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Theorem opprunit 15766
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 2436 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 15734 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  S )
4 eqid 2436 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
5 eqid 2436 . . . . . . . . . 10  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2436 . . . . . . . . . 10  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
73, 4, 5, 6opprmul 15731 . . . . . . . . 9  |-  ( y ( .r `  (oppr `  S
) ) x )  =  ( x ( .r `  S ) y )
8 eqid 2436 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
92, 8, 1, 4opprmul 15731 . . . . . . . . 9  |-  ( x ( .r `  S
) y )  =  ( y ( .r
`  R ) x )
107, 9eqtr2i 2457 . . . . . . . 8  |-  ( y ( .r `  R
) x )  =  ( y ( .r
`  (oppr
`  S ) ) x )
1110eqeq1i 2443 . . . . . . 7  |-  ( ( y ( .r `  R ) x )  =  ( 1r `  R )  <->  ( y
( .r `  (oppr `  S
) ) x )  =  ( 1r `  R ) )
1211rexbii 2730 . . . . . 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 676 . . . . 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 2436 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
152, 14, 8dvdsr 15751 . . . . 5  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
( x  e.  (
Base `  R )  /\  E. y  e.  (
Base `  R )
( y ( .r
`  R ) x )  =  ( 1r
`  R ) ) )
165, 3opprbas 15734 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (oppr
`  S ) )
17 eqid 2436 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
1816, 17, 6dvdsr 15751 . . . . 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 269 . . . 4  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) )
2019anbi2ci 678 . . 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 2436 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
23 eqid 2436 . . . 4  |-  ( ||r `  S
)  =  ( ||r `  S
)
2421, 22, 14, 1, 23isunit 15762 . . 3  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  S
) ( 1r `  R ) ) )
25 eqid 2436 . . . 4  |-  (Unit `  S )  =  (Unit `  S )
261, 22oppr1 15739 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  S
)
2725, 26, 23, 5, 17isunit 15762 . . 3  |-  ( x  e.  (Unit `  S
)  <->  ( x (
||r `  S ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) ) )
2820, 24, 273bitr4i 269 . 2  |-  ( x  e.  U  <->  x  e.  (Unit `  S ) )
2928eqriv 2433 1  |-  U  =  (Unit `  S )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   .rcmulr 13530   1rcur 15662  opprcoppr 15727   ||rcdsr 15743  Unitcui 15744
This theorem is referenced by:  opprirred  15807  irredlmul  15813  opprdrng  15859  ply1divalg2  20061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mgp 15649  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747
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