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Theorem m2detleib 21980

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 2736	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
5		eqid 2736	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6		eqid 2736	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
7		m2detleib.t	. . . 4 ⊢ · = (._r‘𝑅)
8		eqid 2736	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
9	1, 2, 3, 4, 5, 6, 7, 8	mdetleib1 21940	. . 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 482	. 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 2736	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
12		eqid 2736	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
13		ringcmn 20003	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
14	13	adantr 481	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ CMnd)
15		m2detleib.n	. . . . . 6 ⊢ 𝑁 = {1, 2}
16		prfi 9266	. . . . . 6 ⊢ {1, 2} ∈ Fin
17	15, 16	eqeltri 2834	. . . . 5 ⊢ 𝑁 ∈ Fin
18		eqid 2736	. . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
19	18, 4	symgbasfi 19160	. . . . 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 483	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
23	22	adantr 481	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
24	4, 6, 5	zrhpsgnelbas 20998	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) ∈ (Base‘𝑅))
25	17, 24	mp3an2 1449	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) ∈ (Base‘𝑅))
26	25	adantlr 713	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) ∈ (Base‘𝑅))
27		simpr 485	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → 𝑘 ∈ (Base‘(SymGrp‘𝑁)))
28		simpr 485	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
29	28	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → 𝑀 ∈ 𝐵)
30	15, 4, 2, 3, 8	m2detleiblem2 21977	. . . . 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 1371	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
32	11, 7	ringcl 19981	. . . 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 1371	. . 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 5421	. . . . . . . 8 ⊢ ⟨1, 1⟩ ∈ V
35		opex 5421	. . . . . . . 8 ⊢ ⟨2, 2⟩ ∈ V
36	34, 35	pm3.2i 471	. . . . . . 7 ⊢ (⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V)
37		opex 5421	. . . . . . . 8 ⊢ ⟨1, 2⟩ ∈ V
38		opex 5421	. . . . . . . 8 ⊢ ⟨2, 1⟩ ∈ V
39	37, 38	pm3.2i 471	. . . . . . 7 ⊢ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V)
40	36, 39	pm3.2i 471	. . . . . 6 ⊢ ((⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V) ∧ (⟨1, 2⟩ ∈ V ∧ ⟨2, 1⟩ ∈ V))
41		1ne2 12361	. . . . . . . . . 10 ⊢ 1 ≠ 2
42	41	olci 864	. . . . . . . . 9 ⊢ (1 ≠ 1 ∨ 1 ≠ 2)
43		1ex 11151	. . . . . . . . . 10 ⊢ 1 ∈ V
44	43, 43	opthne 5439	. . . . . . . . 9 ⊢ (⟨1, 1⟩ ≠ ⟨1, 2⟩ ↔ (1 ≠ 1 ∨ 1 ≠ 2))
45	42, 44	mpbir 230	. . . . . . . 8 ⊢ ⟨1, 1⟩ ≠ ⟨1, 2⟩
46	41	orci 863	. . . . . . . . 9 ⊢ (1 ≠ 2 ∨ 1 ≠ 1)
47	43, 43	opthne 5439	. . . . . . . . 9 ⊢ (⟨1, 1⟩ ≠ ⟨2, 1⟩ ↔ (1 ≠ 2 ∨ 1 ≠ 1))
48	46, 47	mpbir 230	. . . . . . . 8 ⊢ ⟨1, 1⟩ ≠ ⟨2, 1⟩
49	45, 48	pm3.2i 471	. . . . . . 7 ⊢ (⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩)
50	49	orci 863	. . . . . 6 ⊢ ((⟨1, 1⟩ ≠ ⟨1, 2⟩ ∧ ⟨1, 1⟩ ≠ ⟨2, 1⟩) ∨ (⟨2, 2⟩ ≠ ⟨1, 2⟩ ∧ ⟨2, 2⟩ ≠ ⟨2, 1⟩))
51	40, 50	pm3.2i 471	. . . . 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 4812	. . . . 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 407	. . . 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 4673	. . . 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 12226	. . . . . 6 ⊢ 2 ∈ ℕ
58	18, 4, 15	symg2bas 19174	. . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → (Base‘(SymGrp‘𝑁)) = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}})
59	43, 57, 58	mp2an 690	. . . . 5 ⊢ (Base‘(SymGrp‘𝑁)) = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
60		df-pr 4589	. . . . 5 ⊢ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}} = ({{⟨1, 1⟩, ⟨2, 2⟩}} ∪ {{⟨1, 2⟩, ⟨2, 1⟩}})
61	59, 60	eqtri 2764	. . . 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 19707	. 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 19974	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
65	64	adantr 481	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Mnd)
66		prex 5389	. . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V
67	66	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ V)
68	66	prid1 4723	. . . . . . . . 9 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
69	68, 59	eleqtrri 2837	. . . . . . . 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 20998	. . . . . . . 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 1449	. . . . . . 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 21977	. . . . . . 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 1449	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
76	11, 7	ringcl 19981	. . . . . 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 1371	. . . . 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 6847	. . . . . . 7 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) = ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})))
79		fveq1 6841	. . . . . . . . . 10 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑘‘𝑛) = ({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛))
80	79	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → ((𝑘‘𝑛)𝑀𝑛) = (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))
81	80	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))
82	81	oveq2d 7373	. . . . . . 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 7375	. . . . . 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 19731	. . . . 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 1371	. . . 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 5389	. . . . . 6 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ V
87	86	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 2⟩, ⟨2, 1⟩} ∈ V)
88	86	prid2 4724	. . . . . . . . 9 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
