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Theorem mdetdiaglem 21947

Description: Lemma for mdetdiag 21948. Previously part of proof for mdet1 21950. (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 5820	. . . 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 6844	. . 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 2736	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
8		mdetdiaglem.g	. . . . . . . . . . . 12 ⊢ 𝐻 = (Base‘(SymGrp‘𝑁))
9	7, 8	symgbasf1o 19156	. . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁–1-1-onto→𝑁)
10		f1ofn 6785	. . . . . . . . . . 11 ⊢ (𝑃:𝑁–1-1-onto→𝑁 → 𝑃 Fn 𝑁)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝐻 → 𝑃 Fn 𝑁)
12		fnnfpeq0 7124	. . . . . . . . . 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 482	. . . . . . . 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 2988	. . . . . 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 4306	. . . . . . 7 ⊢ (dom (𝑃 ∖ I ) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ dom (𝑃 ∖ I ))
18		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
19		mdetdiag.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (mulGrp‘𝑅)
20		eqid 2736	. . . . . . . . . . . 12 ⊢ (._r‘𝑅) = (._r‘𝑅)
21	19, 20	mgpplusg 19900	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (+_g‘𝐺)
22	19	crngmgp 19972	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
23	22	3ad2ant1 1133	. . . . . . . . . . . 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 1213	. . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝑅)
28		mdetdiag.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘𝐴)
29	26, 27, 28	matbas2i 21771	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
30	29	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
31		elmapi 8787	. . . . . . . . . . . . . . 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 19902	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝐺)
34	33	eqcomi 2745	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝑅)
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (Base‘𝐺) = (Base‘𝑅))
36	35	feq3d 6655	. . . . . . . . . . . . . 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 19157	. . . . . . . . . . . . . 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 7035	. . . . . . . . . . . 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 485	. . . . . . . . . . . 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 7526	. . . . . . . . . . 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 4431	. . . . . . . . . . . 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 4091	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∖ I ) ⊆ 𝑃
47		dmss 5858	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∖ I ) ⊆ 𝑃 → dom (𝑃 ∖ I ) ⊆ dom 𝑃)
48	46, 47	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑃 ∖ I ) ⊆ dom 𝑃
49	39	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
50	48, 49	fssdm 6688	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
51	50	sseld 3943	. . . . . . . . . . . . . . 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 455	. . . . . . . . . . . . . 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 4769	. . . . . . . . . . . . 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 4441	. . . . . . . . . . . . 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 2742	. . . . . . . . . . 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 19707	. . . . . . . . . 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 19976	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
60	19	ringmgp 19970	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐺 ∈ Mnd)
62	61	3adant3 1132	. . . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . . 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 7526	. . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → (𝑃‘𝑘) = (𝑃‘𝑠))
70		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → 𝑘 = 𝑠)
71	69, 70	oveq12d 7375	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑠 → ((𝑃‘𝑘)𝑀𝑘) = ((𝑃‘𝑠)𝑀𝑠))
72	33, 71	gsumsn 19731	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 7123	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 Fn 𝑁 ∧ 𝑠 ∈ 𝑁) → (𝑠 ∈ dom (𝑃 ∖ I ) ↔ (𝑃‘𝑠) ≠ 𝑠))
77	75, 52, 76	syl2anc 584	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
81	48, 80	fssdm 6688	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
82	81	sseld 3943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁))
83	82	impr 455	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁)
84	79, 83	ffvelcdmd 7036	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁)
85		neeq1 3006	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖 ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑗))
86		oveq1 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑗))
87	86	eqeq1d 2738	. . . . . . . . . . . . . . . . . 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 3007	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠) ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑠))
90		oveq2 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠)𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑠))
91	90	eqeq1d 2738	. . . . . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 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 7372	. . . . . . . . . 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 1133	. . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝐺 ∈ CMnd)
103		simpl2 1192	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑁 ∈ Fin)
104		difss 4091	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∖ {𝑠}) ⊆ 𝑁
105		ssfi 9117	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝑠}) ⊆ 𝑁) → (𝑁 ∖ {𝑠}) ∈ Fin)
106	103, 104, 105	sylancl 586	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑃:𝑁⟶𝑁)
109		eldifi 4086	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑠}) → 𝑘 ∈ 𝑁)
110	109	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑘 ∈ 𝑁)
111	108, 110	ffvelcdmd 7036	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → (𝑃‘𝑘) ∈ 𝑁)
112	107, 111, 110	fovcdmd 7526	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → ((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
113	112	ralrimiva 3143	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → ∀𝑘 ∈ (𝑁 ∖ {𝑠})((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
114	33, 102, 106, 113	gsummptcl 19744	. . . . . . . . . . . 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 20011	. . . . . . . . . . 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 584	. . . . . . . . . 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 2780	. . . . . . . . 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 457	. . . . . . . 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 1936	. . . . . . 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 454	. . . 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 1117	. . 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 7375	. 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 1148	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ CRing ∧ 𝑁 ∈ Fin))
128		simpl 483	. . . 4 ⊢ ((𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁)) → 𝑃 ∈ 𝐻)
129	58	ad2antrr 724	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → 𝑅 ∈ Ring)
130		zrhpsgnmhm 20988	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
131	58, 130	sylan 580	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
132		eqid 2736	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
133	8, 132	mhmf 18607	. . . . . . 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 7035	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅)))
136		eqid 2736	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
137	136, 27	mgpbas 19902	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
138	137	eqcomi 2745	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘𝑅)
139		mdetdiaglem.t	. . . . . 6 ⊢ · = (._r‘𝑅)
140	138, 139, 116	ringrz 20012	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
141	129, 135, 140	syl2anc 584	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
142	127, 128, 141	syl2an 596	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
143	142	3adant2 1131	. 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 2776	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 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 {csn 4586 ↦ cmpt 5188 I cid 5530 × cxp 5631 dom cdm 5633 ↾ cres 5635 ∘ ccom 5637 Fn wfn 6491 ⟶wf 6492 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 Fincfn 8883 Basecbs 17083 ._rcmulr 17134 0_gc0g 17321 Σ_g cgsu 17322 Mndcmnd 18556 MndHom cmhm 18599 SymGrpcsymg 19148 pmSgncpsgn 19271 CMndccmn 19562 mulGrpcmgp 19896 Ringcrg 19964 CRingccrg 19965 ℤRHomczrh 20900 Mat cmat 21754 maDet cmdat 21933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-word 14403 df-lsw 14451 df-concat 14459 df-s1 14484 df-substr 14529 df-pfx 14559 df-splice 14638 df-reverse 14647 df-s2 14737 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-efmnd 18679 df-grp 18751 df-minusg 18752 df-mulg 18873 df-subg 18925 df-ghm 19006 df-gim 19049 df-cntz 19097 df-oppg 19124 df-symg 19149 df-pmtr 19224 df-psgn 19273 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-subrg 20220 df-sra 20633 df-rgmod 20634 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-dsmm 21138 df-frlm 21153 df-mat 21755

This theorem is referenced by: mdetdiag 21948

W3C validator