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Theorem subrguss 19479
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 = (𝑅s 𝐴)
subrguss.2 𝑈 = (Unit‘𝑅)
subrguss.3 𝑉 = (Unit‘𝑆)
Assertion
Ref Expression
subrguss (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)

Proof of Theorem subrguss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥𝑉)
2 subrguss.3 . . . . . . . . 9 𝑉 = (Unit‘𝑆)
3 eqid 2818 . . . . . . . . 9 (1r𝑆) = (1r𝑆)
4 eqid 2818 . . . . . . . . 9 (∥r𝑆) = (∥r𝑆)
5 eqid 2818 . . . . . . . . 9 (oppr𝑆) = (oppr𝑆)
6 eqid 2818 . . . . . . . . 9 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
72, 3, 4, 5, 6isunit 19336 . . . . . . . 8 (𝑥𝑉 ↔ (𝑥(∥r𝑆)(1r𝑆) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑆)))
81, 7sylib 219 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(∥r𝑆)(1r𝑆) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑆)))
98simpld 495 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑆)(1r𝑆))
10 subrguss.1 . . . . . . . 8 𝑆 = (𝑅s 𝐴)
11 eqid 2818 . . . . . . . 8 (1r𝑅) = (1r𝑅)
1210, 11subrg1 19474 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
1312adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (1r𝑅) = (1r𝑆))
149, 13breqtrrd 5085 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑆)(1r𝑅))
15 eqid 2818 . . . . . . . 8 (∥r𝑅) = (∥r𝑅)
1610, 15, 4subrgdvds 19478 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (∥r𝑆) ⊆ (∥r𝑅))
1716adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (∥r𝑆) ⊆ (∥r𝑅))
1817ssbrd 5100 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(∥r𝑆)(1r𝑅) → 𝑥(∥r𝑅)(1r𝑅)))
1914, 18mpd 15 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑅)(1r𝑅))
2010subrgbas 19473 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
2120adantr 481 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝐴 = (Base‘𝑆))
22 eqid 2818 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
2322subrgss 19465 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
2423adantr 481 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝐴 ⊆ (Base‘𝑅))
2521, 24eqsstrrd 4003 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
26 eqid 2818 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
2726, 2unitcl 19338 . . . . . . . 8 (𝑥𝑉𝑥 ∈ (Base‘𝑆))
2827adantl 482 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑆))
2925, 28sseldd 3965 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑅))
3010subrgring 19467 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
31 eqid 2818 . . . . . . . . 9 (invr𝑆) = (invr𝑆)
322, 31, 26ringinvcl 19355 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
3330, 32sylan 580 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
3425, 33sseldd 3965 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑅))
35 eqid 2818 . . . . . . . 8 (oppr𝑅) = (oppr𝑅)
3635, 22opprbas 19308 . . . . . . 7 (Base‘𝑅) = (Base‘(oppr𝑅))
37 eqid 2818 . . . . . . 7 (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅))
38 eqid 2818 . . . . . . 7 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
3936, 37, 38dvdsrmul 19327 . . . . . 6 ((𝑥 ∈ (Base‘𝑅) ∧ ((invr𝑆)‘𝑥) ∈ (Base‘𝑅)) → 𝑥(∥r‘(oppr𝑅))(((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥))
4029, 34, 39syl2anc 584 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r‘(oppr𝑅))(((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥))
41 eqid 2818 . . . . . . 7 (.r𝑅) = (.r𝑅)
4222, 41, 35, 38opprmul 19305 . . . . . 6 (((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)((invr𝑆)‘𝑥))
43 eqid 2818 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
442, 31, 43, 3unitrinv 19357 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝑥𝑉) → (𝑥(.r𝑆)((invr𝑆)‘𝑥)) = (1r𝑆))
4530, 44sylan 580 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑆)((invr𝑆)‘𝑥)) = (1r𝑆))
4610, 41ressmulr 16613 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
4746adantr 481 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (.r𝑅) = (.r𝑆))
4847oveqd 7162 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑅)((invr𝑆)‘𝑥)) = (𝑥(.r𝑆)((invr𝑆)‘𝑥)))
4945, 48, 133eqtr4d 2863 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑅)((invr𝑆)‘𝑥)) = (1r𝑅))
5042, 49syl5eq 2865 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥) = (1r𝑅))
5140, 50breqtrd 5083 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r‘(oppr𝑅))(1r𝑅))
52 subrguss.2 . . . . 5 𝑈 = (Unit‘𝑅)
5352, 11, 15, 35, 37isunit 19336 . . . 4 (𝑥𝑈 ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
5419, 51, 53sylanbrc 583 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥𝑈)
5554ex 413 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝑉𝑥𝑈))
5655ssrdv 3970 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wss 3933   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  s cress 16472  .rcmulr 16554  1rcur 19180  Ringcrg 19226  opprcoppr 19301  rcdsr 19317  Unitcui 19318  invrcinvr 19350  SubRingcsubrg 19460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-subg 18214  df-mgp 19169  df-ur 19181  df-ring 19228  df-oppr 19302  df-dvdsr 19320  df-unit 19321  df-invr 19351  df-subrg 19462
This theorem is referenced by:  subrginv  19480  subrgdv  19481  subrgunit  19482  subrgugrp  19483  issubdrg  19489  zringunit  20563
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