MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opprunit Unicode version

Theorem opprunit 15443
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 2283 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 15411 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  S )
4 eqid 2283 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
5 eqid 2283 . . . . . . . . . 10  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2283 . . . . . . . . . 10  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
73, 4, 5, 6opprmul 15408 . . . . . . . . 9  |-  ( y ( .r `  (oppr `  S
) ) x )  =  ( x ( .r `  S ) y )
8 eqid 2283 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
92, 8, 1, 4opprmul 15408 . . . . . . . . 9  |-  ( x ( .r `  S
) y )  =  ( y ( .r
`  R ) x )
107, 9eqtr2i 2304 . . . . . . . 8  |-  ( y ( .r `  R
) x )  =  ( y ( .r
`  (oppr
`  S ) ) x )
1110eqeq1i 2290 . . . . . . 7  |-  ( ( y ( .r `  R ) x )  =  ( 1r `  R )  <->  ( y
( .r `  (oppr `  S
) ) x )  =  ( 1r `  R ) )
1211rexbii 2568 . . . . . 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 675 . . . . 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 2283 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
152, 14, 8dvdsr 15428 . . . . 5  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
( x  e.  (
Base `  R )  /\  E. y  e.  (
Base `  R )
( y ( .r
`  R ) x )  =  ( 1r
`  R ) ) )
165, 3opprbas 15411 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (oppr
`  S ) )
17 eqid 2283 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
1816, 17, 6dvdsr 15428 . . . . 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 268 . . . 4  |-  ( x ( ||r `
 R ) ( 1r `  R )  <-> 
x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) )
2019anbi2ci 677 . . 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 2283 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
23 eqid 2283 . . . 4  |-  ( ||r `  S
)  =  ( ||r `  S
)
2421, 22, 14, 1, 23isunit 15439 . . 3  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  S
) ( 1r `  R ) ) )
25 eqid 2283 . . . 4  |-  (Unit `  S )  =  (Unit `  S )
261, 22oppr1 15416 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  S
)
2725, 26, 23, 5, 17isunit 15439 . . 3  |-  ( x  e.  (Unit `  S
)  <->  ( x (
||r `  S ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  S ) ) ( 1r `  R
) ) )
2820, 24, 273bitr4i 268 . 2  |-  ( x  e.  U  <->  x  e.  (Unit `  S ) )
2928eqriv 2280 1  |-  U  =  (Unit `  S )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   1rcur 15339  opprcoppr 15404   ||rcdsr 15420  Unitcui 15421
This theorem is referenced by:  opprirred  15484  irredlmul  15490  opprdrng  15536  ply1divalg2  19524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mgp 15326  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424
  Copyright terms: Public domain W3C validator