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Theorem m2detleiblem3 22691

Description: Lemma 3 for m2detleib 22693. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 2-Jan-2019.)

**Hypotheses**
Ref	Expression
m2detleiblem2.n	⊢ 𝑁 = {1, 2}
m2detleiblem2.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
m2detleiblem2.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2detleiblem2.b	⊢ 𝐵 = (Base‘𝐴)
m2detleiblem2.g	⊢ 𝐺 = (mulGrp‘𝑅)
m2detleiblem3.m	⊢ · = (+_g‘𝐺)

**Assertion**
Ref	Expression
m2detleiblem3	⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑃,𝑛 𝑄,𝑛 𝑅,𝑛 Allowed substitution hints: 𝐴(𝑛) · (𝑛) 𝐺(𝑛)

**Proof of Theorem m2detleiblem3**
Step	Hyp	Ref	Expression
1		m2detleiblem2.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
2		eqid 2764	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	mgpbas 20193	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
4		m2detleiblem3.m	. . 3 ⊢ · = (+_g‘𝐺)
5	1	fvexi 6883	. . . 4 ⊢ 𝐺 ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ V)
7		1ex 11178	. . . . . . 7 ⊢ 1 ∈ V
8		2nn 12293	. . . . . . 7 ⊢ 2 ∈ ℕ
9		prex 5397	. . . . . . . . 9 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V
10	9	prid1 4723	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
11		eqid 2764	. . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
12		m2detleiblem2.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
13		m2detleiblem2.n	. . . . . . . . 9 ⊢ 𝑁 = {1, 2}
14	11, 12, 13	symg2bas 19435	. . . . . . . 8 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}})
15	10, 14	eleqtrrid 2871	. . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃)
16	7, 8, 15	mp2an 702	. . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃
17		eleq1 2852	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄 ∈ 𝑃 ↔ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃))
18	16, 17	mpbiri 260	. . . . 5 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → 𝑄 ∈ 𝑃)
19		m2detleiblem2.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
20	13	oveq1i 7408	. . . . . . 7 ⊢ (𝑁 Mat 𝑅) = ({1, 2} Mat 𝑅)
21	19, 20	eqtri 2787	. . . . . 6 ⊢ 𝐴 = ({1, 2} Mat 𝑅)
22		m2detleiblem2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
23	13	fveq2i 6872	. . . . . . . 8 ⊢ (SymGrp‘𝑁) = (SymGrp‘{1, 2})
24	23	fveq2i 6872	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘{1, 2}))
25	12, 24	eqtri 2787	. . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘{1, 2}))
26	21, 22, 25	matepmcl 22524	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
27	18, 26	syl3an2 1178	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
28	13	mpteq1i 5193	. . . . 5 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ {1, 2} ↦ ((𝑄‘𝑛)𝑀𝑛))
29	28	fmpt 7093	. . . 4 ⊢ (∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)):{1, 2}⟶(Base‘𝑅))
30	27, 29	sylib 220	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)):{1, 2}⟶(Base‘𝑅))
31	3, 4, 6, 30	gsumpr12val 18725	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = (((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) · ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2)))
32	7	prid1 4723	. . . . . 6 ⊢ 1 ∈ {1, 2}
33	32, 13	eleqtrri 2863	. . . . 5 ⊢ 1 ∈ 𝑁
34	19, 22, 12	matepmcl 22524	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
35	18, 34	syl3an2 1178	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
36		fveq2 6869	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑄‘𝑛) = (𝑄‘1))
37		id 22	. . . . . . . . 9 ⊢ (𝑛 = 1 → 𝑛 = 1)
38	36, 37	oveq12d 7416	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘1)𝑀1))
39	38	eleq1d 2849	. . . . . . 7 ⊢ (𝑛 = 1 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)))
40	39	rspcva 3581	. . . . . 6 ⊢ ((1 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
41	33, 35, 40	sylancr 596	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
42		eqid 2764	. . . . . 6 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))
43	38, 42	fvmptg 6975	. . . . 5 ⊢ ((1 ∈ 𝑁 ∧ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
44	33, 41, 43	sylancr 596	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
45		fveq1 6868	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = ({⟨1, 1⟩, ⟨2, 2⟩}‘1))
46		1ne2 12430	. . . . . . . 8 ⊢ 1 ≠ 2
47	7, 7	fvpr1 7178	. . . . . . . 8 ⊢ (1 ≠ 2 → ({⟨1, 1⟩, ⟨2, 2⟩}‘1) = 1)
48	46, 47	ax-mp 5	. . . . . . 7 ⊢ ({⟨1, 1⟩, ⟨2, 2⟩}‘1) = 1
49	45, 48	eqtrdi 2815	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = 1)
50	49	3ad2ant2 1148	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘1) = 1)
51	50	oveq1d 7413	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) = (1𝑀1))
52	44, 51	eqtrd 2799	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = (1𝑀1))
53		2ex 12297	. . . . . . 7 ⊢ 2 ∈ V
54	53	prid2 4724	. . . . . 6 ⊢ 2 ∈ {1, 2}
55	54, 13	eleqtrri 2863	. . . . 5 ⊢ 2 ∈ 𝑁
56		fveq2 6869	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝑄‘𝑛) = (𝑄‘2))
57		id 22	. . . . . . . . 9 ⊢ (𝑛 = 2 → 𝑛 = 2)
58	56, 57	oveq12d 7416	. . . . . . . 8 ⊢ (𝑛 = 2 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘2)𝑀2))
59	58	eleq1d 2849	. . . . . . 7 ⊢ (𝑛 = 2 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)))
60	59	rspcva 3581	. . . . . 6 ⊢ ((2 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
61	55, 35, 60	sylancr 596	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
62	58, 42	fvmptg 6975	. . . . 5 ⊢ ((2 ∈ 𝑁 ∧ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
63	55, 61, 62	sylancr 596	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
64		fveq1 6868	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = ({⟨1, 1⟩, ⟨2, 2⟩}‘2))
65	53, 53	fvpr2 7179	. . . . . . . 8 ⊢ (1 ≠ 2 → ({⟨1, 1⟩, ⟨2, 2⟩}‘2) = 2)
66	46, 65	ax-mp 5	. . . . . . 7 ⊢ ({⟨1, 1⟩, ⟨2, 2⟩}‘2) = 2
67	64, 66	eqtrdi 2815	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = 2)
68	67	3ad2ant2 1148	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘2) = 2)
69	68	oveq1d 7413	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) = (2𝑀2))
70	63, 69	eqtrd 2799	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = (2𝑀2))
71	52, 70	oveq12d 7416	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) · ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2)) = ((1𝑀1) · (2𝑀2)))
72	31, 71	eqtrd 2799	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 Vcvv 3456 {cpr 4586 ⟨cop 4590 ↦ cmpt 5183 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 1c1 11076 ℕcn 12212 2c2 12274 Basecbs 17247 +_gcplusg 17288 Σ_g cgsu 17471 SymGrpcsymg 19411 mulGrpcmgp 20188 Ringcrg 20285 Mat cmat 22469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-oadd 8443 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-sup 9390 df-dju 9861 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-xnn0 12557 df-z 12571 df-dec 12691 df-uz 12842 df-fz 13515 df-seq 14017 df-fac 14289 df-bc 14318 df-hash 14346 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-hom 17312 df-cco 17313 df-0g 17472 df-gsum 17473 df-prds 17478 df-pws 17480 df-efmnd 18905 df-symg 19412 df-mgp 20189 df-sra 21242 df-rgmod 21243 df-dsmm 21786 df-frlm 21801 df-mat 22470

This theorem is referenced by: m2detleib 22693

W3C validator