89	88, 59	eleqtrri 2837	. . . . . . . 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 20998	. . . . . . . 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 1449	. . . . . . 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 21977	. . . . . . 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 1449	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
96	11, 7	ringcl 19981	. . . . . 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 1371	. . . . 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 6847	. . . . . . 7 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) = ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})))
99		fveq1 6841	. . . . . . . . . 10 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑘‘𝑛) = ({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛))
100	99	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → ((𝑘‘𝑛)𝑀𝑛) = (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))
101	100	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))
102	101	oveq2d 7373	. . . . . . 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 7375	. . . . . 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 19731	. . . . 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 1371	. . . 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 7375	. . 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 2737	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩})
108		eqid 2736	. . . . . . . 8 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
109	15, 4, 5, 6, 108	m2detleiblem5 21974	. . . . . . 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 2737	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩})
112	8, 7	mgpplusg 19900	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘𝑅))
113	15, 4, 2, 3, 8, 112	m2detleiblem3 21978	. . . . . . 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 1371	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))
115	110, 114	oveq12d 7375	. . . . 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 4723	. . . . . . . . . 10 ⊢ 1 ∈ {1, 2}
117	116, 15	eleqtrri 2837	. . . . . . . . 9 ⊢ 1 ∈ 𝑁
118	117	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝑁)
119	3	eleq2i 2829	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
120	119	biimpi 215	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
121	120	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐴))
122	2, 11	matecl 21774	. . . . . . . 8 ⊢ ((1 ∈ 𝑁 ∧ 1 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (1𝑀1) ∈ (Base‘𝑅))
123	118, 118, 121, 122	syl3anc 1371	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (1𝑀1) ∈ (Base‘𝑅))
124		prid2g 4722	. . . . . . . . . . 11 ⊢ (2 ∈ ℕ → 2 ∈ {1, 2})
125	57, 124	ax-mp 5	. . . . . . . . . 10 ⊢ 2 ∈ {1, 2}
126	125, 15	eleqtrri 2837	. . . . . . . . 9 ⊢ 2 ∈ 𝑁
127	126	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 2 ∈ 𝑁)
128	2, 11	matecl 21774	. . . . . . . 8 ⊢ ((2 ∈ 𝑁 ∧ 2 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (2𝑀2) ∈ (Base‘𝑅))
129	127, 127, 121, 128	syl3anc 1371	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (2𝑀2) ∈ (Base‘𝑅))
130	11, 7	ringcl 19981	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1𝑀1) ∈ (Base‘𝑅) ∧ (2𝑀2) ∈ (Base‘𝑅)) → ((1𝑀1) · (2𝑀2)) ∈ (Base‘𝑅))
131	22, 123, 129, 130	syl3anc 1371	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((1𝑀1) · (2𝑀2)) ∈ (Base‘𝑅))
132	11, 7, 108	ringlidm 19992	. . . . . 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 2776	. . . 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 2737	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩})
136		eqid 2736	. . . . . . 7 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
137	15, 4, 5, 6, 108, 136	m2detleiblem6 21975	. . . . . 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 2737	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩})
140	15, 4, 2, 3, 8, 112	m2detleiblem4 21979	. . . . . 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 1371	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((mulGrp‘𝑅) Σ_g (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))
142	138, 141	oveq12d 7375	. . . 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 7375	. . 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 21774	. . . . . 6 ⊢ ((2 ∈ 𝑁 ∧ 1 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (2𝑀1) ∈ (Base‘𝑅))
145	127, 118, 121, 144	syl3anc 1371	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (2𝑀1) ∈ (Base‘𝑅))
146	2, 11	matecl 21774	. . . . . 6 ⊢ ((1 ∈ 𝑁 ∧ 2 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (1𝑀2) ∈ (Base‘𝑅))
147	118, 127, 121, 146	syl3anc 1371	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (1𝑀2) ∈ (Base‘𝑅))
148	11, 7	ringcl 19981	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (2𝑀1) ∈ (Base‘𝑅) ∧ (1𝑀2) ∈ (Base‘𝑅)) → ((2𝑀1) · (1𝑀2)) ∈ (Base‘𝑅))
149	22, 145, 147, 148	syl3anc 1371	. . . 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 21976	. . . 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 1371	. . 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 2780	. 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 2780	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 ∪ cun 3908 ∩ cin 3909 ∅c0 4282 {csn 4586 {cpr 4588 ⟨cop 4592 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 1c1 11052 ℕcn 12153 2c2 12208 Basecbs 17083 +_gcplusg 17133 ._rcmulr 17134 Σ_g cgsu 17322 Mndcmnd 18556 inv_gcminusg 18749 -_gcsg 18750 SymGrpcsymg 19148 pmSgncpsgn 19271 CMndccmn 19562 mulGrpcmgp 19896 1_rcur 19913 Ringcrg 19964 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-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 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-dju 9837 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-fac 14174 df-bc 14203 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-sbg 18753 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-rnghom 20146 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 df-mdet 21934

This theorem is referenced by: lmat22det 32403

W3C validator