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Theorem m2detleib 22637

Description: Leibniz' Formula for 2x2-matrices. (Contributed by AV, 21-Dec-2018.) (Revised by AV, 26-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)

**Hypotheses**
Ref	Expression
m2detleib.n	⊢ 𝑁 = {1, 2}
m2detleib.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
m2detleib.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2detleib.b	⊢ 𝐵 = (Base‘𝐴)
m2detleib.m	⊢ − = (-_g‘𝑅)
m2detleib.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
m2detleib	⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))

Proof of Theorem m2detleib Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		m2detleib.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
2		m2detleib.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		m2detleib.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
4		eqid 2737	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
5		eqid 2737	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6		eqid 2737	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
7		m2detleib.t	. . . 4 ⊢ · = (._r‘𝑅)
8		eqid 2737	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
9	1, 2, 3, 4, 5, 6, 7, 8	mdetleib1 22597	. . 3 ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σ_g (𝑘 ∈ (Base‘(SymGrp‘𝑁)) ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))))))
10	9	adantl 481	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (𝑅 Σ_g (𝑘 ∈ (Base‘(SymGrp‘𝑁)) ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))))))
11		eqid 2737	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
12		eqid 2737	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
13		ringcmn 20279	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
14	13	adantr 480	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ CMnd)
15		m2detleib.n	. . . . . 6 ⊢ 𝑁 = {1, 2}
16		prfi 9363	. . . . . 6 ⊢ {1, 2} ∈ Fin
17	15, 16	eqeltri 2837	. . . . 5 ⊢ 𝑁 ∈ Fin
18		eqid 2737	. . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
19	18, 4	symgbasfi 19396	. . . . 5 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
20	17, 19	ax-mp 5	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) ∈ Fin
21	20	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘(SymGrp‘𝑁)) ∈ Fin)
22		simpl 482	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
23	22	adantr 480	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
24	4, 6, 5	zrhpsgnelbas 21612	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) ∈ (Base‘𝑅))
25	17, 24	mp3an2 1451	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) ∈ (Base‘𝑅))
26	25	adantlr 715	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) ∈ (Base‘𝑅))
27		simpr 484	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → 𝑘 ∈ (Base‘(SymGrp‘𝑁)))
28		simpr 484	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
29	28	adantr 480	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → 𝑀 ∈ 𝐵)
30	15, 4, 2, 3, 8	m2detleiblem2 22634	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
31	23, 27, 29, 30	syl3anc 1373	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
32	11, 7	ringcl 20247	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅)) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
33	23, 26, 31, 32	syl3anc 1373	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
34		opex 5469	. . . . . . . 8 ⊢ ⟨1, 1⟩ ∈ V
35		opex 5469	. . . . . . . 8 ⊢ ⟨2, 2⟩ ∈ V
36	34, 35	pm3.2i 470	. . . . . . 7 ⊢ (⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V)
37		opex 5469	. . . . . . . 8 ⊢ ⟨1, 2⟩ ∈ V
38		opex 5469	. . . . . . . 8 ⊢ ⟨2, 1⟩ ∈ V
39	37, 38	pm3.2i 470	. . . . . . 7 ⊢ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V)
