88	ad2antrr 724	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝑀 ∈ 𝐵)
90	36, 1, 37, 86, 87, 89	matecld 21919	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → ((𝑞‘𝑥)𝑀𝑥) ∈ (Base‘𝑅))
91	90	ralrimiva 3146	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ∀𝑥 ∈ 𝑁 ((𝑞‘𝑥)𝑀𝑥) ∈ (Base‘𝑅))
92	33, 84, 79, 91	gsummptcl 19829	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥))) ∈ (Base‘𝑅))
93	1, 3	ringcl 20066	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((𝑍 ∘ 𝑆)‘𝑞) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥))) ∈ (Base‘𝑅)) → (((𝑍 ∘ 𝑆)‘𝑞) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)))) ∈ (Base‘𝑅))
94	76, 83, 92, 93	syl3anc 1371	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → (((𝑍 ∘ 𝑆)‘𝑞) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)))) ∈ (Base‘𝑅))
95		eqid 2732	. . . . . . 7 ⊢ (+_g‘(SymGrp‘𝑁)) = (+_g‘(SymGrp‘𝑁))
96	38, 6, 95	symgov 19245	. . . . . 6 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑃(+_g‘(SymGrp‘𝑁))𝑝) = (𝑃 ∘ 𝑝))
97	38, 6, 95	symgcl 19246	. . . . . 6 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑃(+_g‘(SymGrp‘𝑁))𝑝) ∈ 𝐺)
98	96, 97	eqeltrrd 2834	. . . . 5 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑃 ∘ 𝑝) ∈ 𝐺)
99	14, 98	sylan 580	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑃 ∘ 𝑝) ∈ 𝐺)
100	14	adantr 481	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑃 ∈ 𝐺)
101	6	symgfcoeu 32230	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺 ∧ 𝑞 ∈ 𝐺) → ∃!𝑝 ∈ 𝐺 𝑞 = (𝑃 ∘ 𝑝))
102	79, 100, 80, 101	syl3anc 1371	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ∃!𝑝 ∈ 𝐺 𝑞 = (𝑃 ∘ 𝑝))
103	66, 1, 2, 72, 74, 61, 75, 94, 99, 102	gsummptf1o 19825	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑅 Σ_g (𝑞 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑞) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)))))) = (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))))))
104		mdetpmtr.d	. . . . 5 ⊢ 𝐷 = (𝑁 maDet 𝑅)
105	104, 36, 37, 6, 20, 10, 3, 32	mdetleib 22080	. . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σ_g (𝑞 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑞) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)))))))
106	105	ad2antrl 726	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (𝑅 Σ_g (𝑞 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑞) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)))))))
107	23	adantr 481	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑃) ∈ (Base‘𝑅))
108	1, 3	ringass 20069	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (((𝑍 ∘ 𝑆)‘𝑃) ∈ (Base‘𝑅) ∧ ((𝑍 ∘ 𝑆)‘𝑝) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))) ∈ (Base‘𝑅))) → ((((𝑍 ∘ 𝑆)‘𝑃) · ((𝑍 ∘ 𝑆)‘𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))) = (((𝑍 ∘ 𝑆)‘𝑃) · (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))))
109	24, 107, 31, 56, 108	syl13anc 1372	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((((𝑍 ∘ 𝑆)‘𝑃) · ((𝑍 ∘ 𝑆)‘𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))) = (((𝑍 ∘ 𝑆)‘𝑃) · (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))))
110	19	adantr 481	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑃) = (𝑍‘(𝑆‘𝑃)))
111	110, 26	oveq12d 7423	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (((𝑍 ∘ 𝑆)‘𝑃) · ((𝑍 ∘ 𝑆)‘𝑝)) = ((𝑍‘(𝑆‘𝑃)) · (𝑍‘(𝑆‘𝑝))))
112	6, 10	cofipsgn 21137	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ (𝑃 ∘ 𝑝) ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) = (𝑍‘(𝑆‘(𝑃 ∘ 𝑝))))
113	27, 99, 112	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) = (𝑍‘(𝑆‘(𝑃 ∘ 𝑝))))
114	14	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑃 ∈ 𝐺)
115	38, 10, 6	psgnco 21127	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑆‘(𝑃 ∘ 𝑝)) = ((𝑆‘𝑃) · (𝑆‘𝑝)))
116	27, 114, 28, 115	syl3anc 1371	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘(𝑃 ∘ 𝑝)) = ((𝑆‘𝑃) · (𝑆‘𝑝)))
117	116	fveq2d 6892	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑍‘(𝑆‘(𝑃 ∘ 𝑝))) = (𝑍‘((𝑆‘𝑃) · (𝑆‘𝑝))))
118	20	zrhrhm 21052	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑍 ∈ (ℤ_ring RingHom 𝑅))
119	5, 118	syl 17	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑍 ∈ (ℤ_ring RingHom 𝑅))
120	119	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑍 ∈ (ℤ_ring RingHom 𝑅))
121		1z 12588	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
122		neg1z 12594	. . . . . . . . . . . 12 ⊢ -1 ∈ ℤ
123		prssi 4823	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ -1 ∈ ℤ) → {1, -1} ⊆ ℤ)
124	121, 122, 123	mp2an 690	. . . . . . . . . . 11 ⊢ {1, -1} ⊆ ℤ
125	6, 10	psgnran 19377	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺) → (𝑆‘𝑃) ∈ {1, -1})
126	27, 114, 125	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑃) ∈ {1, -1})
127	124, 126	sselid 3979	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑃) ∈ ℤ)
128	6, 10	psgnran 19377	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑝) ∈ {1, -1})
129	9, 128	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑝) ∈ {1, -1})
130	124, 129	sselid 3979	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑝) ∈ ℤ)
131		zringbas 21015	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
132		zringmulr 21018	. . . . . . . . . . 11 ⊢ · = (._r‘ℤ_ring)
