m2detleiblem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > m2detleiblem4		Structured version Visualization version GIF version

Theorem m2detleiblem4 20436

Description: Lemma 4 for m2detleib 20437. (Contributed by AV, 20-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
m2detleiblem4	⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))

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 m2detleiblem4**
Step	Hyp	Ref	Expression
1		m2detleiblem2.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
2		eqid 2622	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	mgpbas 18495	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
4		m2detleiblem3.m	. . 3 ⊢ · = (+_g‘𝐺)
5		fvex 6201	. . . . 5 ⊢ (mulGrp‘𝑅) ∈ V
6	1, 5	eqeltri 2697	. . . 4 ⊢ 𝐺 ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ V)
8		1ex 10035	. . . . . . 7 ⊢ 1 ∈ V
9		2nn 11185	. . . . . . 7 ⊢ 2 ∈ ℕ
10		prex 4909	. . . . . . . . 9 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ V
11	10	prid2 4298	. . . . . . . 8 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}
12		eqid 2622	. . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
13		m2detleiblem2.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
14		m2detleiblem2.n	. . . . . . . . 9 ⊢ 𝑁 = {1, 2}
15	12, 13, 14	symg2bas 17818	. . . . . . . 8 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}})
16	11, 15	syl5eleqr 2708	. . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {⟨1, 2⟩, ⟨2, 1⟩} ∈ 𝑃)
17	8, 9, 16	mp2an 708	. . . . . 6 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ 𝑃
18		eleq1 2689	. . . . . 6 ⊢ (𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑄 ∈ 𝑃 ↔ {⟨1, 2⟩, ⟨2, 1⟩} ∈ 𝑃))
19	17, 18	mpbiri 248	. . . . 5 ⊢ (𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} → 𝑄 ∈ 𝑃)
20		m2detleiblem2.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
21	14	oveq1i 6660	. . . . . . 7 ⊢ (𝑁 Mat 𝑅) = ({1, 2} Mat 𝑅)
22	20, 21	eqtri 2644	. . . . . 6 ⊢ 𝐴 = ({1, 2} Mat 𝑅)
23		m2detleiblem2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
24	14	fveq2i 6194	. . . . . . . 8 ⊢ (SymGrp‘𝑁) = (SymGrp‘{1, 2})
25	24	fveq2i 6194	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘{1, 2}))
26	13, 25	eqtri 2644	. . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘{1, 2}))
27	22, 23, 26	matepmcl 20268	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
28	19, 27	syl3an2 1360	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
29		mpteq1 4737	. . . . . 6 ⊢ (𝑁 = {1, 2} → (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ {1, 2} ↦ ((𝑄‘𝑛)𝑀𝑛)))
30	14, 29	ax-mp 5	. . . . 5 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ {1, 2} ↦ ((𝑄‘𝑛)𝑀𝑛))
31	30	fmpt 6381	. . . 4 ⊢ (∀𝑛 ∈ {1, 2} ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)):{1, 2}⟶(Base‘𝑅))
32	28, 31	sylib 208	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)):{1, 2}⟶(Base‘𝑅))
33	3, 4, 7, 32	gsumpr12val 17282	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = (((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) · ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2)))
34	8	prid1 4297	. . . . . 6 ⊢ 1 ∈ {1, 2}
35	34, 14	eleqtrri 2700	. . . . 5 ⊢ 1 ∈ 𝑁
36	20, 23, 13	matepmcl 20268	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
37	19, 36	syl3an2 1360	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
38		fveq2 6191	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑄‘𝑛) = (𝑄‘1))
39		id 22	. . . . . . . . 9 ⊢ (𝑛 = 1 → 𝑛 = 1)
40	38, 39	oveq12d 6668	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘1)𝑀1))
