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Theorem m2detleib 22547

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 2731	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
5		eqid 2731	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6		eqid 2731	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
7		m2detleib.t	. . . 4 ⊢ · = (._r‘𝑅)
8		eqid 2731	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
9	1, 2, 3, 4, 5, 6, 7, 8	mdetleib1 22507	. . 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 2731	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
12		eqid 2731	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
13		ringcmn 20201	. . . 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 9208	. . . . . 6 ⊢ {1, 2} ∈ Fin
17	15, 16	eqeltri 2827	. . . . 5 ⊢ 𝑁 ∈ Fin
18		eqid 2731	. . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
19	18, 4	symgbasfi 19292	. . . . 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 21532	. . . . . 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 22544	. . . . 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 20169	. . . 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 5404	. . . . . . . 8 ⊢ ⟨1, 1⟩ ∈ V
35		opex 5404	. . . . . . . 8 ⊢ ⟨2, 2⟩ ∈ V
36	34, 35	pm3.2i 470	. . . . . . 7 ⊢ (⟨1, 1⟩ ∈ V ∧ ⟨2, 2⟩ ∈ V)
37		opex 5404	. . . . . . . 8 ⊢ ⟨1, 2⟩ ∈ V
38		opex 5404	. . . . . . . 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 12328	. . . . . . . . . 10 ⊢ 1 ≠ 2
42	41	olci 866	. . . . . . . . 9 ⊢ (1 ≠ 1 ∨ 1 ≠ 2)
43		1ex 11108	. . . . . . . . . 10 ⊢ 1 ∈ V
44	43, 43	opthne 5422	. . . . . . . . 9 ⊢ (⟨1, 1⟩ ≠ ⟨1, 2⟩ ↔ (1 ≠ 1 ∨ 1 ≠ 2))
45	42, 44	mpbir 231	. . . . . . . 8 ⊢ ⟨1, 1⟩ ≠ ⟨1, 2⟩
46	41	orci 865	. . . . . . . . 9 ⊢ (1 ≠ 2 ∨ 1 ≠ 1)
47	43, 43	opthne 5422	. . . . . . . . 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 865	. . . . . 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 4806	. . . . 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 4665	. . . 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 12198	. . . . . 6 ⊢ 2 ∈ ℕ
58	18, 4, 15	symg2bas 19306	. . . . . 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 4579	. . . . 5 ⊢ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}} = ({{⟨1, 1⟩, ⟨2, 2⟩}} ∪ {{⟨1, 2⟩, ⟨2, 1⟩}})
61	59, 60	eqtri 2754	. . . 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 19843	. 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 20162	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
65	64	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Mnd)
66		prex 5375	. . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V
67	66	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ V)
68	66	prid1 4715	. . . . . . . . 9 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
69	68, 59	eleqtrri 2830	. . . . . . . 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 21532	. . . . . . . 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 22544	. . . . . . 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 20169	. . . . . 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 6827	. . . . . . 7 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) = ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})))
79		fveq1 6821	. . . . . . . . . 10 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑘‘𝑛) = ({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛))
80	79	oveq1d 7361	. . . . . . . . 9 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → ((𝑘‘𝑛)𝑀𝑛) = (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛))
81	80	mpteq2dv 5185	. . . . . . . 8 ⊢ (𝑘 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ (({⟨1, 1⟩, ⟨2, 2⟩}‘𝑛)𝑀𝑛)))
82	81	oveq2d 7362	. . . . . . 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 7364	. . . . . 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 19867	. . . . 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 5375	. . . . . 6 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ V
87	86	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 2⟩, ⟨2, 1⟩} ∈ V)
88	86	prid2 4716	. . . . . . . . 9 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
89	88, 59	eleqtrri 2830	. . . . . . . 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 21532	. . . . . . . 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 22544	. . . . . . 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 20169	. . . . . 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 6827	. . . . . . 7 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘𝑘)) = ((ℤRHom‘𝑅)‘((pmSgn‘𝑁)‘{⟨1, 2⟩, ⟨2, 1⟩})))
