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Theorem opprunit 17824
Description: Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
opprunit.1  |-  U  =  (Unit `  R )
opprunit.2  |-  S  =  (oppr
`  R )
Assertion
Ref Expression
opprunit  |-  U  =  (Unit `  S )

Proof of Theorem opprunit
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprunit.2 . . . . . . . . . . 11  |-  S  =  (oppr
`  R )
2 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 17792 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  S )
4 eqid 2429 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
5 eqid 2429 . . . . . . . . . 10  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2429 . . . . . . . . . 10  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
73, 4, 5, 6opprmul 17789 . . . . . . . . 9  |-  ( y ( .r `  (oppr `  S
) ) x )  =  ( x ( .r `  S ) y )
8 eqid 2429 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
92, 8, 1, 4opprmul 17789 . . . . . . . . 9  |-  ( x ( .r `  S
) y )  =  ( y ( .r
`  R ) x )
107, 9eqtr2i 2459 . . . . . . . 8  |-  ( y ( .r `  R
) x )  =  ( y ( .r
`  (oppr
`  S ) ) x )
1110eqeq1i 2436 . . . . . . 7  |-  ( ( y ( .r `  R ) x )  =  ( 1r `  R )  <->  ( y
( .r `  (oppr `  S
) ) x )  =  ( 1r `  R ) )
1211rexbii 2934 . . . . . 6  |-  ( E. y  e.  ( Base `  R ) ( y ( .r `  R
) x )  =  ( 1r `  R
)  <->  E. y  e.  (
Base `  R )
( y ( .r
`  (oppr
`  S ) ) x )  =  ( 1r `  R ) )
1312anbi2i 698 . . . . 5  |-  ( ( x  e.  ( Base `  R )  /\  E. y  e.  ( Base `  R ) ( y ( .r `  R
) x )  =  ( 1r `  R
) )  <->  ( x  e.  ( Base `  R
)  /\  E. y  e.  ( Base `  R
) ( y ( .r `  (oppr `  S
) ) x )  =  ( 1r `  R ) ) )
14 eqid 2429 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
152, 14, 8dvdsr 17809 . . . . 5  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
( x  e.  (
Base `  R )  /\  E. y  e.  (
Base `  R )
( y ( .r
`  R ) x )  =  ( 1r
`  R ) ) )
165, 3opprbas 17792 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (oppr
`  S ) )
17 eqid 2429 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
1816, 17, 6dvdsr 17809 . . . . 5  |-  ( x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
)  <->  ( x  e.  ( Base `  R
)  /\  E. y  e.  ( Base `  R
) ( y ( .r `  (oppr `  S
) ) x )  =  ( 1r `  R ) ) )
1913, 15, 183bitr4i 280 . . . 4  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) )
2019anbi2ci 700 . . 3  |-  ( ( x ( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  S
) ( 1r `  R ) )  <->  ( x
( ||r `
 S ) ( 1r `  R )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  R ) ) )
21 opprunit.1 . . . 4  |-  U  =  (Unit `  R )
22 eqid 2429 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
23 eqid 2429 . . . 4  |-  ( ||r `  S
)  =  ( ||r `  S
)
2421, 22, 14, 1, 23isunit 17820 . . 3  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  S
) ( 1r `  R ) ) )
25 eqid 2429 . . . 4  |-  (Unit `  S )  =  (Unit `  S )
261, 22oppr1 17797 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  S
)
2725, 26, 23, 5, 17isunit 17820 . . 3  |-  ( x  e.  (Unit `  S
)  <->  ( x (
||r `  S ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) ) )
2820, 24, 273bitr4i 280 . 2  |-  ( x  e.  U  <->  x  e.  (Unit `  S ) )
2928eqriv 2425 1  |-  U  =  (Unit `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   .rcmulr 15153   1rcur 17670  opprcoppr 17785   ||rcdsr 17801  Unitcui 17802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-plusg 15165  df-mulr 15166  df-0g 15299  df-mgp 17659  df-ur 17671  df-oppr 17786  df-dvdsr 17804  df-unit 17805
This theorem is referenced by:  opprirred  17865  irredlmul  17871  opprdrng  17934  ply1divalg2  22964
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