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Theorem mdetrlin 22520

Description: The determinant function is additive for each row: The matrices X, Y, Z are identical except for the I's row, and the I's row of the matrix X is the componentwise sum of the I's row of the matrices Y and Z. In this case the determinant of X is the sum of the determinants of Y and Z. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)

**Hypotheses**
Ref	Expression
mdetrlin.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetrlin.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetrlin.b	⊢ 𝐵 = (Base‘𝐴)
mdetrlin.p	⊢ + = (+_g‘𝑅)
mdetrlin.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mdetrlin.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mdetrlin.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
mdetrlin.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
mdetrlin.i	⊢ (𝜑 → 𝐼 ∈ 𝑁)
mdetrlin.eq	⊢ (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁))))
mdetrlin.ne1	⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
mdetrlin.ne2	⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))

**Assertion**
Ref	Expression
mdetrlin	⊢ (𝜑 → (𝐷‘𝑋) = ((𝐷‘𝑌) + (𝐷‘𝑍)))

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

Step	Hyp	Ref	Expression
1		fvex 6903	. . . . . 6 ⊢ (Base‘(SymGrp‘𝑁)) ∈ V
2		ovex 7447	. . . . . . 7 ⊢ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) ∈ V
3		eqid 2725	. . . . . . 7 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))))
4	2, 3	fnmpti 6691	. . . . . 6 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) Fn (Base‘(SymGrp‘𝑁))
5		ovex 7447	. . . . . . 7 ⊢ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) ∈ V
6		eqid 2725	. . . . . . 7 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))
7	5, 6	fnmpti 6691	. . . . . 6 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) Fn (Base‘(SymGrp‘𝑁))
8		ofmpteq 7702	. . . . . 6 ⊢ (((Base‘(SymGrp‘𝑁)) ∈ V ∧ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) Fn (Base‘(SymGrp‘𝑁)) ∧ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) Fn (Base‘(SymGrp‘𝑁))) → ((𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) ∘_f + (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) + ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))))
9	1, 4, 7, 8	mp3an 1457	. . . . 5 ⊢ ((𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) ∘_f + (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) + ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
10		mdetrlin.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing)
11		crngring 20187	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	12	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
14		mdetrlin.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
15		mdetrlin.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (𝑁 Mat 𝑅)
16		mdetrlin.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐴)
17	15, 16	matrcl 22328	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
18	14, 17	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
19	18	simpld 493	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ Fin)
20		zrhpsgnmhm 21518	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
21	12, 19, 20	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
22		eqid 2725	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
23		eqid 2725	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
24		eqid 2725	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
25	23, 24	mgpbas 20082	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
26	22, 25	mhmf 18743	. . . . . . . . . 10 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶(Base‘𝑅))
27	21, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶(Base‘𝑅))
28	27	ffvelcdmda 7087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ (Base‘𝑅))
29	23	crngmgp 20183	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
30	10, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
31	30	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
32	19	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 ∈ Fin)
33	15, 24, 16	matbas2i 22340	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
34		elmapi 8864	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) → 𝑌:(𝑁 × 𝑁)⟶(Base‘𝑅))
35	14, 33, 34	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:(𝑁 × 𝑁)⟶(Base‘𝑅))
36	35	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑌:(𝑁 × 𝑁)⟶(Base‘𝑅))
37		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑟 ∈ 𝑁)
38		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
39	38, 22	symgbasf 19332	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) → 𝑝:𝑁⟶𝑁)
40	39	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁⟶𝑁)
41	40	ffvelcdmda 7087	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑝‘𝑟) ∈ 𝑁)
42	36, 37, 41	fovcdmd 7588	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑟𝑌(𝑝‘𝑟)) ∈ (Base‘𝑅))
