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Theorem mdetdiaglem 20223

Description: Lemma for mdetdiag 20224. Previously part of proof for mdet1 20226. (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 5208	. . . 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 6105	. . 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 2610	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
8		mdetdiaglem.g	. . . . . . . . . . . 12 ⊢ 𝐻 = (Base‘(SymGrp‘𝑁))
9	7, 8	symgbasf1o 17626	. . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁–1-1-onto→𝑁)
10		f1ofn 6051	. . . . . . . . . . 11 ⊢ (𝑃:𝑁–1-1-onto→𝑁 → 𝑃 Fn 𝑁)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝐻 → 𝑃 Fn 𝑁)
12		fnnfpeq0 6349	. . . . . . . . . 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 212	. . . . . . 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 2826	. . . . . 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 3890	. . . . . . 7 ⊢ (dom (𝑃 ∖ I ) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ dom (𝑃 ∖ I ))
18		eqid 2610	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
19		mdetdiag.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (mulGrp‘𝑅)
20		eqid 2610	. . . . . . . . . . . 12 ⊢ (._r‘𝑅) = (._r‘𝑅)
21	19, 20	mgpplusg 18316	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (+_g‘𝐺)
22	19	crngmgp 18378	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
23	22	3ad2ant1 1075	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ CMnd)
24	23	ad2antrr 758	. . . . . . . . . . 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 1094	. . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝑅)
28		mdetdiag.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘𝐴)
29	26, 27, 28	matbas2i 20047	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑_𝑚 (𝑁 × 𝑁)))
30	29	3ad2ant3 1077	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ ((Base‘𝑅) ↑_𝑚 (𝑁 × 𝑁)))
31		elmapi 7765	. . . . . . . . . . . . . . 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 18318	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝐺)
34	33	eqcomi 2619	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝑅)
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (Base‘𝐺) = (Base‘𝑅))
36	35	feq3d 5945	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺) ↔ 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)))
37	32, 36	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺))
38	37	ad3antrrr 762	. . . . . . . . . . . 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 17627	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁⟶𝑁)
40	39	ad2antrl 760	. . . . . . . . . . . . 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 6267	. . . . . . . . . . . 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 476	. . . . . . . . . . . 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 6704	. . . . . . . . . . 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 3992	. . . . . . . . . . . 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 3699	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∖ I ) ⊆ 𝑃
47		dmss 5245	. . . . . . . . . . . . . . . . . 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	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
50		fdm 5964	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃:𝑁⟶𝑁 → dom 𝑃 = 𝑁)
51	49, 50	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → dom 𝑃 = 𝑁)
52	48, 51	syl5sseq 3616	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
53	52	sseld 3567	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁))
54	53	impr 647	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁)
55	54	snssd 4281	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → {𝑠} ⊆ 𝑁)
56		undif 4001	. . . . . . . . . . . . 13 ⊢ ({𝑠} ⊆ 𝑁 ↔ ({𝑠} ∪ (𝑁 ∖ {𝑠})) = 𝑁)
57	55, 56	sylib 207	. . . . . . . . . . . 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)
58	57	eqcomd 2616	. . . . . . . . . . 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})))
59	18, 21, 24, 25, 43, 45, 58	gsummptfidmsplit 18153	. . . . . . . . . 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)))))
60		crngring 18381	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
61	60	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
62	19	ringmgp 18376	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐺 ∈ Mnd)
64	63	3adant3 1074	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd)
65	64	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝐺 ∈ Mnd)
66		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑠 ∈ V
67	66	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)
68	32	ad2antrr 758	. . . . . . . . . . . . . 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))
69	40, 54	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁)
70	68, 69, 54	fovrnd 6704	. . . . . . . . . . . . 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))
71		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → (𝑃‘𝑘) = (𝑃‘𝑠))
72		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑠 → 𝑘 = 𝑠)
73	71, 72	oveq12d 6567	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑠 → ((𝑃‘𝑘)𝑀𝑘) = ((𝑃‘𝑠)𝑀𝑠))
74	33, 73	gsumsn 18177	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ 𝑠 ∈ V ∧ ((𝑃‘𝑠)𝑀𝑠) ∈ (Base‘𝑅)) → (𝐺 Σ_g (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘))) = ((𝑃‘𝑠)𝑀𝑠))
75	65, 67, 70, 74	syl3anc 1318	. . . . . . . . . . . 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))
76		simprr 792	. . . . . . . . . . . . . 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 ))
77	11	ad2antrl 760	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃 Fn 𝑁)
78		fnelnfp 6348	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 Fn 𝑁 ∧ 𝑠 ∈ 𝑁) → (𝑠 ∈ dom (𝑃 ∖ I ) ↔ (𝑃‘𝑠) ≠ 𝑠))
79	77, 54, 78	syl2anc 691	. . . . . . . . . . . . . 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))
80	76, 79	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ≠ 𝑠)
81	39	ad2antrl 760	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃:𝑁⟶𝑁)
82	39	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁)
83	82, 50	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → dom 𝑃 = 𝑁)
84	48, 83	syl5sseq 3616	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁)
85	84	sseld 3567	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁))
86	85	impr 647	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁)
87	81, 86	ffvelrnd 6268	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁)
