Theorem subrguss 20503

Description: A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrguss.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrguss.2	⊢ 𝑈 = (Unit‘𝑅)
subrguss.3	⊢ 𝑉 = (Unit‘𝑆)

Assertion

Ref	Expression
subrguss	⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)

Proof of Theorem subrguss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
2		subrguss.3	. . . . . . . . 9 ⊢ 𝑉 = (Unit‘𝑆)
3		eqid 2730	. . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆)
4		eqid 2730	. . . . . . . . 9 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
5		eqid 2730	. . . . . . . . 9 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
6		eqid 2730	. . . . . . . . 9 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
7	2, 3, 4, 5, 6	isunit 20289	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 ↔ (𝑥(∥r‘𝑆)(1r‘𝑆) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑆)))
8	1, 7	sylib 218	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(∥r‘𝑆)(1r‘𝑆) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑆)))
9	8	simpld 494	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑆))
10		subrguss.1	. . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
11		eqid 2730	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	10, 11	subrg1 20498	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
13	12	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (1r‘𝑅) = (1r‘𝑆))
14	9, 13	breqtrrd 5138	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑅))
15		eqid 2730	. . . . . . . 8 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
16	10, 15, 4	subrgdvds 20502	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
17	16	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
18	17	ssbrd 5153	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(∥r‘𝑆)(1r‘𝑅) → 𝑥(∥r‘𝑅)(1r‘𝑅)))
19	14, 18	mpd 15	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑅)(1r‘𝑅))
20	10	subrgbas 20497	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
21	20	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
22		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
23	22	subrgss 20488	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 ⊆ (Base‘𝑅))
25	21, 24	eqsstrrd 3985	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
26		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
27	26, 2	unitcl 20291	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑆))
28	27	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑆))
29	25, 28	sseldd 3950	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
30	10	subrgring 20490	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
31		eqid 2730	. . . . . . . . 9 ⊢ (invr‘𝑆) = (invr‘𝑆)
32	2, 31, 26	ringinvcl 20308	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
33	30, 32	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
34	25, 33	sseldd 3950	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑅))
35		eqid 2730	. . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
36	35, 22	opprbas 20259	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
37		eqid 2730	. . . . . . 7 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
38		eqid 2730	. . . . . . 7 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
39	36, 37, 38	dvdsrmul 20280	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑅)) → 𝑥(∥r‘(oppr‘𝑅))(((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
40	29, 34, 39	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘(oppr‘𝑅))(((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
41		eqid 2730	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
42	22, 41, 35, 38	opprmul 20256	. . . . . 6 ⊢ (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥))
43		eqid 2730	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
44	2, 31, 43, 3	unitrinv 20310	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)) = (1r‘𝑆))
45	30, 44	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)) = (1r‘𝑆))
46	10, 41	ressmulr 17277	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
47	46	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
48	47	oveqd 7407	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)))
49	45, 48, 13	3eqtr4d 2775	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (1r‘𝑅))
50	42, 49	eqtrid 2777	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
51	40, 50	breqtrd 5136	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))
52		subrguss.2	. . . . 5 ⊢ 𝑈 = (Unit‘𝑅)
53	52, 11, 15, 35, 37	isunit 20289	. . . 4 ⊢ (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)))
54	19, 51, 53	sylanbrc 583	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑈)
55	54	ex 412	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑈))
56	55	ssrdv 3955	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186   ↾_s cress 17207  ._rcmulr 17228  1_rcur 20097  Ringcrg 20149  opp_rcoppr 20252  ∥_rcdsr 20270  Unitcui 20271  inv_rcinvr 20303  SubRingcsubrg 20485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-subg 19062  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-subrg 20486

This theorem is referenced by:  subrginv  20504  subrgdv  20505  subrgunit  20506  subrgugrp  20507  issubdrg  20696  zringunit  21383