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Theorem mdetdiaglem 22591

Description: Lemma for mdetdiag 22592. Previously part of proof for mdet1 22594. (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 5871	. . . 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 6903	. . 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 2726	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
8		mdetdiaglem.g	. . . . . . . . . . . 12 ⊢ 𝐻 = (Base‘(SymGrp‘𝑁))
9	7, 8	symgbasf1o 19372	. . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁–1-1-onto→𝑁)
10		f1ofn 6844	. . . . . . . . . . 11 ⊢ (𝑃:𝑁–1-1-onto→𝑁 → 𝑃 Fn 𝑁)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝐻 → 𝑃 Fn 𝑁)
12		fnnfpeq0 7192	. . . . . . . . . 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 480	. . . . . . . 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 2975	. . . . . 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 4349	. . . . . . 7 ⊢ (dom (𝑃 ∖ I ) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ dom (𝑃 ∖ I ))
18		eqid 2726	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
19		mdetdiag.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (mulGrp‘𝑅)
20		eqid 2726	. . . . . . . . . . . 12 ⊢ (._r‘𝑅) = (._r‘𝑅)
21	19, 20	mgpplusg 20121	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (+_g‘𝐺)
22	19	crngmgp 20224	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
23	22	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ CMnd)
24	23	ad2antrr 724	. . . . . . . . . . 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 1210	. . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝑅)
28		mdetdiag.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘𝐴)
29	26, 27, 28	matbas2i 22415	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
30	29	3ad2ant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
31		elmapi 8878	. . . . . . . . . . . . . . 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 20123	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝐺)
34	33	eqcomi 2735	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝑅)
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (Base‘𝐺) = (Base‘𝑅))
36	35	feq3d 6715	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 19373	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁⟶𝑁)
40	39	ad2antrl 726	. . . . . . . . . . . . 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	ffvelcdmda 7098	. . . . . . . . . . . 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 483	. . . . . . . . . . . 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	fovcdmd 7598	. . . . . . . . . . 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 4476	. . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∖ I ) ⊆ 𝑃
47		dmss 5909	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∖ I ) ⊆ 𝑃 → dom (𝑃 ∖ I ) ⊆ dom 𝑃)
48	46, 47	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑃 ∖ I ) ⊆ dom 𝑃
49	39	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
50	48, 49	fssdm 6747	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
51	50	sseld 3978	. . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . 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 4818	. . . . . . . . . . . . 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 4486	. . . . . . . . . . . . 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 2732	. . . . . . . . . . 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 19928	. . . . . . . . . 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 20228	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
60	19	ringmgp 20222	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐺 ∈ Mnd)
62	61	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd)
63	62	ad2antrr 724	. . . . . . . . . . . . 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 3466	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . 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	ffvelcdmd 7099	. . . . . . . . . . . . . 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	fovcdmd 7598	. . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → (𝑃‘𝑘) = (𝑃‘𝑠))
70		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → 𝑘 = 𝑠)
71	69, 70	oveq12d 7442	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑠 → ((𝑃‘𝑘)𝑀𝑘) = ((𝑃‘𝑠)𝑀𝑠))
72	33, 71	gsumsn 19952	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 7191	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 Fn 𝑁 ∧ 𝑠 ∈ 𝑁) → (𝑠 ∈ dom (𝑃 ∖ I ) ↔ (𝑃‘𝑠) ≠ 𝑠))
77	75, 52, 76	syl2anc 582	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃:𝑁⟶𝑁)
80	39	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
81	48, 80	fssdm 6747	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
82	81	sseld 3978	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁))
83	82	impr 453	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁)
84	79, 83	ffvelcdmd 7099	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁)
85		neeq1 2993	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖 ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑗))
86		oveq1 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑗))
87	86	eqeq1d 2728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑗) = 0 ))
88	85, 87	imbi12d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 )))
89		neeq2 2994	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠) ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑠))
90		oveq2 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠)𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑠))
