Theorem ringidss 14035

Description: A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
ringidss.g	⊢ 𝑀 = ((mulGrp‘𝑅) ↾s 𝐴)
ringidss.b	⊢ 𝐵 = (Base‘𝑅)
ringidss.u	⊢ 1 = (1r‘𝑅)

Assertion

Ref	Expression
ringidss	⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 = (0g‘𝑀))

Proof of Theorem ringidss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2229	. 2 ⊢ (0g‘𝑀) = (0g‘𝑀)
3		eqid 2229	. 2 ⊢ (+g‘𝑀) = (+g‘𝑀)
4		simp3 1023	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 ∈ 𝐴)
5		ringidss.g	. . . . 5 ⊢ 𝑀 = ((mulGrp‘𝑅) ↾s 𝐴)
6	5	a1i 9	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝑀 = ((mulGrp‘𝑅) ↾s 𝐴))
7		eqid 2229	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
8		ringidss.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
9	7, 8	mgpbasg 13932	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(mulGrp‘𝑅)))
10	9	3ad2ant1 1042	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐵 = (Base‘(mulGrp‘𝑅)))
11	7	mgpex 13931	. . . . 5 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ V)
12	11	3ad2ant1 1042	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (mulGrp‘𝑅) ∈ V)
13		simp2 1022	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
14	6, 10, 12, 13	ressbas2d 13144	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 = (Base‘𝑀))
15	4, 14	eleqtrd 2308	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 ∈ (Base‘𝑀))
16	14, 13	eqsstrrd 3262	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (Base‘𝑀) ⊆ 𝐵)
17	16	sselda 3225	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → 𝑦 ∈ 𝐵)
18		eqid 2229	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	7, 18	mgpplusgg 13930	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
20	19	3ad2ant1 1042	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
21		basfn 13134	. . . . . . . . . 10 ⊢ Base Fn V
22		simp1 1021	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝑅 ∈ Ring)
23	22	elexd 2814	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝑅 ∈ V)
24		funfvex 5652	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
25	24	funfni 5429	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
26	21, 23, 25	sylancr 414	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (Base‘𝑅) ∈ V)
27	8, 26	eqeltrid 2316	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐵 ∈ V)
28	27, 13	ssexd 4227	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 ∈ V)
29	6, 20, 28, 12	ressplusgd 13205	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (.r‘𝑅) = (+g‘𝑀))
30	29	adantr 276	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (.r‘𝑅) = (+g‘𝑀))
31	30	oveqd 6030	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = ( 1 (+g‘𝑀)𝑦))
32		ringidss.u	. . . . . 6 ⊢ 1 = (1r‘𝑅)
33	8, 18, 32	ringlidm 14029	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = 𝑦)
34	33	3ad2antl1 1183	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = 𝑦)
35	31, 34	eqtr3d 2264	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (+g‘𝑀)𝑦) = 𝑦)
36	17, 35	syldan 282	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → ( 1 (+g‘𝑀)𝑦) = 𝑦)
37	30	oveqd 6030	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = (𝑦(+g‘𝑀) 1 ))
38	8, 18, 32	ringridm 14030	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = 𝑦)
39	38	3ad2antl1 1183	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = 𝑦)
40	37, 39	eqtr3d 2264	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(+g‘𝑀) 1 ) = 𝑦)
41	17, 40	syldan 282	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀) 1 ) = 𝑦)
42	1, 2, 3, 15, 36, 41	ismgmid2 13456	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 = (0g‘𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ⊆ wss 3198   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013  Basecbs 13075   ↾_s cress 13076  +_gcplusg 13153  ._rcmulr 13154  0_gc0g 13332  mulGrpcmgp 13926  1_rcur 13965  Ringcrg 14002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-3 9196  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-mulr 13167  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-mgp 13927  df-ur 13966  df-ring 14004

This theorem is referenced by:  unitgrpid  14125