mdetpmtr1 - Mathbox for Thierry Arnoux

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Theorem mdetpmtr1 32803

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 2733	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2733	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
3		mdetpmtr.t	. . 3 ⊢ · = (._r‘𝑅)
4		crngring 20068	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	ad2antrr 725	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ Ring)
6		mdetpmtr.g	. . . . 5 ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))
7	6	fvexi 6906	. . . 4 ⊢ 𝐺 ∈ V
8	7	a1i 11	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐺 ∈ V)
9		simplr 768	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑁 ∈ Fin)
10		mdetpmtr.s	. . . . . . 7 ⊢ 𝑆 = (pmSgn‘𝑁)
11	10, 6	psgndmfi 32257	. . . . . 6 ⊢ (𝑁 ∈ Fin → 𝑆 Fn 𝐺)
12		fnfun 6650	. . . . . 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 772	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑃 ∈ 𝐺)
15		fndm 6653	. . . . . . 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 2837	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑃 ∈ dom 𝑆)
18		fvco 6990	. . . . 5 ⊢ ((Fun 𝑆 ∧ 𝑃 ∈ dom 𝑆) → ((𝑍 ∘ 𝑆)‘𝑃) = (𝑍‘(𝑆‘𝑃)))
19	13, 17, 18	syl2anc 585	. . . 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 21147	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺) → (𝑍‘(𝑆‘𝑃)) ∈ (Base‘𝑅))
22	5, 9, 14, 21	syl3anc 1372	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑍‘(𝑆‘𝑃)) ∈ (Base‘𝑅))
23	19, 22	eqeltrd 2834	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → ((𝑍 ∘ 𝑆)‘𝑃) ∈ (Base‘𝑅))
24	5	adantr 482	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑅 ∈ Ring)
25	6, 10	cofipsgn 21146	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑝) = (𝑍‘(𝑆‘𝑝)))
26	9, 25	sylan 581	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑝) = (𝑍‘(𝑆‘𝑝)))
27		simpllr 775	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑁 ∈ Fin)
28		simpr 486	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑝 ∈ 𝐺)
29	6, 10, 20	zrhpsgnelbas 21147	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑝 ∈ 𝐺) → (𝑍‘(𝑆‘𝑝)) ∈ (Base‘𝑅))
30	24, 27, 28, 29	syl3anc 1372	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑍‘(𝑆‘𝑝)) ∈ (Base‘𝑅))
31	26, 30	eqeltrd 2834	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑝) ∈ (Base‘𝑅))
32		eqid 2733	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
33	32, 1	mgpbas 19993	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
34	32	crngmgp 20064	. . . . . 6 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
35	34	ad3antrrr 729	. . . . 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 2733	. . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
39	38, 6	symgfv 19247	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝐺 ∧ 𝑥 ∈ 𝑁) → (𝑝‘𝑥) ∈ 𝑁)
40	39	adantll 713	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → (𝑝‘𝑥) ∈ 𝑁)
41		simpr 486	. . . . . . 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 766	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ CRing)
44		simp1rr 1240	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ 𝐺)
45		simp2 1138	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
46	38, 6	symgfv 19247	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑖 ∈ 𝑁) → (𝑃‘𝑖) ∈ 𝑁)
47	44, 45, 46	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑖) ∈ 𝑁)
48		simp3 1139	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
49		simp1rl 1239	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
50	36, 1, 37, 47, 48, 49	matecld 21928	. . . . . . . . . 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 21925	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗)) ∈ 𝐵)
52	42, 51	eqeltrid 2838	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐸 ∈ 𝐵)
53	52	ad2antrr 725	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝐸 ∈ 𝐵)
54	36, 1, 37, 40, 41, 53	matecld 21928	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → ((𝑝‘𝑥)𝐸𝑥) ∈ (Base‘𝑅))
55	54	ralrimiva 3147	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ∀𝑥 ∈ 𝑁 ((𝑝‘𝑥)𝐸𝑥) ∈ (Base‘𝑅))
56	33, 35, 27, 55	gsummptcl 19835	. . . 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 20073	. . . 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 1372	. . 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 2733	. . . 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 19246	. . . . 5 ⊢ (𝑁 ∈ Fin → 𝐺 ∈ Fin)
61	9, 60	syl 17	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐺 ∈ Fin)
62		ovexd 7444	. . . 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 6907	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (0_g‘𝑅) ∈ V)
