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Theorem mdetdiaglem 21655

Description: Lemma for mdetdiag 21656. Previously part of proof for mdet1 21658. (Contributed by SO, 10-Jul-2018.) (Revised by AV, 17-Aug-2019.)

**Hypotheses**
Ref	Expression
mdetdiag.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetdiag.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetdiag.b	⊢ 𝐵 = (Base‘𝐴)
mdetdiag.g	⊢ 𝐺 = (mulGrp‘𝑅)
mdetdiag.0	⊢ 0 = (0_g‘𝑅)
mdetdiaglem.g	⊢ 𝐻 = (Base‘(SymGrp‘𝑁))
mdetdiaglem.z	⊢ 𝑍 = (ℤRHom‘𝑅)
mdetdiaglem.s	⊢ 𝑆 = (pmSgn‘𝑁)
mdetdiaglem.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
mdetdiaglem	⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 )

Distinct variable groups: 𝐵,𝑘 𝑘,𝐺 𝑘,𝐻 𝑖,𝑀,𝑗,𝑘 𝑖,𝑁,𝑗,𝑘 𝑃,𝑖,𝑗,𝑘 𝑅,𝑘 0 ,𝑖,𝑗,𝑘 Allowed substitution hints: 𝐴(𝑖,𝑗,𝑘) 𝐵(𝑖,𝑗) 𝐷(𝑖,𝑗,𝑘) 𝑅(𝑖,𝑗) 𝑆(𝑖,𝑗,𝑘) · (𝑖,𝑗,𝑘) 𝐺(𝑖,𝑗) 𝐻(𝑖,𝑗) 𝑍(𝑖,𝑗,𝑘)

