Theorem subrguss 20237

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 485	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
2		subrguss.3	. . . . . . . . 9 ⊢ 𝑉 = (Unit‘𝑆)
3		eqid 2736	. . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆)
4		eqid 2736	. . . . . . . . 9 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
5		eqid 2736	. . . . . . . . 9 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
6		eqid 2736	. . . . . . . . 9 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
7	2, 3, 4, 5, 6	isunit 20086	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 ↔ (𝑥(∥r‘𝑆)(1r‘𝑆) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑆)))
8	1, 7	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(∥r‘𝑆)(1r‘𝑆) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑆)))
9	8	simpld 495	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑆))
10		subrguss.1	. . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
11		eqid 2736	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	10, 11	subrg1 20232	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
13	12	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (1r‘𝑅) = (1r‘𝑆))
14	9, 13	breqtrrd 5133	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑅))
15		eqid 2736	. . . . . . . 8 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
16	10, 15, 4	subrgdvds 20236	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
17	16	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
18	17	ssbrd 5148	. . . . 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 20231	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
21	20	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
22		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
23	22	subrgss 20223	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
24	23	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 ⊆ (Base‘𝑅))
25	21, 24	eqsstrrd 3983	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
26		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
27	26, 2	unitcl 20088	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑆))
28	27	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑆))
29	25, 28	sseldd 3945	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
30	10	subrgring 20225	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
31		eqid 2736	. . . . . . . . 9 ⊢ (invr‘𝑆) = (invr‘𝑆)
32	2, 31, 26	ringinvcl 20105	. . . . . . . 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 3945	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑅))
35		eqid 2736	. . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
36	35, 22	opprbas 20056	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
37		eqid 2736	. . . . . . 7 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
38		eqid 2736	. . . . . . 7 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
39	36, 37, 38	dvdsrmul 20077	. . . . . 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 2736	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
42	22, 41, 35, 38	opprmul 20052	. . . . . 6 ⊢ (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥))
43		eqid 2736	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
44	2, 31, 43, 3	unitrinv 20107	. . . . . . . 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 17188	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
47	46	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
48	47	oveqd 7374	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)))
49	45, 48, 13	3eqtr4d 2786	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (1r‘𝑅))
50	42, 49	eqtrid 2788	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
51	40, 50	breqtrd 5131	. . . 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 20086	. . . 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 413	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑈))
56	55	ssrdv 3950	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ⊆ wss 3910   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083   ↾_s cress 17112  ._rcmulr 17134  1_rcur 19913  Ringcrg 19964  opp_rcoppr 20048  ∥_rcdsr 20067  Unitcui 20068  inv_rcinvr 20100  SubRingcsubrg 20218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-subrg 20220

This theorem is referenced by:  subrginv  20238  subrgdv  20239  subrgunit  20240  subrgugrp  20241  issubdrg  20247  zringunit  20887