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Theorem subrgunit 19000
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 18997 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
54sselda 3744 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝑈)
6 eqid 2760 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
76, 3unitcl 18859 . . . . 5 (𝑋𝑉𝑋 ∈ (Base‘𝑆))
87adantl 473 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋 ∈ (Base‘𝑆))
91subrgbas 18991 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
109adantr 472 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝐴 = (Base‘𝑆))
118, 10eleqtrrd 2842 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝐴)
121subrgring 18985 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
13 eqid 2760 . . . . . 6 (invr𝑆) = (invr𝑆)
143, 13, 6ringinvcl 18876 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
1512, 14sylan 489 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
16 subrgunit.4 . . . . 5 𝐼 = (invr𝑅)
171, 16, 3, 13subrginv 18998 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) = ((invr𝑆)‘𝑋))
1815, 17, 103eltr4d 2854 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) ∈ 𝐴)
195, 11, 183jca 1123 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴))
20 simpr2 1236 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝐴)
219adantr 472 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
2220, 21eleqtrd 2841 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
23 simpr3 1238 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ 𝐴)
2423, 21eleqtrd 2841 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ (Base‘𝑆))
25 eqid 2760 . . . . . 6 (∥r𝑆) = (∥r𝑆)
26 eqid 2760 . . . . . 6 (.r𝑆) = (.r𝑆)
276, 25, 26dvdsrmul 18848 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
2822, 24, 27syl2anc 696 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
29 subrgrcl 18987 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3029adantr 472 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑅 ∈ Ring)
31 simpr1 1234 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑈)
32 eqid 2760 . . . . . . 7 (.r𝑅) = (.r𝑅)
33 eqid 2760 . . . . . . 7 (1r𝑅) = (1r𝑅)
342, 16, 32, 33unitlinv 18877 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
3530, 31, 34syl2anc 696 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
361, 32ressmulr 16208 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3736adantr 472 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (.r𝑅) = (.r𝑆))
3837oveqd 6830 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = ((𝐼𝑋)(.r𝑆)𝑋))
391, 33subrg1 18992 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
4039adantr 472 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (1r𝑅) = (1r𝑆))
4135, 38, 403eqtr3d 2802 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑆)𝑋) = (1r𝑆))
4228, 41breqtrd 4830 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)(1r𝑆))
43 eqid 2760 . . . . . . 7 (oppr𝑆) = (oppr𝑆)
4443, 6opprbas 18829 . . . . . 6 (Base‘𝑆) = (Base‘(oppr𝑆))
45 eqid 2760 . . . . . 6 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
46 eqid 2760 . . . . . 6 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
4744, 45, 46dvdsrmul 18848 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
4822, 24, 47syl2anc 696 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
496, 26, 43, 46opprmul 18826 . . . . 5 ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (𝑋(.r𝑆)(𝐼𝑋))
502, 16, 32, 33unitrinv 18878 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5130, 31, 50syl2anc 696 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5237oveqd 6830 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (𝑋(.r𝑆)(𝐼𝑋)))
5351, 52, 403eqtr3d 2802 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑆)(𝐼𝑋)) = (1r𝑆))
5449, 53syl5eq 2806 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (1r𝑆))
5548, 54breqtrd 4830 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))(1r𝑆))
56 eqid 2760 . . . 4 (1r𝑆) = (1r𝑆)
573, 56, 25, 43, 45isunit 18857 . . 3 (𝑋𝑉 ↔ (𝑋(∥r𝑆)(1r𝑆) ∧ 𝑋(∥r‘(oppr𝑆))(1r𝑆)))
5842, 55, 57sylanbrc 701 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑉)
5919, 58impbida 913 1 (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  s cress 16060  .rcmulr 16144  1rcur 18701  Ringcrg 18747  opprcoppr 18822  rcdsr 18838  Unitcui 18839  invrcinvr 18871  SubRingcsubrg 18978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-tpos 7521  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-subg 17792  df-mgp 18690  df-ur 18702  df-ring 18749  df-oppr 18823  df-dvdsr 18841  df-unit 18842  df-invr 18872  df-subrg 18980
This theorem is referenced by:  issubdrg  19007  gzrngunit  20014  zringunit  20038  cphreccllem  23178
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