Proof of Theorem mdetdiaglem Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetdiaglem.z	. . . . . 6 ⊢ 𝑍 = (ℤRHom‘𝑅)
2	1	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → 𝑍 = (ℤRHom‘𝑅))
3		mdetdiaglem.s	. . . . . 6 ⊢ 𝑆 = (pmSgn‘𝑁)
4	3	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → 𝑆 = (pmSgn‘𝑁))
5	2, 4	coeq12d 5762	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (𝑍 ∘ 𝑆) = ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)))
6	5	fveq1d 6758	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((𝑍 ∘ 𝑆)‘𝑃) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃))
7		eqid 2738	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
8		mdetdiaglem.g	. . . . . . . . . . . 12 ⊢ 𝐻 = (Base‘(SymGrp‘𝑁))
9	7, 8	symgbasf1o 18897	. . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁–1-1-onto→𝑁)
10		f1ofn 6701	. . . . . . . . . . 11 ⊢ (𝑃:𝑁–1-1-onto→𝑁 → 𝑃 Fn 𝑁)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝐻 → 𝑃 Fn 𝑁)
12		fnnfpeq0 7032	. . . . . . . . . 10 ⊢ (𝑃 Fn 𝑁 → (dom (𝑃 ∖ I ) = ∅ ↔ 𝑃 = ( I ↾ 𝑁)))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐻 → (dom (𝑃 ∖ I ) = ∅ ↔ 𝑃 = ( I ↾ 𝑁)))
14	13	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (dom (𝑃 ∖ I ) = ∅ ↔ 𝑃 = ( I ↾ 𝑁)))
15	14	bicomd 222	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑃 = ( I ↾ 𝑁) ↔ dom (𝑃 ∖ I ) = ∅))
16	15	necon3bid 2987	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑃 ≠ ( I ↾ 𝑁) ↔ dom (𝑃 ∖ I ) ≠ ∅))
17		n0 4277	. . . . . . 7 ⊢ (dom (𝑃 ∖ I ) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ dom (𝑃 ∖ I ))
18		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
19		mdetdiag.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (mulGrp‘𝑅)
20		eqid 2738	. . . . . . . . . . . 12 ⊢ (._r‘𝑅) = (._r‘𝑅)
21	19, 20	mgpplusg 19639	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (+_g‘𝐺)
22	19	crngmgp 19706	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
23	22	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ CMnd)
24	23	ad2antrr 722	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝐺 ∈ CMnd)
25		simpll2 1211	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑁 ∈ Fin)
26		mdetdiag.a	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 = (𝑁 Mat 𝑅)
27		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝑅)
28		mdetdiag.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘𝐴)
29	26, 27, 28	matbas2i 21479	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
30	29	3ad2ant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
31		elmapi 8595	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
32	30, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
33	19, 27	mgpbas 19641	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝐺)
34	33	eqcomi 2747	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝑅)
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (Base‘𝐺) = (Base‘𝑅))
36	35	feq3d 6571	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺) ↔ 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)))
37	32, 36	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺))
38	37	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺))
39	7, 8	symgbasf 18898	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁⟶𝑁)
40	39	ad2antrl 724	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃:𝑁⟶𝑁)
41	40	ffvelrnda 6943	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → (𝑃‘𝑘) ∈ 𝑁)
42		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
43	38, 41, 42	fovrnd 7422	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → ((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝐺))
44		disjdif 4402	. . . . . . . . . . . 12 ⊢ ({𝑠} ∩ (𝑁 ∖ {𝑠})) = ∅
45	44	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ({𝑠} ∩ (𝑁 ∖ {𝑠})) = ∅)
46		difss 4062	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∖ I ) ⊆ 𝑃
47		dmss 5800	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∖ I ) ⊆ 𝑃 → dom (𝑃 ∖ I ) ⊆ dom 𝑃)
48	46, 47	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑃 ∖ I ) ⊆ dom 𝑃
49	39	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
50	48, 49	fssdm 6604	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
51	50	sseld 3916	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁))
52	51	impr 454	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁)
53	52	snssd 4739	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → {𝑠} ⊆ 𝑁)
54		undif 4412	. . . . . . . . . . . . 13 ⊢ ({𝑠} ⊆ 𝑁 ↔ ({𝑠} ∪ (𝑁 ∖ {𝑠})) = 𝑁)
55	53, 54	sylib 217	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ({𝑠} ∪ (𝑁 ∖ {𝑠})) = 𝑁)
56	55	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑁 = ({𝑠} ∪ (𝑁 ∖ {𝑠})))
57	18, 21, 24, 25, 43, 45, 56	gsummptfidmsplit 19446	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = ((𝐺 Σ_g (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘)))(._r‘𝑅)(𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))))
58		crngring 19710	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
60	19	ringmgp 19704	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐺 ∈ Mnd)
62	61	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd)
63	62	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝐺 ∈ Mnd)
64		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑠 ∈ V
65	64	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ V)
66	32	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
67	40, 52	ffvelrnd 6944	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁)
68	66, 67, 52	fovrnd 7422	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝑃‘𝑠)𝑀𝑠) ∈ (Base‘𝑅))
69		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → (𝑃‘𝑘) = (𝑃‘𝑠))
70		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → 𝑘 = 𝑠)
71	69, 70	oveq12d 7273	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑠 → ((𝑃‘𝑘)𝑀𝑘) = ((𝑃‘𝑠)𝑀𝑠))
72	33, 71	gsumsn 19470	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ 𝑠 ∈ V ∧ ((𝑃‘𝑠)𝑀𝑠) ∈ (Base‘𝑅)) → (𝐺 Σ_g (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘))) = ((𝑃‘𝑠)𝑀𝑠))
73	63, 65, 68, 72	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σ_g (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘))) = ((𝑃‘𝑠)𝑀𝑠))
74		simprr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ dom (𝑃 ∖ I ))
75	11	ad2antrl 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃 Fn 𝑁)
76		fnelnfp 7031	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 Fn 𝑁 ∧ 𝑠 ∈ 𝑁) → (𝑠 ∈ dom (𝑃 ∖ I ) ↔ (𝑃‘𝑠) ≠ 𝑠))
77	75, 52, 76	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑠 ∈ dom (𝑃 ∖ I ) ↔ (𝑃‘𝑠) ≠ 𝑠))
78	74, 77	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ≠ 𝑠)
79	39	ad2antrl 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃:𝑁⟶𝑁)
80	39	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
81	48, 80	fssdm 6604	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
82	81	sseld 3916	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁))
83	82	impr 454	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁)