99		fveq1 6821	. . . . . . . . . 10 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑘‘𝑛) = ({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛))
100	99	oveq1d 7361	. . . . . . . . 9 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → ((𝑘‘𝑛)𝑀𝑛) = (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛))
101	100	mpteq2dv 5185	. . . . . . . 8 ⊢ (𝑘 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑛 ∈ 𝑁 ↦ ((𝑘‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ (({⟨1, 2⟩, ⟨2, 1⟩}‘𝑛)𝑀𝑛)))
102	101	oveq2d 7362	. . . . . . 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 7364	. . . . . 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 19867	. . . . 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 7364	. . 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 2732	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩})
108		eqid 2731	. . . . . . . 8 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
109	15, 4, 5, 6, 108	m2detleiblem5 22541	. . . . . . 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 2732	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 1⟩, ⟨2, 2⟩} = {⟨1, 1⟩, ⟨2, 2⟩})
112	8, 7	mgpplusg 20063	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘𝑅))
113	15, 4, 2, 3, 8, 112	m2detleiblem3 22545	. . . . . . 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 7364	. . . . 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 4715	. . . . . . . . . 10 ⊢ 1 ∈ {1, 2}
117	116, 15	eleqtrri 2830	. . . . . . . . 9 ⊢ 1 ∈ 𝑁
118	117	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝑁)
119	3	eleq2i 2823	. . . . . . . . . 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 22341	. . . . . . . 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 4714	. . . . . . . . . . 11 ⊢ (2 ∈ ℕ → 2 ∈ {1, 2})
125	57, 124	ax-mp 5	. . . . . . . . . 10 ⊢ 2 ∈ {1, 2}
126	125, 15	eleqtrri 2830	. . . . . . . . 9 ⊢ 2 ∈ 𝑁
127	126	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 2 ∈ 𝑁)
128	2, 11	matecl 22341	. . . . . . . 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 20169	. . . . . . 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 20188	. . . . . 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 2766	. . . 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 2732	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩})
136		eqid 2731	. . . . . . 7 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
137	15, 4, 5, 6, 108, 136	m2detleiblem6 22542	. . . . . 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 2732	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {⟨1, 2⟩, ⟨2, 1⟩} = {⟨1, 2⟩, ⟨2, 1⟩})
140	15, 4, 2, 3, 8, 112	m2detleiblem4 22546	. . . . . 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 7364	. . . 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 7364	. . 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 22341	. . . . . 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 22341	. . . . . 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 20169	. . . . 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 22543	. . . 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 2770	. 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 2770	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 Vcvv 3436 ∪ cun 3900 ∩ cin 3901 ∅c0 4283 {csn 4576 {cpr 4578 ⟨cop 4582 ↦ cmpt 5172 ‘cfv 6481 (class class class)co 7346 Fincfn 8869 1c1 11007 ℕcn 12125 2c2 12180 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Σ_g cgsu 17344 Mndcmnd 18642 inv_gcminusg 18847 -_gcsg 18848 SymGrpcsymg 19282 pmSgncpsgn 19402 CMndccmn 19693 mulGrpcmgp 20059 1_rcur 20100 Ringcrg 20152 ℤRHomczrh 21437 Mat cmat 22323 maDet cmdat 22500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1513 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-ot 4585 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14504 df-substr 14549 df-pfx 14579 df-splice 14657 df-reverse 14666 df-s2 14755 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-efmnd 18777 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19126 df-gim 19172 df-cntz 19230 df-oppg 19259 df-symg 19283 df-pmtr 19355 df-psgn 19404 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-ring 20154 df-cring 20155 df-rhm 20391 df-subrng 20462 df-subrg 20486 df-sra 21108 df-rgmod 21109 df-cnfld 21293 df-zring 21385 df-zrh 21441 df-dsmm 21670 df-frlm 21685 df-mat 22324 df-mdet 22501

This theorem is referenced by: lmat22det 33833

W3C validator