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Theorem opprunit 17111
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 2467 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 17079 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  S )
4 eqid 2467 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
5 eqid 2467 . . . . . . . . . 10  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2467 . . . . . . . . . 10  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
73, 4, 5, 6opprmul 17076 . . . . . . . . 9  |-  ( y ( .r `  (oppr `  S
) ) x )  =  ( x ( .r `  S ) y )
8 eqid 2467 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
92, 8, 1, 4opprmul 17076 . . . . . . . . 9  |-  ( x ( .r `  S
) y )  =  ( y ( .r
`  R ) x )
107, 9eqtr2i 2497 . . . . . . . 8  |-  ( y ( .r `  R
) x )  =  ( y ( .r
`  (oppr
`  S ) ) x )
1110eqeq1i 2474 . . . . . . 7  |-  ( ( y ( .r `  R ) x )  =  ( 1r `  R )  <->  ( y
( .r `  (oppr `  S
) ) x )  =  ( 1r `  R ) )
1211rexbii 2965 . . . . . 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 694 . . . . 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 2467 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
152, 14, 8dvdsr 17096 . . . . 5  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
( x  e.  (
Base `  R )  /\  E. y  e.  (
Base `  R )
( y ( .r
`  R ) x )  =  ( 1r
`  R ) ) )
165, 3opprbas 17079 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (oppr
`  S ) )
17 eqid 2467 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
1816, 17, 6dvdsr 17096 . . . . 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 277 . . . 4  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) )
2019anbi2ci 696 . . 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 2467 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
23 eqid 2467 . . . 4  |-  ( ||r `  S
)  =  ( ||r `  S
)
2421, 22, 14, 1, 23isunit 17107 . . 3  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  S
) ( 1r `  R ) ) )
25 eqid 2467 . . . 4  |-  (Unit `  S )  =  (Unit `  S )
261, 22oppr1 17084 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  S
)
2725, 26, 23, 5, 17isunit 17107 . . 3  |-  ( x  e.  (Unit `  S
)  <->  ( x (
||r `  S ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) ) )
2820, 24, 273bitr4i 277 . 2  |-  ( x  e.  U  <->  x  e.  (Unit `  S ) )
2928eqriv 2463 1  |-  U  =  (Unit `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   .rcmulr 14556   1rcur 16955  opprcoppr 17072   ||rcdsr 17088  Unitcui 17089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-plusg 14568  df-mulr 14569  df-0g 14697  df-mgp 16944  df-ur 16956  df-oppr 17073  df-dvdsr 17091  df-unit 17092
This theorem is referenced by:  opprirred  17152  irredlmul  17158  opprdrng  17220  ply1divalg2  22302
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