43	42	ralrimiva 3136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑟 ∈ 𝑁 (𝑟𝑌(𝑝‘𝑟)) ∈ (Base‘𝑅))
44	25, 31, 32, 43	gsummptcl 19924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) ∈ (Base‘𝑅))
45		mdetrlin.z	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
46	15, 24, 16	matbas2i 22340	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ 𝐵 → 𝑍 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
47		elmapi 8864	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) → 𝑍:(𝑁 × 𝑁)⟶(Base‘𝑅))
48	45, 46, 47	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍:(𝑁 × 𝑁)⟶(Base‘𝑅))
49	48	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑍:(𝑁 × 𝑁)⟶(Base‘𝑅))
50	49, 37, 41	fovcdmd 7588	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑟𝑍(𝑝‘𝑟)) ∈ (Base‘𝑅))
51	50	ralrimiva 3136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑟 ∈ 𝑁 (𝑟𝑍(𝑝‘𝑟)) ∈ (Base‘𝑅))
52	25, 31, 32, 51	gsummptcl 19924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ (Base‘𝑅))
53		mdetrlin.p	. . . . . . . . 9 ⊢ + = (+_g‘𝑅)
54		eqid 2725	. . . . . . . . 9 ⊢ (._r‘𝑅) = (._r‘𝑅)
55	24, 53, 54	ringdi 20202	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ (Base‘𝑅))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)(((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) = (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) + ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
56	13, 28, 44, 52, 55	syl13anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)(((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) = (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) + ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))
57		cmnmnd 19754	. . . . . . . . . . . . 13 ⊢ ((mulGrp‘𝑅) ∈ CMnd → (mulGrp‘𝑅) ∈ Mnd)
58	31, 57	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ Mnd)
59		mdetrlin.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ 𝑁)
60	59	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝐼 ∈ 𝑁)
61	35	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑌:(𝑁 × 𝑁)⟶(Base‘𝑅))
62	40, 60	ffvelcdmd 7088	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑝‘𝐼) ∈ 𝑁)
63	61, 60, 62	fovcdmd 7588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑌(𝑝‘𝐼)) ∈ (Base‘𝑅))
64		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝐼 → 𝑟 = 𝐼)
65		fveq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝐼 → (𝑝‘𝑟) = (𝑝‘𝐼))
66	64, 65	oveq12d 7432	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → (𝑟𝑌(𝑝‘𝑟)) = (𝐼𝑌(𝑝‘𝐼)))
67	25, 66	gsumsn 19911	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑁 ∧ (𝐼𝑌(𝑝‘𝐼)) ∈ (Base‘𝑅)) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) = (𝐼𝑌(𝑝‘𝐼)))
68	58, 60, 63, 67	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) = (𝐼𝑌(𝑝‘𝐼)))
69	68, 63	eqeltrd 2825	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) ∈ (Base‘𝑅))
70	48	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑍:(𝑁 × 𝑁)⟶(Base‘𝑅))
71	70, 60, 62	fovcdmd 7588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑍(𝑝‘𝐼)) ∈ (Base‘𝑅))
72	64, 65	oveq12d 7432	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐼 → (𝑟𝑍(𝑝‘𝑟)) = (𝐼𝑍(𝑝‘𝐼)))
73	25, 72	gsumsn 19911	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑁 ∧ (𝐼𝑍(𝑝‘𝐼)) ∈ (Base‘𝑅)) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) = (𝐼𝑍(𝑝‘𝐼)))
74	58, 60, 71, 73	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) = (𝐼𝑍(𝑝‘𝐼)))
75	74, 71	eqeltrd 2825	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ (Base‘𝑅))
76		difssd 4123	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑁 ∖ {𝐼}) ⊆ 𝑁)
77	32, 76	ssfid 9288	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑁 ∖ {𝐼}) ∈ Fin)
78		eldifi 4117	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (𝑁 ∖ {𝐼}) → 𝑟 ∈ 𝑁)
79		mdetrlin.x	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
80	15, 24, 16	matbas2i 22340	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
81		elmapi 8864	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶(Base‘𝑅))
82	79, 80, 81	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶(Base‘𝑅))
83	82	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶(Base‘𝑅))
84	83, 37, 41	fovcdmd 7588	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ 𝑁) → (𝑟𝑋(𝑝‘𝑟)) ∈ (Base‘𝑅))
85	78, 84	sylan2 591	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟𝑋(𝑝‘𝑟)) ∈ (Base‘𝑅))
86	85	ralrimiva 3136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑟 ∈ (𝑁 ∖ {𝐼})(𝑟𝑋(𝑝‘𝑟)) ∈ (Base‘𝑅))
87	25, 31, 77, 86	gsummptcl 19924	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))) ∈ (Base‘𝑅))
88	24, 53, 54	ringdir 20203	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))) ∈ (Base‘𝑅))) → ((((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))) = ((((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))) + (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))))))
