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Theorem subrgunit 18719
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 18716 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
54sselda 3583 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝑈)
6 eqid 2621 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
76, 3unitcl 18580 . . . . 5 (𝑋𝑉𝑋 ∈ (Base‘𝑆))
87adantl 482 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋 ∈ (Base‘𝑆))
91subrgbas 18710 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
109adantr 481 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝐴 = (Base‘𝑆))
118, 10eleqtrrd 2701 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝐴)
121subrgring 18704 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
13 eqid 2621 . . . . . 6 (invr𝑆) = (invr𝑆)
143, 13, 6ringinvcl 18597 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
1512, 14sylan 488 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
16 subrgunit.4 . . . . 5 𝐼 = (invr𝑅)
171, 16, 3, 13subrginv 18717 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) = ((invr𝑆)‘𝑋))
1815, 17, 103eltr4d 2713 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) ∈ 𝐴)
195, 11, 183jca 1240 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴))
20 simpr2 1066 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝐴)
219adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
2220, 21eleqtrd 2700 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
23 simpr3 1067 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ 𝐴)
2423, 21eleqtrd 2700 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ (Base‘𝑆))
25 eqid 2621 . . . . . 6 (∥r𝑆) = (∥r𝑆)
26 eqid 2621 . . . . . 6 (.r𝑆) = (.r𝑆)
276, 25, 26dvdsrmul 18569 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
2822, 24, 27syl2anc 692 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
29 subrgrcl 18706 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3029adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑅 ∈ Ring)
31 simpr1 1065 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑈)
32 eqid 2621 . . . . . . 7 (.r𝑅) = (.r𝑅)
33 eqid 2621 . . . . . . 7 (1r𝑅) = (1r𝑅)
342, 16, 32, 33unitlinv 18598 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
3530, 31, 34syl2anc 692 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
361, 32ressmulr 15927 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3736adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (.r𝑅) = (.r𝑆))
3837oveqd 6621 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = ((𝐼𝑋)(.r𝑆)𝑋))
391, 33subrg1 18711 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
4039adantr 481 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (1r𝑅) = (1r𝑆))
4135, 38, 403eqtr3d 2663 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑆)𝑋) = (1r𝑆))
4228, 41breqtrd 4639 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)(1r𝑆))
43 eqid 2621 . . . . . . 7 (oppr𝑆) = (oppr𝑆)
4443, 6opprbas 18550 . . . . . 6 (Base‘𝑆) = (Base‘(oppr𝑆))
45 eqid 2621 . . . . . 6 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
46 eqid 2621 . . . . . 6 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
4744, 45, 46dvdsrmul 18569 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
4822, 24, 47syl2anc 692 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
496, 26, 43, 46opprmul 18547 . . . . 5 ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (𝑋(.r𝑆)(𝐼𝑋))
502, 16, 32, 33unitrinv 18599 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5130, 31, 50syl2anc 692 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5237oveqd 6621 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (𝑋(.r𝑆)(𝐼𝑋)))
5351, 52, 403eqtr3d 2663 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑆)(𝐼𝑋)) = (1r𝑆))
5449, 53syl5eq 2667 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (1r𝑆))
5548, 54breqtrd 4639 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))(1r𝑆))
56 eqid 2621 . . . 4 (1r𝑆) = (1r𝑆)
573, 56, 25, 43, 45isunit 18578 . . 3 (𝑋𝑉 ↔ (𝑋(∥r𝑆)(1r𝑆) ∧ 𝑋(∥r‘(oppr𝑆))(1r𝑆)))
5842, 55, 57sylanbrc 697 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑉)
5919, 58impbida 876 1 (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  s cress 15782  .rcmulr 15863  1rcur 18422  Ringcrg 18468  opprcoppr 18543  rcdsr 18559  Unitcui 18560  invrcinvr 18592  SubRingcsubrg 18697
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-subg 17512  df-mgp 18411  df-ur 18423  df-ring 18470  df-oppr 18544  df-dvdsr 18562  df-unit 18563  df-invr 18593  df-subrg 18699
This theorem is referenced by:  issubdrg  18726  gzrngunit  19731  zringunit  19755  cphreccllem  22886
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