40	36, 39	pm3.2i 470	. . . . . 6 ⊢ ((⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V) ∧ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V))
41		1ne2 12474	. . . . . . . . . 10 ⊢ 1 ≠ 2
42	41	olci 867	. . . . . . . . 9 ⊢ (1 ≠ 1 ∨ 1 ≠ 2)
43		1ex 11257	. . . . . . . . . 10 ⊢ 1 ∈ V
44	43, 43	opthne 5487	. . . . . . . . 9 ⊢ (⟨1, 1⟩ ≠ ⟨1, 2⟩ ↔ (1 ≠ 1 ∨ 1 ≠ 2))
45	42, 44	mpbir 231	. . . . . . . 8 ⊢ ⟨1, 1⟩ ≠ ⟨1, 2⟩
46	41	orci 866	. . . . . . . . 9 ⊢ (1 ≠ 2 ∨ 1 ≠ 1)
47	43, 43	opthne 5487	. . . . . . . . 9 ⊢ (⟨1, 1⟩ ≠ ⟨2, 1⟩ ↔ (1 ≠ 2 ∨ 1 ≠ 1))
48	46, 47	mpbir 231	. . . . . . . 8 ⊢ ⟨1, 1⟩ ≠ ⟨2, 1⟩
49	45, 48	pm3.2i 470	. . . . . . 7 ⊢ (⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩)
50	49	orci 866	. . . . . 6 ⊢ ((⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩) ∨ (⟨2, 2⟩ ≠ ⟨1, 2⟩ ∧ ⟨2, 2⟩ ≠ ⟨2, 1⟩))
51	40, 50	pm3.2i 470	. . . . 5 ⊢ (((⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V) ∧ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V)) ∧ ((⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩) ∨ (⟨2, 2⟩ ≠ ⟨1, 2⟩ ∧ ⟨2, 2⟩ ≠ ⟨2, 1⟩)))
52	51	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V) ∧ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V)) ∧ ((⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩) ∨ (⟨2, 2⟩ ≠ ⟨1, 2⟩ ∧ ⟨2, 2⟩ ≠ ⟨2, 1⟩))))
53		prneimg 4854	. . . . 5 ⊢ (((⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V) ∧ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V)) → (((⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩) ∨ (⟨2, 2⟩ ≠ ⟨1, 2⟩ ∧ ⟨2, 2⟩ ≠ ⟨2, 1⟩)) → {⟨1, 1⟩, ⟨2, 2⟩} ≠ {⟨1, 2⟩, ⟨2, 1⟩}))
54	53	imp 406	. . . 4 ⊢ ((((⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V) ∧ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V)) ∧ ((⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩) ∨ (⟨2, 2⟩ ≠ ⟨1, 2⟩ ∧ ⟨2, 2⟩ ≠ ⟨2, 1⟩))) → {⟨1, 1⟩, ⟨2, 2⟩} ≠ {⟨1, 2⟩, ⟨2, 1⟩})
55		disjsn2 4712	. . . 4 ⊢ ({⟨1, 1⟩, ⟨2, 2⟩} ≠ {⟨1, 2⟩, ⟨2, 1⟩} → ({{⟨1, 1⟩, ⟨2, 2⟩}} ∩ {{⟨1, 2⟩, ⟨2, 1⟩}}) = ∅)
56	52, 54, 55	3syl 18	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ({{⟨1, 1⟩, ⟨2, 2⟩}} ∩ {{⟨1, 2⟩, ⟨2, 1⟩}}) = ∅)
57		2nn 12339	. . . . . 6 ⊢ 2 ∈ ℕ
58	18, 4, 15	symg2bas 19410	. . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → (Base‘(SymGrp‘𝑁)) = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}})
59	43, 57, 58	mp2an 692	. . . . 5 ⊢ (Base‘(SymGrp‘𝑁)) = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
60		df-pr 4629	. . . . 5 ⊢ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}} = ({{⟨1, 1⟩, ⟨2, 2⟩}} ∪ {{⟨1, 2⟩, ⟨2, 1⟩}})
61	59, 60	eqtri 2765	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) = ({{⟨1, 1⟩, ⟨2, 2⟩}} ∪ {{⟨1, 2⟩, ⟨2, 1⟩}})
62	61	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘(SymGrp‘𝑁)) = ({{⟨1, 1⟩, ⟨2, 2⟩}} ∪ {{⟨1, 2⟩, ⟨2, 1⟩}}))
63	11, 12, 14, 21, 33, 56, 62	gsummptfidmsplit 19948	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 Σ_g (𝑘 ∈ (Base‘(SymGrp‘𝑁)) ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))))) = ((𝑅 Σ_g (𝑘 ∈ {{⟨1, 1⟩, ⟨2, 2⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))))))(+_g‘𝑅)(𝑅 Σ_g (𝑘 ∈ {{⟨1, 2⟩, ⟨2, 1⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))))))))
64		ringmnd 20240	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
65	64	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Mnd)
66		prex 5437	. . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V
67	66	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ V)
68	66	prid1 4762	. . . . . . . . 9 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
69	68, 59	eleqtrri 2840	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ (Base‘(SymGrp‘𝑁))
70	69	a1i 11	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 1⟩, ⟨2, 2⟩} ∈ (Base‘(SymGrp‘𝑁)))
71	4, 6, 5	zrhpsgnelbas 21612	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ {⟨1, 1⟩, ⟨2, 2⟩} ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) ∈ (Base‘𝑅))
