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Theorem m2detleiblem3 22549

Description: Lemma 3 for m2detleib 22551. (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 2729	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	mgpbas 20065	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
4		m2detleiblem3.m	. . 3 ⊢ · = (+_g‘𝐺)
5	1	fvexi 6854	. . . 4 ⊢ 𝐺 ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ V)
7		1ex 11146	. . . . . . 7 ⊢ 1 ∈ V
8		2nn 12235	. . . . . . 7 ⊢ 2 ∈ ℕ
9		prex 5387	. . . . . . . . 9 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V
10	9	prid1 4722	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
11		eqid 2729	. . . . . . . . 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 19307	. . . . . . . 8 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}})
15	10, 14	eleqtrrid 2835	. . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃)
16	7, 8, 15	mp2an 692	. . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃
17		eleq1 2816	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄 ∈ 𝑃 ↔ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃))
18	16, 17	mpbiri 258	. . . . 5 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → 𝑄 ∈ 𝑃)
19		m2detleiblem2.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
20	13	oveq1i 7379	. . . . . . 7 ⊢ (𝑁 Mat 𝑅) = ({1, 2} Mat 𝑅)
21	19, 20	eqtri 2752	. . . . . 6 ⊢ 𝐴 = ({1, 2} Mat 𝑅)
22		m2detleiblem2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
23	13	fveq2i 6843	. . . . . . . 8 ⊢ (SymGrp‘𝑁) = (SymGrp‘{1, 2})
24	23	fveq2i 6843	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘{1, 2}))
25	12, 24	eqtri 2752	. . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘{1, 2}))
26	21, 22, 25	matepmcl 22382	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
27	18, 26	syl3an2 1164	. . . 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 7064	. . . 4 ⊢ (∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)):{1, 2}⟶(Base‘𝑅))
30	27, 29	sylib 218	. . 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 18598	. 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 4722	. . . . . 6 ⊢ 1 ∈ {1, 2}
33	32, 13	eleqtrri 2827	. . . . 5 ⊢ 1 ∈ 𝑁
34	19, 22, 12	matepmcl 22382	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
35	18, 34	syl3an2 1164	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
36		fveq2 6840	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑄‘𝑛) = (𝑄‘1))
37		id 22	. . . . . . . . 9 ⊢ (𝑛 = 1 → 𝑛 = 1)
38	36, 37	oveq12d 7387	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘1)𝑀1))
39	38	eleq1d 2813	. . . . . . 7 ⊢ (𝑛 = 1 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)))
40	39	rspcva 3583	. . . . . 6 ⊢ ((1 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
41	33, 35, 40	sylancr 587	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
42		eqid 2729	. . . . . 6 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))
43	38, 42	fvmptg 6948	. . . . 5 ⊢ ((1 ∈ 𝑁 ∧ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
44	33, 41, 43	sylancr 587	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
45		fveq1 6839	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = ({⟨1, 1⟩, ⟨2, 2⟩}‘1))
46		1ne2 12365	. . . . . . . 8 ⊢ 1 ≠ 2
47	7, 7	fvpr1 7148	. . . . . . . 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 2780	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = 1)
50	49	3ad2ant2 1134	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘1) = 1)
51	50	oveq1d 7384	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) = (1𝑀1))
52	44, 51	eqtrd 2764	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = (1𝑀1))
53		2ex 12239	. . . . . . 7 ⊢ 2 ∈ V
54	53	prid2 4723	. . . . . 6 ⊢ 2 ∈ {1, 2}
55	54, 13	eleqtrri 2827	. . . . 5 ⊢ 2 ∈ 𝑁
56		fveq2 6840	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝑄‘𝑛) = (𝑄‘2))
57		id 22	. . . . . . . . 9 ⊢ (𝑛 = 2 → 𝑛 = 2)
58	56, 57	oveq12d 7387	. . . . . . . 8 ⊢ (𝑛 = 2 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘2)𝑀2))
59	58	eleq1d 2813	. . . . . . 7 ⊢ (𝑛 = 2 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)))
60	59	rspcva 3583	. . . . . 6 ⊢ ((2 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
61	55, 35, 60	sylancr 587	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
62	58, 42	fvmptg 6948	. . . . 5 ⊢ ((2 ∈ 𝑁 ∧ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
63	55, 61, 62	sylancr 587	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
64		fveq1 6839	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = ({⟨1, 1⟩, ⟨2, 2⟩}‘2))
65	53, 53	fvpr2 7149	. . . . . . . 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 2780	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = 2)
68	67	3ad2ant2 1134	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘2) = 2)
69	68	oveq1d 7384	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) = (2𝑀2))
70	63, 69	eqtrd 2764	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = (2𝑀2))
71	52, 70	oveq12d 7387	. 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 2764	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3444 {cpr 4587 ⟨cop 4591 ↦ cmpt 5183 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 1c1 11045 ℕcn 12162 2c2 12217 Basecbs 17155 +_gcplusg 17196 Σ_g cgsu 17379 SymGrpcsymg 19283 mulGrpcmgp 20060 Ringcrg 20153 Mat cmat 22327

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-seq 13943 df-fac 14215 df-bc 14244 df-hash 14272 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17380 df-gsum 17381 df-prds 17386 df-pws 17388 df-efmnd 18778 df-symg 19284 df-mgp 20061 df-sra 21112 df-rgmod 21113 df-dsmm 21674 df-frlm 21689 df-mat 22328

This theorem is referenced by: m2detleib 22551

W3C validator