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Theorem subrguss 18735
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 477 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥𝑉)
2 subrguss.3 . . . . . . . . 9 𝑉 = (Unit‘𝑆)
3 eqid 2621 . . . . . . . . 9 (1r𝑆) = (1r𝑆)
4 eqid 2621 . . . . . . . . 9 (∥r𝑆) = (∥r𝑆)
5 eqid 2621 . . . . . . . . 9 (oppr𝑆) = (oppr𝑆)
6 eqid 2621 . . . . . . . . 9 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
72, 3, 4, 5, 6isunit 18597 . . . . . . . 8 (𝑥𝑉 ↔ (𝑥(∥r𝑆)(1r𝑆) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑆)))
81, 7sylib 208 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(∥r𝑆)(1r𝑆) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑆)))
98simpld 475 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑆)(1r𝑆))
10 subrguss.1 . . . . . . . 8 𝑆 = (𝑅s 𝐴)
11 eqid 2621 . . . . . . . 8 (1r𝑅) = (1r𝑅)
1210, 11subrg1 18730 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
1312adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (1r𝑅) = (1r𝑆))
149, 13breqtrrd 4651 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑆)(1r𝑅))
15 eqid 2621 . . . . . . . 8 (∥r𝑅) = (∥r𝑅)
1610, 15, 4subrgdvds 18734 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (∥r𝑆) ⊆ (∥r𝑅))
1716adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (∥r𝑆) ⊆ (∥r𝑅))
1817ssbrd 4666 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(∥r𝑆)(1r𝑅) → 𝑥(∥r𝑅)(1r𝑅)))
1914, 18mpd 15 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑅)(1r𝑅))
2010subrgbas 18729 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
2120adantr 481 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝐴 = (Base‘𝑆))
22 eqid 2621 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
2322subrgss 18721 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
2423adantr 481 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝐴 ⊆ (Base‘𝑅))
2521, 24eqsstr3d 3625 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
26 eqid 2621 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
2726, 2unitcl 18599 . . . . . . . 8 (𝑥𝑉𝑥 ∈ (Base‘𝑆))
2827adantl 482 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑆))
2925, 28sseldd 3589 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑅))
3010subrgring 18723 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
31 eqid 2621 . . . . . . . . 9 (invr𝑆) = (invr𝑆)
322, 31, 26ringinvcl 18616 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
3330, 32sylan 488 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
3425, 33sseldd 3589 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑅))
35 eqid 2621 . . . . . . . 8 (oppr𝑅) = (oppr𝑅)
3635, 22opprbas 18569 . . . . . . 7 (Base‘𝑅) = (Base‘(oppr𝑅))
37 eqid 2621 . . . . . . 7 (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅))
38 eqid 2621 . . . . . . 7 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
3936, 37, 38dvdsrmul 18588 . . . . . 6 ((𝑥 ∈ (Base‘𝑅) ∧ ((invr𝑆)‘𝑥) ∈ (Base‘𝑅)) → 𝑥(∥r‘(oppr𝑅))(((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥))
4029, 34, 39syl2anc 692 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r‘(oppr𝑅))(((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥))
41 eqid 2621 . . . . . . 7 (.r𝑅) = (.r𝑅)
4222, 41, 35, 38opprmul 18566 . . . . . 6 (((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)((invr𝑆)‘𝑥))
43 eqid 2621 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
442, 31, 43, 3unitrinv 18618 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝑥𝑉) → (𝑥(.r𝑆)((invr𝑆)‘𝑥)) = (1r𝑆))
4530, 44sylan 488 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑆)((invr𝑆)‘𝑥)) = (1r𝑆))
4610, 41ressmulr 15946 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
4746adantr 481 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (.r𝑅) = (.r𝑆))
4847oveqd 6632 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑅)((invr𝑆)‘𝑥)) = (𝑥(.r𝑆)((invr𝑆)‘𝑥)))
4945, 48, 133eqtr4d 2665 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑅)((invr𝑆)‘𝑥)) = (1r𝑅))
5042, 49syl5eq 2667 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥) = (1r𝑅))
5140, 50breqtrd 4649 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r‘(oppr𝑅))(1r𝑅))
52 subrguss.2 . . . . 5 𝑈 = (Unit‘𝑅)
5352, 11, 15, 35, 37isunit 18597 . . . 4 (𝑥𝑈 ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
5419, 51, 53sylanbrc 697 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥𝑈)
5554ex 450 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝑉𝑥𝑈))
5655ssrdv 3594 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3560   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  s cress 15801  .rcmulr 15882  1rcur 18441  Ringcrg 18487  opprcoppr 18562  rcdsr 18578  Unitcui 18579  invrcinvr 18611  SubRingcsubrg 18716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-tpos 7312  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-subg 17531  df-mgp 18430  df-ur 18442  df-ring 18489  df-oppr 18563  df-dvdsr 18581  df-unit 18582  df-invr 18612  df-subrg 18718
This theorem is referenced by:  subrginv  18736  subrgdv  18737  subrgunit  18738  subrgugrp  18739  issubdrg  18745  zringunit  19776
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