64	59, 61, 62, 63	fsuppmptdm 9374	. . 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 20129	. 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 2904	. . . 4 ⊢ Ⅎ𝑞(((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) · ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))))
67		fveq2 6892	. . . . 5 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → ((𝑍 ∘ 𝑆)‘𝑞) = ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)))
68		fveq1 6891	. . . . . . . 8 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → (𝑞‘𝑥) = ((𝑃 ∘ 𝑝)‘𝑥))
69	68	oveq1d 7424	. . . . . . 7 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → ((𝑞‘𝑥)𝑀𝑥) = (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))
70	69	mpteq2dv 5251	. . . . . 6 ⊢ (𝑞 = (𝑃 ∘ 𝑝) → (𝑥 ∈ 𝑁 ↦ ((𝑞‘𝑥)𝑀𝑥)) = (𝑥 ∈ 𝑁 ↦ (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥)))
71	70	oveq2d 7425	. . . . 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 7427	. . . 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 20099	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
74	5, 73	syl 17	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ CMnd)
75		ssidd 4006	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (Base‘𝑅) ⊆ (Base‘𝑅))
76	5	adantr 482	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑅 ∈ Ring)
77	6, 10	cofipsgn 21146	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑞 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑞) = (𝑍‘(𝑆‘𝑞)))
78	9, 77	sylan 581	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑞) = (𝑍‘(𝑆‘𝑞)))
79		simpllr 775	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑁 ∈ Fin)
80		simpr 486	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑞 ∈ 𝐺)
81	6, 10, 20	zrhpsgnelbas 21147	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑞 ∈ 𝐺) → (𝑍‘(𝑆‘𝑞)) ∈ (Base‘𝑅))
82	76, 79, 80, 81	syl3anc 1372	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → (𝑍‘(𝑆‘𝑞)) ∈ (Base‘𝑅))
83	78, 82	eqeltrd 2834	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑞) ∈ (Base‘𝑅))
84	34	ad3antrrr 729	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → (mulGrp‘𝑅) ∈ CMnd)
85	38, 6	symgfv 19247	. . . . . . . . 9 ⊢ ((𝑞 ∈ 𝐺 ∧ 𝑥 ∈ 𝑁) → (𝑞‘𝑥) ∈ 𝑁)
86	85	adantll 713	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → (𝑞‘𝑥) ∈ 𝑁)
87		simpr 486	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁)
88		simprl 770	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑀 ∈ 𝐵)
89	88	ad2antrr 725	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → 𝑀 ∈ 𝐵)
90	36, 1, 37, 86, 87, 89	matecld 21928	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → ((𝑞‘𝑥)𝑀𝑥) ∈ (Base‘𝑅))
91	90	ralrimiva 3147	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → ∀𝑥 ∈ 𝑁 ((𝑞‘𝑥)𝑀𝑥) ∈ (Base‘𝑅))
92	33, 84, 79, 91	gsummptcl 19835	. . . . 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 20073	. . . . 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 1372	. . . 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 2733	. . . . . . 7 ⊢ (+_g‘(SymGrp‘𝑁)) = (+_g‘(SymGrp‘𝑁))
96	38, 6, 95	symgov 19251	. . . . . 6 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑃(+_g‘(SymGrp‘𝑁))𝑝) = (𝑃 ∘ 𝑝))
97	38, 6, 95	symgcl 19252	. . . . . 6 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑃(+_g‘(SymGrp‘𝑁))𝑝) ∈ 𝐺)
98	96, 97	eqeltrrd 2835	. . . . 5 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑃 ∘ 𝑝) ∈ 𝐺)
99	14, 98	sylan 581	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑃 ∘ 𝑝) ∈ 𝐺)
100	14	adantr 482	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑞 ∈ 𝐺) → 𝑃 ∈ 𝐺)
101	6	symgfcoeu 32243	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺 ∧ 𝑞 ∈ 𝐺) → ∃!𝑝 ∈ 𝐺 𝑞 = (𝑃 ∘ 𝑝))
102	79, 100, 80, 101	syl3anc 1372	. . . 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 19831	. . 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 22089	. . . 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 727	. . 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 482	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑃) ∈ (Base‘𝑅))
108	1, 3	ringass 20076	. . . . . . 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 1373	. . . . . 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 482	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘𝑃) = (𝑍‘(𝑆‘𝑃)))
111	110, 26	oveq12d 7427	. . . . . . . 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 21146	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ (𝑃 ∘ 𝑝) ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) = (𝑍‘(𝑆‘(𝑃 ∘ 𝑝))))
113	27, 99, 112	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → ((𝑍 ∘ 𝑆)‘(𝑃 ∘ 𝑝)) = (𝑍‘(𝑆‘(𝑃 ∘ 𝑝))))
114	14	adantr 482	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑃 ∈ 𝐺)