89	13, 69, 75, 87, 88	syl13anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))) = ((((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))) + (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))))))
90	23, 54	mgpplusg 20080	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
91		disjdif 4465	. . . . . . . . . . . 12 ⊢ ({𝐼} ∩ (𝑁 ∖ {𝐼})) = ∅
92	91	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} ∩ (𝑁 ∖ {𝐼})) = ∅)
93	59	snssd 4806	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝐼} ⊆ 𝑁)
94	93	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → {𝐼} ⊆ 𝑁)
95		undif 4475	. . . . . . . . . . . . 13 ⊢ ({𝐼} ⊆ 𝑁 ↔ ({𝐼} ∪ (𝑁 ∖ {𝐼})) = 𝑁)
96	94, 95	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} ∪ (𝑁 ∖ {𝐼})) = 𝑁)
97	96	eqcomd 2731	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 = ({𝐼} ∪ (𝑁 ∖ {𝐼})))
98	25, 90, 31, 32, 84, 92, 97	gsummptfidmsplit 19887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
99		mdetrlin.eq	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁))))
100	99	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑋 ↾ ({𝐼} × 𝑁)) = ((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁))))
101	100	oveqd 7431	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)))
102		xpss1 5689	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝐼} ⊆ 𝑁 → ({𝐼} × 𝑁) ⊆ (𝑁 × 𝑁))
103	94, 102	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} × 𝑁) ⊆ (𝑁 × 𝑁))
104	61, 103	fssresd 6757	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑌 ↾ ({𝐼} × 𝑁)):({𝐼} × 𝑁)⟶(Base‘𝑅))
105	104	ffnd 6716	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑌 ↾ ({𝐼} × 𝑁)) Fn ({𝐼} × 𝑁))
106	70, 103	fssresd 6757	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑍 ↾ ({𝐼} × 𝑁)):({𝐼} × 𝑁)⟶(Base‘𝑅))
107	106	ffnd 6716	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑍 ↾ ({𝐼} × 𝑁)) Fn ({𝐼} × 𝑁))
108		snex 5425	. . . . . . . . . . . . . . . . 17 ⊢ {𝐼} ∈ V
109		xpexg 7748	. . . . . . . . . . . . . . . . 17 ⊢ (({𝐼} ∈ V ∧ 𝑁 ∈ Fin) → ({𝐼} × 𝑁) ∈ V)
110	108, 32, 109	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ({𝐼} × 𝑁) ∈ V)
111		snidg 4656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ 𝑁 → 𝐼 ∈ {𝐼})
112	60, 111	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝐼 ∈ {𝐼})
113	112, 62	opelxpd 5709	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ⟨𝐼, (𝑝‘𝐼)⟩ ∈ ({𝐼} × 𝑁))
114		fnfvof 7697	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑌 ↾ ({𝐼} × 𝑁)) Fn ({𝐼} × 𝑁) ∧ (𝑍 ↾ ({𝐼} × 𝑁)) Fn ({𝐼} × 𝑁)) ∧ (({𝐼} × 𝑁) ∈ V ∧ ⟨𝐼, (𝑝‘𝐼)⟩ ∈ ({𝐼} × 𝑁))) → (((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁)))‘⟨𝐼, (𝑝‘𝐼)⟩) = (((𝑌 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩) + ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)))
115	105, 107, 110, 113, 114	syl22anc 837	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁)))‘⟨𝐼, (𝑝‘𝐼)⟩) = (((𝑌 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩) + ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)))
116		df-ov 7417	. . . . . . . . . . . . . . 15 ⊢ (𝐼((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)) = (((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁)))‘⟨𝐼, (𝑝‘𝐼)⟩)
117		df-ov 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝐼(𝑌 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = ((𝑌 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)
118		df-ov 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩)
119	117, 118	oveq12i 7426	. . . . . . . . . . . . . . 15 ⊢ ((𝐼(𝑌 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) + (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))) = (((𝑌 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩) + ((𝑍 ↾ ({𝐼} × 𝑁))‘⟨𝐼, (𝑝‘𝐼)⟩))
120	115, 116, 119	3eqtr4g 2790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼((𝑌 ↾ ({𝐼} × 𝑁)) ∘_f + (𝑍 ↾ ({𝐼} × 𝑁)))(𝑝‘𝐼)) = ((𝐼(𝑌 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) + (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))))
121	101, 120	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = ((𝐼(𝑌 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) + (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))))
122		ovres 7582	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ {𝐼} ∧ (𝑝‘𝐼) ∈ 𝑁) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑋(𝑝‘𝐼)))
123	112, 62, 122	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑋 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑋(𝑝‘𝐼)))
124		ovres 7582	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ {𝐼} ∧ (𝑝‘𝐼) ∈ 𝑁) → (𝐼(𝑌 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑌(𝑝‘𝐼)))
125	112, 62, 124	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑌 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑌(𝑝‘𝐼)))
126		ovres 7582	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ {𝐼} ∧ (𝑝‘𝐼) ∈ 𝑁) → (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑍(𝑝‘𝐼)))
