Theorem subrg1 14477

Description: A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Hypotheses

Ref	Expression
subrg1.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrg1.2	⊢ 1 = (1r‘𝑅)

Assertion

Ref	Expression
subrg1	⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆))

Proof of Theorem subrg1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrg1.2	. 2 ⊢ 1 = (1r‘𝑅)
2		eqid 2234	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
3	2	subrg1cl 14475	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐴)
4		subrg1.1	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
5	4	subrgbas 14476	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
6	3, 5	eleqtrd 2313	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ (Base‘𝑆))
7		eqid 2234	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7	subrgss 14468	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
9	5, 8	eqsstrrd 3279	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
10	9	sselda 3242	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
11		subrgrcl 14472	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
12		eqid 2234	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
13	7, 12, 2	ringidmlem 14265	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥))
14	11, 13	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥))
15	4, 12	ressmulrg 13442	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
16	11, 15	mpdan 421	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
17	16	oveqd 6075	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((1r‘𝑅)(.r‘𝑅)𝑥) = ((1r‘𝑅)(.r‘𝑆)𝑥))
18	17	eqeq1d 2243	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ↔ ((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥))
19	16	oveqd 6075	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)(1r‘𝑅)) = (𝑥(.r‘𝑆)(1r‘𝑅)))
20	19	eqeq1d 2243	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥 ↔ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥))
21	18, 20	anbi12d 473	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥) ↔ (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)))
22	21	biimpa 296	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥)) → (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥))
23	14, 22	syldan 282	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥))
24	10, 23	syldan 282	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥))
25	24	ralrimiva 2617	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ (Base‘𝑆)(((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥))
26	4	subrgring 14470	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
27		eqid 2234	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
28		eqid 2234	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
29		eqid 2234	. . . . 5 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	27, 28, 29	isringid 14268	. . . 4 ⊢ (𝑆 ∈ Ring → (((1r‘𝑅) ∈ (Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)(((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) ↔ (1r‘𝑆) = (1r‘𝑅)))
31	26, 30	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (((1r‘𝑅) ∈ (Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)(((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) ↔ (1r‘𝑆) = (1r‘𝑅)))
32	6, 25, 31	mpbi2and 952	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑆) = (1r‘𝑅))
33	1, 32	eqtr4id 2286	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ‘cfv 5357  (class class class)co 6058  Basecbs 13296   ↾_s cress 13297  ._rcmulr 13375  1_rcur 14202  Ringcrg 14239  SubRingcsubrg 14463

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-subg 13923  df-mgp 14160  df-ur 14203  df-ring 14241  df-subrg 14465

This theorem is referenced by:  subrguss  14482  subrginv  14483  subrgunit  14485  subrgnzr  14488  subsubrg  14491  sralmod  14724  zring1  14875