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Theorem m2detleiblem3 22797

Description: Lemma 3 for m2detleib 22799. (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 2762	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	mgpbas 20227	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
4		m2detleiblem3.m	. . 3 ⊢ · = (+_g‘𝐺)
5	1	fvexi 6895	. . . 4 ⊢ 𝐺 ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ V)
7		1ex 11209	. . . . . . 7 ⊢ 1 ∈ V
8		2nn 12320	. . . . . . 7 ⊢ 2 ∈ ℕ
9		prex 5408	. . . . . . . . 9 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V
10	9	prid1 4727	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
11		eqid 2762	. . . . . . . . 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 19469	. . . . . . . 8 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}})
15	10, 14	eleqtrrid 2869	. . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃)
16	7, 8, 15	mp2an 704	. . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃
17		eleq1 2850	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄 ∈ 𝑃 ↔ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃))
18	16, 17	mpbiri 261	. . . . 5 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → 𝑄 ∈ 𝑃)
19		m2detleiblem2.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
20	13	oveq1i 7422	. . . . . . 7 ⊢ (𝑁 Mat 𝑅) = ({1, 2} Mat 𝑅)
21	19, 20	eqtri 2785	. . . . . 6 ⊢ 𝐴 = ({1, 2} Mat 𝑅)
22		m2detleiblem2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
23	13	fveq2i 6884	. . . . . . . 8 ⊢ (SymGrp‘𝑁) = (SymGrp‘{1, 2})
24	23	fveq2i 6884	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘{1, 2}))
25	12, 24	eqtri 2785	. . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘{1, 2}))
26	21, 22, 25	matepmcl 22630	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
27	18, 26	syl3an2 1181	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
28	13	mpteq1i 5201	. . . . 5 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ {1, 2} ↦ ((𝑄‘𝑛)𝑀𝑛))
29	28	fmpt 7105	. . . 4 ⊢ (∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)):{1, 2}⟶(Base‘𝑅))
30	27, 29	sylib 221	. . 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 18753	. 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 4727	. . . . . 6 ⊢ 1 ∈ {1, 2}
33	32, 13	eleqtrri 2861	. . . . 5 ⊢ 1 ∈ 𝑁
34	19, 22, 12	matepmcl 22630	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
35	18, 34	syl3an2 1181	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
36		fveq2 6881	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑄‘𝑛) = (𝑄‘1))
37		id 23	. . . . . . . . 9 ⊢ (𝑛 = 1 → 𝑛 = 1)
38	36, 37	oveq12d 7430	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘1)𝑀1))
39	38	eleq1d 2847	. . . . . . 7 ⊢ (𝑛 = 1 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)))
40	39	rspcva 3578	. . . . . 6 ⊢ ((1 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
41	33, 35, 40	sylancr 598	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
42		eqid 2762	. . . . . 6 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))
43	38, 42	fvmptg 6987	. . . . 5 ⊢ ((1 ∈ 𝑁 ∧ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
44	33, 41, 43	sylancr 598	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
45		fveq1 6880	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = ({⟨1, 1⟩, ⟨2, 2⟩}‘1))
46		1ne2 12457	. . . . . . . 8 ⊢ 1 ≠ 2
47	7, 7	fvpr1 7190	. . . . . . . 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 2813	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘1) = 1)
50	49	3ad2ant2 1151	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘1) = 1)
51	50	oveq1d 7427	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) = (1𝑀1))
52	44, 51	eqtrd 2797	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = (1𝑀1))
53		2ex 12324	. . . . . . 7 ⊢ 2 ∈ V
54	53	prid2 4728	. . . . . 6 ⊢ 2 ∈ {1, 2}
55	54, 13	eleqtrri 2861	. . . . 5 ⊢ 2 ∈ 𝑁
56		fveq2 6881	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝑄‘𝑛) = (𝑄‘2))
57		id 23	. . . . . . . . 9 ⊢ (𝑛 = 2 → 𝑛 = 2)
58	56, 57	oveq12d 7430	. . . . . . . 8 ⊢ (𝑛 = 2 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘2)𝑀2))
59	58	eleq1d 2847	. . . . . . 7 ⊢ (𝑛 = 2 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)))
60	59	rspcva 3578	. . . . . 6 ⊢ ((2 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
61	55, 35, 60	sylancr 598	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
62	58, 42	fvmptg 6987	. . . . 5 ⊢ ((2 ∈ 𝑁 ∧ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
63	55, 61, 62	sylancr 598	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
64		fveq1 6880	. . . . . . 7 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = ({⟨1, 1⟩, ⟨2, 2⟩}‘2))
65	53, 53	fvpr2 7191	. . . . . . . 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 2813	. . . . . 6 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄‘2) = 2)
68	67	3ad2ant2 1151	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘2) = 2)
69	68	oveq1d 7427	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) = (2𝑀2))
70	63, 69	eqtrd 2797	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = (2𝑀2))
71	52, 70	oveq12d 7430	. 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 2797	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

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 Vcvv 3454 {cpr 4590 ⟨cop 4594 ↦ cmpt 5191 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 1c1 11107 ℕcn 12239 2c2 12301 Basecbs 17275 +_gcplusg 17316 Σ_g cgsu 17499 SymGrpcsymg 19445 mulGrpcmgp 20222 Ringcrg 20321 Mat cmat 22575

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-ot 4597 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-sup 9400 df-dju 9894 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-xnn0 12584 df-z 12598 df-dec 12718 df-uz 12869 df-fz 13542 df-seq 14045 df-fac 14317 df-bc 14346 df-hash 14374 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-hom 17340 df-cco 17341 df-0g 17500 df-gsum 17501 df-prds 17506 df-pws 17508 df-efmnd 18934 df-symg 19446 df-mgp 20223 df-sra 21305 df-rgmod 21306 df-dsmm 21893 df-frlm 21908 df-mat 22576

This theorem is used by: m2detleib 22799

W3C validator