133	131, 132, 3	rhmmul 20256	. . . . . . . . . 10 ⊢ ((𝑍 ∈ (ℤ_ring RingHom 𝑅) ∧ (𝑆‘𝑃) ∈ ℤ ∧ (𝑆‘𝑝) ∈ ℤ) → (𝑍‘((𝑆‘𝑃) · (𝑆‘𝑝))) = ((𝑍‘(𝑆‘𝑃)) · (𝑍‘(𝑆‘𝑝))))
134	120, 127, 130, 133	syl3anc 1371	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑍‘((𝑆‘𝑃) · (𝑆‘𝑝))) = ((𝑍‘(𝑆‘𝑃)) · (𝑍‘(𝑆‘𝑝))))
135	113, 117, 134	3eqtrrd 2777	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍‘(𝑆‘𝑃)) · (𝑍‘(𝑆‘𝑝))) = ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)))
136	111, 135	eqtrd 2772	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (((𝑍 ∘ 𝑆)‘𝑃) · ((𝑍 ∘ 𝑆)‘𝑝)) = ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)))
137	42	a1i 11	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗)))
138		simprl 769	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑖 = (𝑝‘𝑥))
139	138	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → (𝑃‘𝑖) = (𝑃‘(𝑝‘𝑥)))
140		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑝 ∈ 𝐺)
141	38, 6	symgbasf 19237	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐺 → 𝑝:𝑁⟶𝑁)
142		ffun 6717	. . . . . . . . . . . . . 14 ⊢ (𝑝:𝑁⟶𝑁 → Fun 𝑝)
143	140, 141, 142	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → Fun 𝑝)
144		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑥 ∈ 𝑁)
145		fdm 6723	. . . . . . . . . . . . . . 15 ⊢ (𝑝:𝑁⟶𝑁 → dom 𝑝 = 𝑁)
146	140, 141, 145	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → dom 𝑝 = 𝑁)
147	144, 146	eleqtrrd 2836	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑥 ∈ dom 𝑝)
148		fvco 6986	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑝 ∧ 𝑥 ∈ dom 𝑝) → ((𝑃 ∘ 𝑝)‘𝑥) = (𝑃‘(𝑝‘𝑥)))
149	143, 147, 148	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → ((𝑃 ∘ 𝑝)‘𝑥) = (𝑃‘(𝑝‘𝑥)))
150	139, 149	eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → (𝑃‘𝑖) = ((𝑃 ∘ 𝑝)‘𝑥))
151		simprr 771	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑗 = 𝑥)
152	150, 151	oveq12d 7423	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → ((𝑃‘𝑖)𝑀𝑗) = (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))
153		ovexd 7440	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥) ∈ V)
154	137, 152, 40, 41, 153	ovmpod 7556	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → ((𝑝‘𝑥)𝐸𝑥) = (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))
155	154	mpteq2dva 5247	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)) = (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))
156	155	oveq2d 7421	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))) = ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))))
157	136, 156	oveq12d 7423	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((((𝑍 ∘ 𝑆)‘𝑃) · ((𝑍 ∘ 𝑆)‘𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))) = (((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))))
158	109, 157	eqtr3d 2774	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (((𝑍 ∘ 𝑆)‘𝑃) · (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))) = (((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))))
159	158	mpteq2dva 5247	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑃) · (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))))) = (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))))))
160	159	oveq2d 7421	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑃) · (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))))) = (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))))))
161	103, 106, 160	3eqtr4d 2782	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑃) · (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))))))
162	104, 36, 37, 6, 20, 10, 3, 32	mdetleib 22080	. . . 4 ⊢ (𝐸 ∈ 𝐵 → (𝐷‘𝐸) = (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))))))
163	52, 162	syl 17	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝐸) = (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥)))))))
164	163	oveq2d 7421	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝑅 Σ_g (𝑝 ∈ 𝐺 ↦ (((𝑍 ∘ 𝑆)‘𝑝) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝐸𝑥))))))))
165	65, 161, 164	3eqtr4d 2782	1 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃!wreu 3374 Vcvv 3474 ⊆ wss 3947 {cpr 4629 ↦ cmpt 5230 dom cdm 5675 ∘ ccom 5679 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 Fincfn 8935 1c1 11107 · cmul 11111 -cneg 11441 ℤcz 12554 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 0_gc0g 17381 Σ_g cgsu 17382 SymGrpcsymg 19228 pmSgncpsgn 19351 CMndccmn 19642 mulGrpcmgp 19981 Ringcrg 20049 CRingccrg 20050 RingHom crh 20240 ℤ_ringczring 21009 ℤRHomczrh 21040 Mat cmat 21898 maDet cmdat 22077

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-word 14461 df-lsw 14509 df-concat 14517 df-s1 14542 df-substr 14587 df-pfx 14617 df-splice 14696 df-reverse 14705 df-s2 14795 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-efmnd 18746 df-grp 18818 df-minusg 18819 df-mulg 18945 df-subg 18997 df-ghm 19084 df-gim 19127 df-cntz 19175 df-oppg 19204 df-symg 19229 df-pmtr 19304 df-psgn 19353 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-rnghom 20243 df-drng 20309 df-subrg 20353 df-sra 20777 df-rgmod 20778 df-cnfld 20937 df-zring 21010 df-zrh 21044 df-dsmm 21278 df-frlm 21293 df-mat 21899 df-mdet 22078

This theorem is referenced by: mdetpmtr2 32792 mdetpmtr12 32793

W3C validator