72	17, 71	mp3an2 1451	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 1⟩, ⟨2, 2⟩} ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) ∈ (Base‘𝑅))
73	70, 72	sylan2 593	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) ∈ (Base‘𝑅))
74	15, 4, 2, 3, 8	m2detleiblem2 22634	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 1⟩, ⟨2, 2⟩} ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
75	69, 74	mp3an2 1451	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
76	11, 7	ringcl 20247	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅)) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
77	22, 73, 75, 76	syl3anc 1373	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
78		2fveq3 6911	. . . . . . 7 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) = ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})))
79		fveq1 6905	. . . . . . . . . 10 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑘‘𝑛) = ({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛))
80	79	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → ((𝑘‘𝑛)𝑀𝑛) = (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))
81	80	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))
82	81	oveq2d 7447	. . . . . . 7 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))) = ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))))
83	78, 82	oveq12d 7449	. . . . . 6 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))) = (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))))
84	11, 83	gsumsn 19972	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V ∧ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅)) → (𝑅 Σ_g (𝑘 ∈ {{⟨1, 1⟩, ⟨2, 2⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))))) = (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))))
85	65, 67, 77, 84	syl3anc 1373	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 Σ_g (𝑘 ∈ {{⟨1, 1⟩, ⟨2, 2⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))))) = (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))))
86		prex 5437	. . . . . 6 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ V
87	86	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 2⟩, ⟨2, 1⟩} ∈ V)
88	86	prid2 4763	. . . . . . . . 9 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
89	88, 59	eleqtrri 2840	. . . . . . . 8 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ (Base‘(SymGrp‘𝑁))
90	89	a1i 11	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 2⟩, ⟨2, 1⟩} ∈ (Base‘(SymGrp‘𝑁)))
91	4, 6, 5	zrhpsgnelbas 21612	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ {⟨1, 2⟩, ⟨2, 1⟩} ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) ∈ (Base‘𝑅))
92	17, 91	mp3an2 1451	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 2⟩, ⟨2, 1⟩} ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) ∈ (Base‘𝑅))
93	90, 92	sylan2 593	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) ∈ (Base‘𝑅))
94	15, 4, 2, 3, 8	m2detleiblem2 22634	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 2⟩, ⟨2, 1⟩} ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
95	89, 94	mp3an2 1451	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
96	11, 7	ringcl 20247	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) ∈ (Base‘𝑅) ∧ ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅)) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
97	22, 93, 95, 96	syl3anc 1373	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
98		2fveq3 6911	. . . . . . 7 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) = ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})))
99		fveq1 6905	. . . . . . . . . 10 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑘‘𝑛) = ({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛))
