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Theorem isunit 15455
 Description: Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)
Hypotheses
Ref Expression
unit.1 Unit
unit.2
unit.3 r
unit.4 oppr
unit.5 r
Assertion
Ref Expression
isunit

Proof of Theorem isunit
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elfvdm 5570 . . . 4 Unit Unit
2 unit.1 . . . 4 Unit
31, 2eleq2s 2388 . . 3 Unit
4 elex 2809 . . 3 Unit
53, 4syl 15 . 2
6 df-br 4040 . . . 4
7 elfvdm 5570 . . . . . 6 r r
8 unit.3 . . . . . 6 r
97, 8eleq2s 2388 . . . . 5 r
10 elex 2809 . . . . 5 r
119, 10syl 15 . . . 4
126, 11sylbi 187 . . 3
14 fveq2 5541 . . . . . . . . . . 11 r r
1514, 8syl6eqr 2346 . . . . . . . . . 10 r
16 fveq2 5541 . . . . . . . . . . . . 13 oppr oppr
17 unit.4 . . . . . . . . . . . . 13 oppr
1816, 17syl6eqr 2346 . . . . . . . . . . . 12 oppr
1918fveq2d 5545 . . . . . . . . . . 11 roppr r
20 unit.5 . . . . . . . . . . 11 r
2119, 20syl6eqr 2346 . . . . . . . . . 10 roppr
2215, 21ineq12d 3384 . . . . . . . . 9 r roppr
2322cnveqd 4873 . . . . . . . 8 r roppr
2423imaeq1d 5027 . . . . . . 7 r roppr
25 fveq2 5541 . . . . . . . . . 10
26 unit.2 . . . . . . . . . 10
2725, 26syl6eqr 2346 . . . . . . . . 9
2827sneqd 3666 . . . . . . . 8
2928imaeq2d 5028 . . . . . . 7
3024, 29eqtrd 2328 . . . . . 6 r roppr
31 df-unit 15440 . . . . . 6 Unit r roppr
32 fvex 5555 . . . . . . . . . 10 r
338, 32eqeltri 2366 . . . . . . . . 9
3433inex1 4171 . . . . . . . 8
3534cnvex 5225 . . . . . . 7
36 imaexg 5042 . . . . . . 7
3735, 36ax-mp 8 . . . . . 6
3830, 31, 37fvmpt 5618 . . . . 5 Unit
392, 38syl5eq 2340 . . . 4
4039eleq2d 2363 . . 3
41 inss1 3402 . . . . . 6
428reldvdsr 15442 . . . . . 6
43 relss 4791 . . . . . 6
4441, 42, 43mp2 17 . . . . 5
45 eliniseg2 5069 . . . . 5
4644, 45ax-mp 8 . . . 4
47 brin 4086 . . . 4
4846, 47bitri 240 . . 3
4940, 48syl6bb 252 . 2
505, 13, 49pm5.21nii 342 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   wceq 1632   wcel 1696  cvv 2801   cin 3164   wss 3165  csn 3653  cop 3656   class class class wbr 4039  ccnv 4704   cdm 4705  cima 4708   wrel 4710  cfv 5271  cur 15355  opprcoppr 15420  rcdsr 15436  Unitcui 15437 This theorem is referenced by:  1unit  15456  unitcl  15457  opprunit  15459  crngunit  15460  unitmulcl  15462  unitgrp  15465  unitnegcl  15479  unitpropd  15495  isdrng2  15538  subrguss  15576  subrgunit  15579  fidomndrng  16064 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-dvdsr 15439  df-unit 15440
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