115	38, 10, 6	psgnco 21136	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺 ∧ 𝑝 ∈ 𝐺) → (𝑆‘(𝑃 ∘ 𝑝)) = ((𝑆‘𝑃) · (𝑆‘𝑝)))
116	27, 114, 28, 115	syl3anc 1372	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘(𝑃 ∘ 𝑝)) = ((𝑆‘𝑃) · (𝑆‘𝑝)))
117	116	fveq2d 6896	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑍‘(𝑆‘(𝑃 ∘ 𝑝))) = (𝑍‘((𝑆‘𝑃) · (𝑆‘𝑝))))
118	20	zrhrhm 21061	. . . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → 𝑍 ∈ (ℤ_ring RingHom 𝑅))
121		1z 12592	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
122		neg1z 12598	. . . . . . . . . . . 12 ⊢ -1 ∈ ℤ
123		prssi 4825	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ -1 ∈ ℤ) → {1, -1} ⊆ ℤ)
124	121, 122, 123	mp2an 691	. . . . . . . . . . 11 ⊢ {1, -1} ⊆ ℤ
125	6, 10	psgnran 19383	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ 𝐺) → (𝑆‘𝑃) ∈ {1, -1})
126	27, 114, 125	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑃) ∈ {1, -1})
127	124, 126	sselid 3981	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑃) ∈ ℤ)
128	6, 10	psgnran 19383	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑝) ∈ {1, -1})
129	9, 128	sylan 581	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑝) ∈ {1, -1})
130	124, 129	sselid 3981	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) → (𝑆‘𝑝) ∈ ℤ)
131		zringbas 21023	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
132		zringmulr 21027	. . . . . . . . . . 11 ⊢ · = (._r‘ℤ_ring)
133	131, 132, 3	rhmmul 20264	. . . . . . . . . 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 1372	. . . . . . . . 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 2778	. . . . . . . 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 2773	. . . . . . 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 770	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑖 = (𝑝‘𝑥))
139	138	fveq2d 6896	. . . . . . . . . . . 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 775	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑝 ∈ 𝐺)
141	38, 6	symgbasf 19243	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐺 → 𝑝:𝑁⟶𝑁)
142		ffun 6721	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑥 ∈ 𝑁)
145		fdm 6727	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑥 ∈ dom 𝑝)
148		fvco 6990	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑝 ∧ 𝑥 ∈ dom 𝑝) → ((𝑃 ∘ 𝑝)‘𝑥) = (𝑃‘(𝑝‘𝑥)))
149	143, 147, 148	syl2anc 585	. . . . . . . . . . . 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 2776	. . . . . . . . . . 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 772	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) ∧ (𝑖 = (𝑝‘𝑥) ∧ 𝑗 = 𝑥)) → 𝑗 = 𝑥)
152	150, 151	oveq12d 7427	. . . . . . . . . 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 7444	. . . . . . . . . 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 7560	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑝 ∈ 𝐺) ∧ 𝑥 ∈ 𝑁) → ((𝑝‘𝑥)𝐸𝑥) = (((𝑃 ∘ 𝑝)‘𝑥)𝑀𝑥))
155	154	mpteq2dva 5249	. . . . . . . 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 7425	. . . . . . 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 7427	. . . . . 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 2775	. . . . 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 5249	. . . 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 7425	. . 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 2783	. 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 22089	. . . 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 7425	. 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 2783	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃!wreu 3375 Vcvv 3475 ⊆ wss 3949 {cpr 4631 ↦ cmpt 5232 dom cdm 5677 ∘ ccom 5681 Fun wfun 6538 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 Fincfn 8939 1c1 11111 · cmul 11115 -cneg 11445 ℤcz 12558 Basecbs 17144 +_gcplusg 17197 ._rcmulr 17198 0_gc0g 17385 Σ_g cgsu 17386 SymGrpcsymg 19234 pmSgncpsgn 19357 CMndccmn 19648 mulGrpcmgp 19987 Ringcrg 20056 CRingccrg 20057 RingHom crh 20248 ℤ_ringczring 21017 ℤRHomczrh 21049 Mat cmat 21907 maDet cmdat 22086

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-word 14465 df-lsw 14513 df-concat 14521 df-s1 14546 df-substr 14591 df-pfx 14621 df-splice 14700 df-reverse 14709 df-s2 14799 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-efmnd 18750 df-grp 18822 df-minusg 18823 df-mulg 18951 df-subg 19003 df-ghm 19090 df-gim 19133 df-cntz 19181 df-oppg 19210 df-symg 19235 df-pmtr 19310 df-psgn 19359 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-sra 20785 df-rgmod 20786 df-cnfld 20945 df-zring 21018 df-zrh 21053 df-dsmm 21287 df-frlm 21302 df-mat 21908 df-mdet 22087

This theorem is referenced by: mdetpmtr2 32804 mdetpmtr12 32805

W3C validator