100	99	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → ((𝑘‘𝑛)𝑀𝑛) = (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))
101	100	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))
102	101	oveq2d 7447	. . . . . . 7 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))) = ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))))
103	98, 102	oveq12d 7449	. . . . . 6 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))) = (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))))
104	11, 103	gsumsn 19972	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ {⟨1, 2⟩, ⟨2, 1⟩} ∈ V ∧ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅)) → (𝑅 Σ_g (𝑘 ∈ {{⟨1, 2⟩, ⟨2, 1⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))))) = (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))))
105	65, 87, 97, 104	syl3anc 1373	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 Σ_g (𝑘 ∈ {{⟨1, 2⟩, ⟨2, 1⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)))))) = (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))))
106	85, 105	oveq12d 7449	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑅 Σ_g (𝑘 ∈ {{⟨1, 1⟩, ⟨2, 2⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))))))(+_g‘𝑅)(𝑅 Σ_g (𝑘 ∈ {{⟨1, 2⟩, ⟨2, 1⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))))))) = ((((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))))(+_g‘𝑅)(((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))))))
107		eqidd 2738	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩})
108		eqid 2737	. . . . . . . 8 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
109	15, 4, 5, 6, 108	m2detleiblem5 22631	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩}) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) = (1_r‘𝑅))
110	107, 109	sylan2 593	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) = (1_r‘𝑅))
111		eqidd 2738	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩})
112	8, 7	mgpplusg 20141	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘𝑅))
113	15, 4, 2, 3, 8, 112	m2detleiblem3 22635	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))
114	22, 111, 28, 113	syl3anc 1373	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))
115	110, 114	oveq12d 7449	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))) = ((1_r‘𝑅) · ((1𝑀1) · (2𝑀2))))
116	43	prid1 4762	. . . . . . . . . 10 ⊢ 1 ∈ {1, 2}
117	116, 15	eleqtrri 2840	. . . . . . . . 9 ⊢ 1 ∈ 𝑁
118	117	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝑁)
119	3	eleq2i 2833	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
120	119	biimpi 216	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
121	120	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐴))
122	2, 11	matecl 22431	. . . . . . . 8 ⊢ ((1 ∈ 𝑁 ∧ 1 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (1𝑀1) ∈ (Base‘𝑅))
123	118, 118, 121, 122	syl3anc 1373	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (1𝑀1) ∈ (Base‘𝑅))
124		prid2g 4761	. . . . . . . . . . 11 ⊢ (2 ∈ ℕ → 2 ∈ {1, 2})
125	57, 124	ax-mp 5	. . . . . . . . . 10 ⊢ 2 ∈ {1, 2}
126	125, 15	eleqtrri 2840	. . . . . . . . 9 ⊢ 2 ∈ 𝑁
127	126	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 2 ∈ 𝑁)
128	2, 11	matecl 22431	. . . . . . . 8 ⊢ ((2 ∈ 𝑁 ∧ 2 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (2𝑀2) ∈ (Base‘𝑅))
129	127, 127, 121, 128	syl3anc 1373	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (2𝑀2) ∈ (Base‘𝑅))
130	11, 7	ringcl 20247	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1𝑀1) ∈ (Base‘𝑅) ∧ (2𝑀2) ∈ (Base‘𝑅)) → ((1𝑀1) · (2𝑀2)) ∈ (Base‘𝑅))
131	22, 123, 129, 130	syl3anc 1373	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((1𝑀1) · (2𝑀2)) ∈ (Base‘𝑅))