41	40	eleq1d 2686	. . . . . . 7 ⊢ (𝑛 = 1 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)))
42	41	rspcva 3307	. . . . . 6 ⊢ ((1 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
43	35, 37, 42	sylancr 695	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) ∈ (Base‘𝑅))
44		eqid 2622	. . . . . 6 ⊢ (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)) = (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))
45	40, 44	fvmptg 6280	. . . . 5 ⊢ ((1 ∈ 𝑁 ∧ ((𝑄‘1)𝑀1) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
46	35, 43, 45	sylancr 695	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = ((𝑄‘1)𝑀1))
47		fveq1 6190	. . . . . . 7 ⊢ (𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑄‘1) = ({⟨1, 2⟩, ⟨2, 1⟩}‘1))
48		1ne2 11240	. . . . . . . 8 ⊢ 1 ≠ 2
49		2ex 11092	. . . . . . . . 9 ⊢ 2 ∈ V
50	8, 49	fvpr1 6456	. . . . . . . 8 ⊢ (1 ≠ 2 → ({⟨1, 2⟩, ⟨2, 1⟩}‘1) = 2)
51	48, 50	ax-mp 5	. . . . . . 7 ⊢ ({⟨1, 2⟩, ⟨2, 1⟩}‘1) = 2
52	47, 51	syl6eq 2672	. . . . . 6 ⊢ (𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑄‘1) = 2)
53	52	3ad2ant2 1083	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘1) = 2)
54	53	oveq1d 6665	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘1)𝑀1) = (2𝑀1))
55	46, 54	eqtrd 2656	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) = (2𝑀1))
56	49	prid2 4298	. . . . . 6 ⊢ 2 ∈ {1, 2}
57	56, 14	eleqtrri 2700	. . . . 5 ⊢ 2 ∈ 𝑁
58		fveq2 6191	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝑄‘𝑛) = (𝑄‘2))
59		id 22	. . . . . . . . 9 ⊢ (𝑛 = 2 → 𝑛 = 2)
60	58, 59	oveq12d 6668	. . . . . . . 8 ⊢ (𝑛 = 2 → ((𝑄‘𝑛)𝑀𝑛) = ((𝑄‘2)𝑀2))
61	60	eleq1d 2686	. . . . . . 7 ⊢ (𝑛 = 2 → (((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅) ↔ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)))
62	61	rspcva 3307	. . . . . 6 ⊢ ((2 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
63	57, 37, 62	sylancr 695	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) ∈ (Base‘𝑅))
64	60, 44	fvmptg 6280	. . . . 5 ⊢ ((2 ∈ 𝑁 ∧ ((𝑄‘2)𝑀2) ∈ (Base‘𝑅)) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
65	57, 63, 64	sylancr 695	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = ((𝑄‘2)𝑀2))
66		fveq1 6190	. . . . . . 7 ⊢ (𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑄‘2) = ({⟨1, 2⟩, ⟨2, 1⟩}‘2))
67	49, 8	fvpr2 6457	. . . . . . . 8 ⊢ (1 ≠ 2 → ({⟨1, 2⟩, ⟨2, 1⟩}‘2) = 1)
68	48, 67	ax-mp 5	. . . . . . 7 ⊢ ({⟨1, 2⟩, ⟨2, 1⟩}‘2) = 1
69	66, 68	syl6eq 2672	. . . . . 6 ⊢ (𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} → (𝑄‘2) = 1)
70	69	3ad2ant2 1083	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝑄‘2) = 1)
71	70	oveq1d 6665	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑄‘2)𝑀2) = (1𝑀2))
72	65, 71	eqtrd 2656	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2) = (1𝑀2))
73	55, 72	oveq12d 6668	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘1) · ((𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))‘2)) = ((2𝑀1) · (1𝑀2)))
74	33, 73	eqtrd 2656	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 {cpr 4179 ⟨cop 4183 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1c1 9937 ℕcn 11020 2c2 11070 Basecbs 15857 +_gcplusg 15941 Σ_g cgsu 16101 SymGrpcsymg 17797 mulGrpcmgp 18489 Ringcrg 18547 Mat cmat 20213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-fac 13061 df-bc 13090 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-symg 17798 df-mgp 18490 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mat 20214

This theorem is referenced by: m2detleib 20437

W3C validator