88		neeq1 2844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖 ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑗))
89		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑃‘𝑠) → (𝑖𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑗))
90	89	eqeq1d 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑗) = 0 ))
91	88, 90	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 )))
92		neeq2 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠) ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑠))
93		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠)𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑠))
94	93	eqeq1d 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠)𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑠) = 0 ))
95	92, 94	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )))
96	91, 95	rspc2v 3293	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃‘𝑠) ∈ 𝑁 ∧ 𝑠 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )))
97	87, 86, 96	syl2anc 691	. . . . . . . . . . . . . . 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 )))
98	97	impancom 455	. . . . . . . . . . . . . 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 )))
99	98	imp 444	. . . . . . . . . . . . 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 ))
100	80, 99	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 )
101	75, 100	eqtrd 2644	. . . . . . . . . . 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 )
102	101	oveq1d 6564	. . . . . . . . . 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)))))
103	60	3ad2ant1 1075	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
104	103	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑅 ∈ Ring)
105	23	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝐺 ∈ CMnd)
106		simpl2 1058	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑁 ∈ Fin)
107		difss 3699	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∖ {𝑠}) ⊆ 𝑁
108		ssfi 8065	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝑠}) ⊆ 𝑁) → (𝑁 ∖ {𝑠}) ∈ Fin)
109	106, 107, 108	sylancl 693	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑁 ∖ {𝑠}) ∈ Fin)
110	32	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
111	82	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑃:𝑁⟶𝑁)
112		eldifi 3694	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑠}) → 𝑘 ∈ 𝑁)
113	112	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑘 ∈ 𝑁)
114	111, 113	ffvelrnd 6268	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → (𝑃‘𝑘) ∈ 𝑁)
115	110, 114, 113	fovrnd 6704	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → ((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
116	115	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → ∀𝑘 ∈ (𝑁 ∖ {𝑠})((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅))
117	33, 105, 109, 116	gsummptcl 18189	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝐺 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅))
118	117	ad2ant2r 779	. . . . . . . . . . 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))
119		mdetdiag.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑅)
120	27, 20, 119	ringlz 18410	. . . . . . . . . . 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 )
121	104, 118, 120	syl2anc 691	. . . . . . . . . 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 )
122	59, 102, 121	3eqtrd 2648	. . . . . . . . 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 )
123	122	expr 641	. . . . . . . 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 ))
124	123	exlimdv 1848	. . . . . . 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 ))
125	17, 124	syl5bi 231	. . . . . 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 ))
126	16, 125	sylbid 229	. . . . 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 ))
127	126	expimpd 627	. . . 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 ))
128	127	3impia 1253	. . 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 )
129	6, 128	oveq12d 6567	. 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 ))
130		3simpa 1051	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ CRing ∧ 𝑁 ∈ Fin))
131		simpl 472	. . . 4 ⊢ ((𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁)) → 𝑃 ∈ 𝐻)
132	60	ad2antrr 758	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → 𝑅 ∈ Ring)
133		zrhpsgnmhm 19749	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
134	60, 133	sylan 487	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
135		eqid 2610	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
136	8, 135	mhmf 17163	. . . . . . 7 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):𝐻⟶(Base‘(mulGrp‘𝑅)))
137	134, 136	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):𝐻⟶(Base‘(mulGrp‘𝑅)))
138	137	ffvelrnda 6267	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅)))
139		eqid 2610	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
140	139, 27	mgpbas 18318	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
141	140	eqcomi 2619	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘𝑅)
142		mdetdiaglem.t	. . . . . 6 ⊢ · = (._r‘𝑅)
143	141, 142, 119	ringrz 18411	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
144	132, 138, 143	syl2anc 691	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
145	130, 131, 144	syl2an 493	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 )
146	145	3adant2 1073	. 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 )
147	129, 146	eqtrd 2644	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ↾ cres 5040 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 Fincfn 7841 Basecbs 15695 ._rcmulr 15769 0_gc0g 15923 Σ_g cgsu 15924 Mndcmnd 17117 MndHom cmhm 17156 SymGrpcsymg 17620 pmSgncpsgn 17732 CMndccmn 18016 mulGrpcmgp 18312 Ringcrg 18370 CRingccrg 18371 ℤRHomczrh 19667 Mat cmat 20032 maDet cmdat 20209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-xor 1457 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-reverse 13160 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-subg 17414 df-ghm 17481 df-gim 17524 df-cntz 17573 df-oppg 17599 df-symg 17621 df-pmtr 17685 df-psgn 17734 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-subrg 18601 df-sra 18993 df-rgmod 18994 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-dsmm 19895 df-frlm 19910 df-mat 20033

This theorem is referenced by: mdetdiag 20224

W3C validator