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Theorem subrgunit 19546
Description: An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrgugrp.1 𝑆 = (𝑅s 𝐴)
subrgugrp.2 𝑈 = (Unit‘𝑅)
subrgugrp.3 𝑉 = (Unit‘𝑆)
subrgunit.4 𝐼 = (invr𝑅)
Assertion
Ref Expression
subrgunit (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))

Proof of Theorem subrgunit
StepHypRef Expression
1 subrgugrp.1 . . . . 5 𝑆 = (𝑅s 𝐴)
2 subrgugrp.2 . . . . 5 𝑈 = (Unit‘𝑅)
3 subrgugrp.3 . . . . 5 𝑉 = (Unit‘𝑆)
41, 2, 3subrguss 19543 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
54sselda 3915 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝑈)
6 eqid 2798 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
76, 3unitcl 19405 . . . . 5 (𝑋𝑉𝑋 ∈ (Base‘𝑆))
87adantl 485 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋 ∈ (Base‘𝑆))
91subrgbas 19537 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
109adantr 484 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝐴 = (Base‘𝑆))
118, 10eleqtrrd 2893 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝐴)
121subrgring 19531 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
13 eqid 2798 . . . . . 6 (invr𝑆) = (invr𝑆)
143, 13, 6ringinvcl 19422 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
1512, 14sylan 583 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
16 subrgunit.4 . . . . 5 𝐼 = (invr𝑅)
171, 16, 3, 13subrginv 19544 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) = ((invr𝑆)‘𝑋))
1815, 17, 103eltr4d 2905 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) ∈ 𝐴)
195, 11, 183jca 1125 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴))
20 simpr2 1192 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝐴)
219adantr 484 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
2220, 21eleqtrd 2892 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
23 simpr3 1193 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ 𝐴)
2423, 21eleqtrd 2892 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ (Base‘𝑆))
25 eqid 2798 . . . . . 6 (∥r𝑆) = (∥r𝑆)
26 eqid 2798 . . . . . 6 (.r𝑆) = (.r𝑆)
276, 25, 26dvdsrmul 19394 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
2822, 24, 27syl2anc 587 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
29 subrgrcl 19533 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3029adantr 484 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑅 ∈ Ring)
31 simpr1 1191 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑈)
32 eqid 2798 . . . . . . 7 (.r𝑅) = (.r𝑅)
33 eqid 2798 . . . . . . 7 (1r𝑅) = (1r𝑅)
342, 16, 32, 33unitlinv 19423 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
3530, 31, 34syl2anc 587 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
361, 32ressmulr 16617 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3736adantr 484 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (.r𝑅) = (.r𝑆))
3837oveqd 7152 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = ((𝐼𝑋)(.r𝑆)𝑋))
391, 33subrg1 19538 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
4039adantr 484 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (1r𝑅) = (1r𝑆))
4135, 38, 403eqtr3d 2841 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑆)𝑋) = (1r𝑆))
4228, 41breqtrd 5056 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)(1r𝑆))
43 eqid 2798 . . . . . . 7 (oppr𝑆) = (oppr𝑆)
4443, 6opprbas 19375 . . . . . 6 (Base‘𝑆) = (Base‘(oppr𝑆))
45 eqid 2798 . . . . . 6 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
46 eqid 2798 . . . . . 6 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
4744, 45, 46dvdsrmul 19394 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
4822, 24, 47syl2anc 587 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
496, 26, 43, 46opprmul 19372 . . . . 5 ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (𝑋(.r𝑆)(𝐼𝑋))
502, 16, 32, 33unitrinv 19424 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5130, 31, 50syl2anc 587 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5237oveqd 7152 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (𝑋(.r𝑆)(𝐼𝑋)))
5351, 52, 403eqtr3d 2841 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑆)(𝐼𝑋)) = (1r𝑆))
5449, 53syl5eq 2845 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (1r𝑆))
5548, 54breqtrd 5056 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))(1r𝑆))
56 eqid 2798 . . . 4 (1r𝑆) = (1r𝑆)
573, 56, 25, 43, 45isunit 19403 . . 3 (𝑋𝑉 ↔ (𝑋(∥r𝑆)(1r𝑆) ∧ 𝑋(∥r‘(oppr𝑆))(1r𝑆)))
5842, 55, 57sylanbrc 586 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑉)
5919, 58impbida 800 1 (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  s cress 16476  .rcmulr 16558  1rcur 19244  Ringcrg 19290  opprcoppr 19368  rcdsr 19384  Unitcui 19385  invrcinvr 19417  SubRingcsubrg 19524
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-subg 18268  df-mgp 19233  df-ur 19245  df-ring 19292  df-oppr 19369  df-dvdsr 19387  df-unit 19388  df-invr 19418  df-subrg 19526
This theorem is referenced by:  issubdrg  19553  gzrngunit  20157  zringunit  20181  cphreccllem  23783
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