127	112, 62, 126	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) = (𝐼𝑍(𝑝‘𝐼)))
128	125, 127	oveq12d 7432	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((𝐼(𝑌 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼)) + (𝐼(𝑍 ↾ ({𝐼} × 𝑁))(𝑝‘𝐼))) = ((𝐼𝑌(𝑝‘𝐼)) + (𝐼𝑍(𝑝‘𝐼))))
129	121, 123, 128	3eqtr3d 2773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) = ((𝐼𝑌(𝑝‘𝐼)) + (𝐼𝑍(𝑝‘𝐼))))
130	82	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑋:(𝑁 × 𝑁)⟶(Base‘𝑅))
131	130, 60, 62	fovcdmd 7588	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝐼𝑋(𝑝‘𝐼)) ∈ (Base‘𝑅))
132	64, 65	oveq12d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝐼 → (𝑟𝑋(𝑝‘𝑟)) = (𝐼𝑋(𝑝‘𝐼)))
133	25, 132	gsumsn 19911	. . . . . . . . . . . . 13 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑁 ∧ (𝐼𝑋(𝑝‘𝐼)) ∈ (Base‘𝑅)) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟)))) = (𝐼𝑋(𝑝‘𝐼)))
134	58, 60, 131, 133	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟)))) = (𝐼𝑋(𝑝‘𝐼)))
135	68, 74	oveq12d 7432	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((𝐼𝑌(𝑝‘𝐼)) + (𝐼𝑍(𝑝‘𝐼))))
136	129, 134, 135	3eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))))
137	136	oveq1d 7429	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑋(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))) = ((((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
138	98, 137	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))) = ((((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟)))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
139	25, 90, 31, 32, 42, 92, 97	gsummptfidmsplit 19887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑌(𝑝‘𝑟))))))
140		mdetrlin.ne1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
141	140	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
142	141	oveqd 7431	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟(𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)))
143		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → 𝑟 ∈ (𝑁 ∖ {𝐼}))
144	78, 41	sylan2 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑝‘𝑟) ∈ 𝑁)
145		ovres 7582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (𝑁 ∖ {𝐼}) ∧ (𝑝‘𝑟) ∈ 𝑁) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
146	143, 144, 145	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
147		ovres 7582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (𝑁 ∖ {𝐼}) ∧ (𝑝‘𝑟) ∈ 𝑁) → (𝑟(𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑌(𝑝‘𝑟)))
148	143, 144, 147	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑌(𝑝‘𝑟)))
149	142, 146, 148	3eqtr3rd 2774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟𝑌(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
150	149	mpteq2dva 5241	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑌(𝑝‘𝑟))) = (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))
151	150	oveq2d 7430	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑌(𝑝‘𝑟)))) = ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))))
152	151	oveq2d 7430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑌(𝑝‘𝑟))))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
153	139, 152	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
154	25, 90, 31, 32, 50, 92, 97	gsummptfidmsplit 19887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))))
155		mdetrlin.ne2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
156	155	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
157	156	oveqd 7431	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)))
158		ovres 7582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (𝑁 ∖ {𝐼}) ∧ (𝑝‘𝑟) ∈ 𝑁) → (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
159	143, 144, 158	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟(𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁))(𝑝‘𝑟)) = (𝑟𝑍(𝑝‘𝑟)))
160	157, 146, 159	3eqtr3rd 2774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑟 ∈ (𝑁 ∖ {𝐼})) → (𝑟𝑍(𝑝‘𝑟)) = (𝑟𝑋(𝑝‘𝑟)))
161	160	mpteq2dva 5241	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))) = (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))
162	161	oveq2d 7430	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟)))) = ((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))))
163	162	oveq2d 7430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑍(𝑝‘𝑟))))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
164	154, 163	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) = (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))))
165	153, 164	oveq12d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑌(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟))))) + (((mulGrp‘𝑅) Σ_g (𝑟 ∈ {𝐼} ↦ (𝑟𝑍(𝑝‘𝑟))))(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑟𝑋(𝑝‘𝑟)))))))
166	89, 138, 165	3eqtr4rd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) = ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))))