132	11, 7, 108	ringlidm 20266	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((1𝑀1) · (2𝑀2)) ∈ (Base‘𝑅)) → ((1_r‘𝑅) · ((1𝑀1) · (2𝑀2))) = ((1𝑀1) · (2𝑀2)))
133	131, 132	syldan 591	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((1_r‘𝑅) · ((1𝑀1) · (2𝑀2))) = ((1𝑀1) · (2𝑀2)))
134	115, 133	eqtrd 2777	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))) = ((1𝑀1) · (2𝑀2)))
135		eqidd 2738	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩})
136		eqid 2737	. . . . . . 7 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
137	15, 4, 5, 6, 108, 136	m2detleiblem6 22632	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩}) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) = ((inv_g‘𝑅)‘(1_r‘𝑅)))
138	135, 137	sylan2 593	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) = ((inv_g‘𝑅)‘(1_r‘𝑅)))
139		eqidd 2738	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩})
140	15, 4, 2, 3, 8, 112	m2detleiblem4 22636	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))
141	22, 139, 28, 140	syl3anc 1373	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))
142	138, 141	oveq12d 7449	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))) = (((inv_g‘𝑅)‘(1_r‘𝑅)) · ((2𝑀1) · (1𝑀2))))
143	134, 142	oveq12d 7449	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))))(+_g‘𝑅)(((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))))) = (((1𝑀1) · (2𝑀2))(+_g‘𝑅)(((inv_g‘𝑅)‘(1_r‘𝑅)) · ((2𝑀1) · (1𝑀2)))))
144	2, 11	matecl 22431	. . . . . 6 ⊢ ((2 ∈ 𝑁 ∧ 1 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (2𝑀1) ∈ (Base‘𝑅))
145	127, 118, 121, 144	syl3anc 1373	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (2𝑀1) ∈ (Base‘𝑅))
146	2, 11	matecl 22431	. . . . . 6 ⊢ ((1 ∈ 𝑁 ∧ 2 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (1𝑀2) ∈ (Base‘𝑅))
147	118, 127, 121, 146	syl3anc 1373	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (1𝑀2) ∈ (Base‘𝑅))
148	11, 7	ringcl 20247	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (2𝑀1) ∈ (Base‘𝑅) ∧ (1𝑀2) ∈ (Base‘𝑅)) → ((2𝑀1) · (1𝑀2)) ∈ (Base‘𝑅))
149	22, 145, 147, 148	syl3anc 1373	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((2𝑀1) · (1𝑀2)) ∈ (Base‘𝑅))
150		m2detleib.m	. . . . 5 ⊢ − = (-_g‘𝑅)
151	15, 4, 5, 6, 108, 136, 7, 150	m2detleiblem7 22633	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((1𝑀1) · (2𝑀2)) ∈ (Base‘𝑅) ∧ ((2𝑀1) · (1𝑀2)) ∈ (Base‘𝑅)) → (((1𝑀1) · (2𝑀2))(+_g‘𝑅)(((inv_g‘𝑅)‘(1_r‘𝑅)) · ((2𝑀1) · (1𝑀2)))) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))
152	22, 131, 149, 151	syl3anc 1373	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((1𝑀1) · (2𝑀2))(+_g‘𝑅)(((inv_g‘𝑅)‘(1_r‘𝑅)) · ((2𝑀1) · (1𝑀2)))) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))
153	106, 143, 152	3eqtrd 2781	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑅 Σ_g (𝑘 ∈ {{⟨1, 1⟩, ⟨2, 2⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))))))(+_g‘𝑅)(𝑅 Σ_g (𝑘 ∈ {{⟨1, 2⟩, ⟨2, 1⟩}} ↦ (((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) · ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))))))) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))
154	10, 63, 153	3eqtrd 2781	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 {cpr 4628 ⟨cop 4632 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 ℕcn 12266 2c2 12321 Basecbs 17247 +_gcplusg 17297 ._rcmulr 17298 Σ_g cgsu 17485 Mndcmnd 18747 inv_gcminusg 18952 -_gcsg 18953 SymGrpcsymg 19386 pmSgncpsgn 19507 CMndccmn 19798 mulGrpcmgp 20137 1_rcur 20178 Ringcrg 20230 ℤRHomczrh 21510 Mat cmat 22411 maDet cmdat 22590

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-gim 19277 df-cntz 19335 df-oppg 19364 df-symg 19387 df-pmtr 19460 df-psgn 19509 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-dsmm 21752 df-frlm 21767 df-mat 22412 df-mdet 22591

This theorem is referenced by: lmat22det 33821

W3C validator