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Theorem m2detleiblem3 22572

Description: Lemma 3 for m2detleib 22574. (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 2737	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	mgpbas 20084	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
4		m2detleiblem3.m	. . 3 ⊢ · = (+_g‘𝐺)
5	1	fvexi 6846	. . . 4 ⊢ 𝐺 ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ V)
7		1ex 11129	. . . . . . 7 ⊢ 1 ∈ V
8		2nn 12219	. . . . . . 7 ⊢ 2 ∈ ℕ
9		prex 5373	. . . . . . . . 9 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V
10	9	prid1 4707	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
11		eqid 2737	. . . . . . . . 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 19326	. . . . . . . 8 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}})
15	10, 14	eleqtrrid 2844	. . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃)
16	7, 8, 15	mp2an 693	. . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃
17		eleq1 2825	. . . . . 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 7368	. . . . . . 7 ⊢ (𝑁 Mat 𝑅) = ({1, 2} Mat 𝑅)
21	19, 20	eqtri 2760	. . . . . 6 ⊢ 𝐴 = ({1, 2} Mat 𝑅)
22		m2detleiblem2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
23	13	fveq2i 6835	. . . . . . . 8 ⊢ (SymGrp‘𝑁) = (SymGrp‘{1, 2})
24	23	fveq2i 6835	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘{1, 2}))
25	12, 24	eqtri 2760	. . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘{1, 2}))
26	21, 22, 25	matepmcl 22405	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
27	18, 26	syl3an2 1165	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
28	13	mpteq1i 5177	. . . . 5 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ {1, 2} ↦ ((𝑄‘𝑛)𝑀𝑛))
29	28	fmpt 7054	. . . 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 18615	. 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 4707	. . . . . 6 ⊢ 1 ∈ {1, 2}
33	32, 13	eleqtrri 2836	. . . . 5 ⊢ 1 ∈ 𝑁
34	19, 22, 12	matepmcl 22405	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
35	18, 34	syl3an2 1165	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
36		fveq2 6832	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑄‘𝑛) = (𝑄‘1))
37		id 22	. . . . . . . . 9 ⊢ (𝑛 = 1 → 𝑛 = 1)
38	36, 37	oveq12d 7376	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘1)𝑀1))
39	38	eleq1d 2822	. . . . . . 7 ⊢ (𝑛 = 1 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)))
40	39	rspcva 3563	. . . . . 6 ⊢ ((1 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
41	33, 35, 40	sylancr 588	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
42		eqid 2737	. . . . . 6 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))
43	38, 42	fvmptg 6937	. . . . 5 ⊢ ((1 ∈ 𝑁 ∧ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
44	33, 41, 43	sylancr 588	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
45		fveq1 6831	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = ({⟨1, 1⟩, ⟨2, 2⟩}‘1))
46		1ne2 12349	. . . . . . . 8 ⊢ 1 ≠ 2
47	7, 7	fvpr1 7138	. . . . . . . 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 2788	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = 1)
50	49	3ad2ant2 1135	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘1) = 1)
51	50	oveq1d 7373	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) = (1𝑀1))
52	44, 51	eqtrd 2772	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = (1𝑀1))
53		2ex 12223	. . . . . . 7 ⊢ 2 ∈ V
54	53	prid2 4708	. . . . . 6 ⊢ 2 ∈ {1, 2}
55	54, 13	eleqtrri 2836	. . . . 5 ⊢ 2 ∈ 𝑁
56		fveq2 6832	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝑄‘𝑛) = (𝑄‘2))
57		id 22	. . . . . . . . 9 ⊢ (𝑛 = 2 → 𝑛 = 2)
58	56, 57	oveq12d 7376	. . . . . . . 8 ⊢ (𝑛 = 2 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘2)𝑀2))
59	58	eleq1d 2822	. . . . . . 7 ⊢ (𝑛 = 2 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)))
60	59	rspcva 3563	. . . . . 6 ⊢ ((2 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
61	55, 35, 60	sylancr 588	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
62	58, 42	fvmptg 6937	. . . . 5 ⊢ ((2 ∈ 𝑁 ∧ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
63	55, 61, 62	sylancr 588	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
64		fveq1 6831	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = ({⟨1, 1⟩, ⟨2, 2⟩}‘2))
65	53, 53	fvpr2 7139	. . . . . . . 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 2788	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = 2)
68	67	3ad2ant2 1135	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘2) = 2)
69	68	oveq1d 7373	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) = (2𝑀2))
70	63, 69	eqtrd 2772	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = (2𝑀2))
71	52, 70	oveq12d 7376	. 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 2772	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3430 {cpr 4570 ⟨cop 4574 ↦ cmpt 5167 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 1c1 11028 ℕcn 12146 2c2 12201 Basecbs 17137 +_gcplusg 17178 Σ_g cgsu 17361 SymGrpcsymg 19302 mulGrpcmgp 20079 Ringcrg 20172 Mat cmat 22350

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-dju 9814 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-xnn0 12476 df-z 12490 df-dec 12609 df-uz 12753 df-fz 13425 df-seq 13926 df-fac 14198 df-bc 14227 df-hash 14255 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-hom 17202 df-cco 17203 df-0g 17362 df-gsum 17363 df-prds 17368 df-pws 17370 df-efmnd 18795 df-symg 19303 df-mgp 20080 df-sra 21127 df-rgmod 21128 df-dsmm 21689 df-frlm 21704 df-mat 22351

This theorem is referenced by: m2detleib 22574

W3C validator