84	79, 83	ffvelrnd 6944	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁)
85		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖 ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑗))
86		oveq1 7262	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑗))
87	86	eqeq1d 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑗) = 0 ))
88	85, 87	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 )))
89		neeq2 3006	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠) ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑠))
90		oveq2 7263	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠)𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑠))
91	90	eqeq1d 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠)𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑠) = 0 ))
92	89, 91	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )))
93	88, 92	rspc2v 3562	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃‘𝑠) ∈ 𝑁 ∧ 𝑠 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )))
94	84, 83, 93	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )))
95	94	impancom 451	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → ((𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I )) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )))
96	95	imp 406	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 ))
97	78, 96	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝑃‘𝑠)𝑀𝑠) = 0 )
98	73, 97	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σ_g (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )
99	98	oveq1d 7270	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝐺 Σ_g (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘)))(._r‘𝑅)(𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))) = ( 0 (._r‘𝑅)(𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))))
100	58	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
101	100	ad2antrr 722	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑅 ∈ Ring)
102	23	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝐺 ∈ CMnd)
103		simpl2 1190	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑁 ∈ Fin)
104		difss 4062	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∖ {𝑠}) ⊆ 𝑁
105		ssfi 8918	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝑠}) ⊆ 𝑁) → (𝑁 ∖ {𝑠}) ∈ Fin)
106	103, 104, 105	sylancl 585	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑁 ∖ {𝑠}) ∈ Fin)
107	32	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
108	80	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑃:𝑁⟶𝑁)
109		eldifi 4057	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑠}) → 𝑘 ∈ 𝑁)
110	109	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑘 ∈ 𝑁)
111	108, 110	ffvelrnd 6944	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → (𝑃‘𝑘) ∈ 𝑁)
112	107, 111, 110	fovrnd 7422	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → ((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
113	112	ralrimiva 3107	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → ∀𝑘 ∈ (𝑁 ∖ {𝑠})((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
114	33, 102, 106, 113	gsummptcl 19483	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅))
115	114	ad2ant2r 743	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅))
116		mdetdiag.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑅)
117	27, 20, 116	ringlz 19741	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅)) → ( 0 (._r‘𝑅)(𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 )
118	101, 115, 117	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ( 0 (._r‘𝑅)(𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 )
119	57, 99, 118	3eqtrd 2782	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )
120	119	expr 456	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ))
121	120	exlimdv 1937	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (∃𝑠 𝑠 ∈ dom (𝑃 ∖ I ) → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ))
122	17, 121	syl5bi 241	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (dom (𝑃 ∖ I ) ≠ ∅ → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ))
123	16, 122	sylbid 239	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑃 ≠ ( I ↾ 𝑁) → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ))
124	123	expimpd 453	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → ((𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁)) → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ))
125	124	3impia 1115	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )
126	6, 125	oveq12d 7273	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘)))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ))
127		3simpa 1146	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ CRing ∧ 𝑁 ∈ Fin))
128		simpl 482	. . . 4 ⊢ ((𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁)) → 𝑃 ∈ 𝐻)
129	58	ad2antrr 722	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → 𝑅 ∈ Ring)
130		zrhpsgnmhm 20701	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
131	58, 130	sylan 579	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
132		eqid 2738	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
133	8, 132	mhmf 18350	. . . . . . 7 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):𝐻⟶(Base‘(mulGrp‘𝑅)))
134	131, 133	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):𝐻⟶(Base‘(mulGrp‘𝑅)))
135	134	ffvelrnda 6943	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅)))
136		eqid 2738	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
137	136, 27	mgpbas 19641	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
138	137	eqcomi 2747	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘𝑅)
139		mdetdiaglem.t	. . . . . 6 ⊢ · = (._r‘𝑅)
140	138, 139, 116	ringrz 19742	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
141	129, 135, 140	syl2anc 583	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
142	127, 128, 141	syl2an 595	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
143	142	3adant2 1129	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
144	126, 143	eqtrd 2778	1 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 ↦ cmpt 5153 I cid 5479 × cxp 5578 dom cdm 5580 ↾ cres 5582 ∘ ccom 5584 Fn wfn 6413 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 Basecbs 16840 ._rcmulr 16889 0_gc0g 17067 Σ_g cgsu 17068 Mndcmnd 18300 MndHom cmhm 18343 SymGrpcsymg 18889 pmSgncpsgn 19012 CMndccmn 19301 mulGrpcmgp 19635 Ringcrg 19698 CRingccrg 19699 ℤRHomczrh 20613 Mat cmat 21464 maDet cmdat 21641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-splice 14391 df-reverse 14400 df-s2 14489 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-efmnd 18423 df-grp 18495 df-minusg 18496 df-mulg 18616 df-subg 18667 df-ghm 18747 df-gim 18790 df-cntz 18838 df-oppg 18865 df-symg 18890 df-pmtr 18965 df-psgn 19014 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-dsmm 20849 df-frlm 20864 df-mat 21465

This theorem is referenced by: mdetdiag 21656

W3C validator