Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  subrguss Structured version   Visualization version   GIF version

Theorem subrguss 18618
 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 476 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥𝑉)
2 subrguss.3 . . . . . . . . 9 𝑉 = (Unit‘𝑆)
3 eqid 2610 . . . . . . . . 9 (1r𝑆) = (1r𝑆)
4 eqid 2610 . . . . . . . . 9 (∥r𝑆) = (∥r𝑆)
5 eqid 2610 . . . . . . . . 9 (oppr𝑆) = (oppr𝑆)
6 eqid 2610 . . . . . . . . 9 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
72, 3, 4, 5, 6isunit 18480 . . . . . . . 8 (𝑥𝑉 ↔ (𝑥(∥r𝑆)(1r𝑆) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑆)))
81, 7sylib 207 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(∥r𝑆)(1r𝑆) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑆)))
98simpld 474 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑆)(1r𝑆))
10 subrguss.1 . . . . . . . 8 𝑆 = (𝑅s 𝐴)
11 eqid 2610 . . . . . . . 8 (1r𝑅) = (1r𝑅)
1210, 11subrg1 18613 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
1312adantr 480 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (1r𝑅) = (1r𝑆))
149, 13breqtrrd 4611 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑆)(1r𝑅))
15 eqid 2610 . . . . . . . 8 (∥r𝑅) = (∥r𝑅)
1610, 15, 4subrgdvds 18617 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (∥r𝑆) ⊆ (∥r𝑅))
1716adantr 480 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (∥r𝑆) ⊆ (∥r𝑅))
1817ssbrd 4626 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(∥r𝑆)(1r𝑅) → 𝑥(∥r𝑅)(1r𝑅)))
1914, 18mpd 15 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r𝑅)(1r𝑅))
2010subrgbas 18612 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
2120adantr 480 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝐴 = (Base‘𝑆))
22 eqid 2610 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
2322subrgss 18604 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
2423adantr 480 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝐴 ⊆ (Base‘𝑅))
2521, 24eqsstr3d 3603 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
26 eqid 2610 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
2726, 2unitcl 18482 . . . . . . . 8 (𝑥𝑉𝑥 ∈ (Base‘𝑆))
2827adantl 481 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑆))
2925, 28sseldd 3569 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑅))
3010subrgring 18606 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
31 eqid 2610 . . . . . . . . 9 (invr𝑆) = (invr𝑆)
322, 31, 26ringinvcl 18499 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
3330, 32sylan 487 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
3425, 33sseldd 3569 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑅))
35 eqid 2610 . . . . . . . 8 (oppr𝑅) = (oppr𝑅)
3635, 22opprbas 18452 . . . . . . 7 (Base‘𝑅) = (Base‘(oppr𝑅))
37 eqid 2610 . . . . . . 7 (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅))
38 eqid 2610 . . . . . . 7 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
3936, 37, 38dvdsrmul 18471 . . . . . 6 ((𝑥 ∈ (Base‘𝑅) ∧ ((invr𝑆)‘𝑥) ∈ (Base‘𝑅)) → 𝑥(∥r‘(oppr𝑅))(((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥))
4029, 34, 39syl2anc 691 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r‘(oppr𝑅))(((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥))
41 eqid 2610 . . . . . . 7 (.r𝑅) = (.r𝑅)
4222, 41, 35, 38opprmul 18449 . . . . . 6 (((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)((invr𝑆)‘𝑥))
43 eqid 2610 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
442, 31, 43, 3unitrinv 18501 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝑥𝑉) → (𝑥(.r𝑆)((invr𝑆)‘𝑥)) = (1r𝑆))
4530, 44sylan 487 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑆)((invr𝑆)‘𝑥)) = (1r𝑆))
4610, 41ressmulr 15829 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
4746adantr 480 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (.r𝑅) = (.r𝑆))
4847oveqd 6566 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑅)((invr𝑆)‘𝑥)) = (𝑥(.r𝑆)((invr𝑆)‘𝑥)))
4945, 48, 133eqtr4d 2654 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (𝑥(.r𝑅)((invr𝑆)‘𝑥)) = (1r𝑅))
5042, 49syl5eq 2656 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → (((invr𝑆)‘𝑥)(.r‘(oppr𝑅))𝑥) = (1r𝑅))
5140, 50breqtrd 4609 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥(∥r‘(oppr𝑅))(1r𝑅))
52 subrguss.2 . . . . 5 𝑈 = (Unit‘𝑅)
5352, 11, 15, 35, 37isunit 18480 . . . 4 (𝑥𝑈 ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
5419, 51, 53sylanbrc 695 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝑉) → 𝑥𝑈)
5554ex 449 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝑉𝑥𝑈))
5655ssrdv 3574 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  .rcmulr 15769  1rcur 18324  Ringcrg 18370  opprcoppr 18445  ∥rcdsr 18461  Unitcui 18462  invrcinvr 18494  SubRingcsubrg 18599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-subrg 18601 This theorem is referenced by:  subrginv  18619  subrgdv  18620  subrgunit  18621  subrgugrp  18622  issubdrg  18628  zringunit  19655
 Copyright terms: Public domain W3C validator