167	166	oveq2d 7430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)(((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) + ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))
168	56, 167	eqtr3d 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) + ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))
169	168	mpteq2dva 5241	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ (((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) + ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))))))
170	9, 169	eqtrid 2777	. . . 4 ⊢ (𝜑 → ((𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) ∘_f + (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟)))))))
171	170	oveq2d 7430	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) ∘_f + (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))))
172		ringcmn 20220	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
173	10, 11, 172	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
174	38, 22	symgbasfi 19335	. . . . 5 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
175	19, 174	syl 17	. . . 4 ⊢ (𝜑 → (Base‘(SymGrp‘𝑁)) ∈ Fin)
176	24, 54	ringcl 20192	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))) ∈ (Base‘𝑅)) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) ∈ (Base‘𝑅))
177	13, 28, 44, 176	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))) ∈ (Base‘𝑅))
178	24, 54	ringcl 20192	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))) ∈ (Base‘𝑅)) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) ∈ (Base‘𝑅))
179	13, 28, 52, 178	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))) ∈ (Base‘𝑅))
180	24, 53, 173, 175, 177, 179, 3, 6	gsummptfidmadd2 19883	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟)))))) ∘_f + (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))) = ((𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))))) + (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))))
181	171, 180	eqtr3d 2767	. 2 ⊢ (𝜑 → (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))) = ((𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))))) + (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))))
182		mdetrlin.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
183		eqid 2725	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
184		eqid 2725	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
185	182, 15, 16, 22, 183, 184, 54, 23	mdetleib2 22506	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐷‘𝑋) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))))
186	10, 79, 185	syl2anc 582	. 2 ⊢ (𝜑 → (𝐷‘𝑋) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑋(𝑝‘𝑟))))))))
187	182, 15, 16, 22, 183, 184, 54, 23	mdetleib2 22506	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑌 ∈ 𝐵) → (𝐷‘𝑌) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))))))
188	10, 14, 187	syl2anc 582	. . 3 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))))))
189	182, 15, 16, 22, 183, 184, 54, 23	mdetleib2 22506	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑍 ∈ 𝐵) → (𝐷‘𝑍) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))))
190	10, 45, 189	syl2anc 582	. . 3 ⊢ (𝜑 → (𝐷‘𝑍) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟))))))))
191	188, 190	oveq12d 7432	. 2 ⊢ (𝜑 → ((𝐷‘𝑌) + (𝐷‘𝑍)) = ((𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑌(𝑝‘𝑟))))))) + (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑟 ∈ 𝑁 ↦ (𝑟𝑍(𝑝‘𝑟)))))))))
192	181, 186, 191	3eqtr4d 2775	1 ⊢ (𝜑 → (𝐷‘𝑋) = ((𝐷‘𝑌) + (𝐷‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∖ cdif 3936 ∪ cun 3937 ∩ cin 3938 ⊆ wss 3939 ∅c0 4316 {csn 4622 ⟨cop 4628 ↦ cmpt 5224 × cxp 5668 ↾ cres 5672 ∘ ccom 5674 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7414 ∘_f cof 7678 ↑_m cmap 8841 Fincfn 8960 Basecbs 17177 +_gcplusg 17230 ._rcmulr 17231 Σ_g cgsu 17419 Mndcmnd 18691 MndHom cmhm 18735 SymGrpcsymg 19323 pmSgncpsgn 19446 CMndccmn 19737 mulGrpcmgp 20076 Ringcrg 20175 CRingccrg 20176 ℤRHomczrh 21427 Mat cmat 22323 maDet cmdat 22502

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-addf 11215 ax-mulf 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-ot 4631 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-xnn0 12573 df-z 12587 df-dec 12706 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-word 14495 df-lsw 14543 df-concat 14551 df-s1 14576 df-substr 14621 df-pfx 14651 df-splice 14730 df-reverse 14739 df-s2 14829 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-efmnd 18823 df-grp 18895 df-minusg 18896 df-mulg 19026 df-subg 19080 df-ghm 19170 df-gim 19215 df-cntz 19270 df-oppg 19299 df-symg 19324 df-pmtr 19399 df-psgn 19448 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-drng 20628 df-sra 21060 df-rgmod 21061 df-cnfld 21282 df-zring 21375 df-zrh 21431 df-dsmm 21668 df-frlm 21683 df-mat 22324 df-mdet 22503

This theorem is referenced by: mdetrlin2 22525 mdetuni0 22539 mdetmul 22541

W3C validator