91	90	eqeq1d 2728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠)𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑠) = 0 ))
92	89, 91	imbi12d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )))
93	88, 92	rspc2v 3619	. . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . 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 405	. . . . . . . . . . . . 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 2766	. . . . . . . . . . 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 7439	. . . . . . . . . 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 1130	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
101	100	ad2antrr 724	. . . . . . . . . . 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 479	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝐺 ∈ CMnd)
103		simpl2 1189	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑁 ∈ Fin)
104		difss 4131	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∖ {𝑠}) ⊆ 𝑁
105		ssfi 9211	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝑠}) ⊆ 𝑁) → (𝑁 ∖ {𝑠}) ∈ Fin)
106	103, 104, 105	sylancl 584	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑁 ∖ {𝑠}) ∈ Fin)
107	32	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
108	80	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑃:𝑁⟶𝑁)
109		eldifi 4126	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑠}) → 𝑘 ∈ 𝑁)
110	109	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑘 ∈ 𝑁)
111	108, 110	ffvelcdmd 7099	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → (𝑃‘𝑘) ∈ 𝑁)
112	107, 111, 110	fovcdmd 7598	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → ((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
113	112	ralrimiva 3136	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → ∀𝑘 ∈ (𝑁 ∖ {𝑠})((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
114	33, 102, 106, 113	gsummptcl 19965	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅))
115	114	ad2ant2r 745	. . . . . . . . . . 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 20272	. . . . . . . . . . 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 582	. . . . . . . . . 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 2770	. . . . . . . . 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 455	. . . . . . . 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 1929	. . . . . . 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	biimtrid 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 452	. . . 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 1114	. . 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 7442	. 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 1145	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ CRing ∧ 𝑁 ∈ Fin))
128		simpl 481	. . . 4 ⊢ ((𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁)) → 𝑃 ∈ 𝐻)
129	58	ad2antrr 724	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → 𝑅 ∈ Ring)
130		zrhpsgnmhm 21580	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
131	58, 130	sylan 578	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
132		eqid 2726	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
133	8, 132	mhmf 18779	. . . . . . 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	ffvelcdmda 7098	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅)))
136		eqid 2726	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
137	136, 27	mgpbas 20123	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
138	137	eqcomi 2735	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘𝑅)
139		mdetdiaglem.t	. . . . . 6 ⊢ · = (._r‘𝑅)
140	138, 139, 116	ringrz 20273	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
141	129, 135, 140	syl2anc 582	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
142	127, 128, 141	syl2an 594	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
143	142	3adant2 1128	. 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 2766	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 394 ∧ w3a 1084 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 Vcvv 3462 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 {csn 4633 ↦ cmpt 5236 I cid 5579 × cxp 5680 dom cdm 5682 ↾ cres 5684 ∘ ccom 5686 Fn wfn 6549 ⟶wf 6550 –1-1-onto→wf1o 6553 ‘cfv 6554 (class class class)co 7424 ↑_m cmap 8855 Fincfn 8974 Basecbs 17213 ._rcmulr 17267 0_gc0g 17454 Σ_g cgsu 17455 Mndcmnd 18727 MndHom cmhm 18771 SymGrpcsymg 19364 pmSgncpsgn 19487 CMndccmn 19778 mulGrpcmgp 20117 Ringcrg 20216 CRingccrg 20217 ℤRHomczrh 21489 Mat cmat 22398 maDet cmdat 22577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-addf 11237 ax-mulf 11238

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-ot 4642 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12597 df-z 12611 df-dec 12730 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-word 14523 df-lsw 14571 df-concat 14579 df-s1 14604 df-substr 14649 df-pfx 14679 df-splice 14758 df-reverse 14767 df-s2 14857 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-efmnd 18859 df-grp 18931 df-minusg 18932 df-mulg 19062 df-subg 19117 df-ghm 19207 df-gim 19253 df-cntz 19311 df-oppg 19340 df-symg 19365 df-pmtr 19440 df-psgn 19489 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-rhm 20454 df-subrng 20528 df-subrg 20553 df-drng 20709 df-sra 21151 df-rgmod 21152 df-cnfld 21344 df-zring 21437 df-zrh 21493 df-dsmm 21730 df-frlm 21745 df-mat 22399

This theorem is referenced by: mdetdiag 22592

W3C validator