MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oppr1 Structured version   Unicode version

Theorem oppr1 16852
Description: Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprbas.1  |-  O  =  (oppr
`  R )
oppr1.2  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
oppr1  |-  .1.  =  ( 1r `  O )

Proof of Theorem oppr1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2454 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
3 opprbas.1 . . . . . . . . 9  |-  O  =  (oppr
`  R )
4 eqid 2454 . . . . . . . . 9  |-  ( .r
`  O )  =  ( .r `  O
)
51, 2, 3, 4opprmul 16844 . . . . . . . 8  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
65eqeq1i 2461 . . . . . . 7  |-  ( ( x ( .r `  O ) y )  =  y  <->  ( y
( .r `  R
) x )  =  y )
71, 2, 3, 4opprmul 16844 . . . . . . . 8  |-  ( y ( .r `  O
) x )  =  ( x ( .r
`  R ) y )
87eqeq1i 2461 . . . . . . 7  |-  ( ( y ( .r `  O ) x )  =  y  <->  ( x
( .r `  R
) y )  =  y )
96, 8anbi12ci 698 . . . . . 6  |-  ( ( ( x ( .r
`  O ) y )  =  y  /\  ( y ( .r
`  O ) x )  =  y )  <-> 
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) )
109ralbii 2839 . . . . 5  |-  ( A. y  e.  ( Base `  R ) ( ( x ( .r `  O ) y )  =  y  /\  (
y ( .r `  O ) x )  =  y )  <->  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) )
1110anbi2i 694 . . . 4  |-  ( ( x  e.  ( Base `  R )  /\  A. y  e.  ( Base `  R ) ( ( x ( .r `  O ) y )  =  y  /\  (
y ( .r `  O ) x )  =  y ) )  <-> 
( x  e.  (
Base `  R )  /\  A. y  e.  (
Base `  R )
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) ) )
1211iotabii 5514 . . 3  |-  ( iota
x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  O
) y )  =  y  /\  ( y ( .r `  O
) x )  =  y ) ) )  =  ( iota x
( x  e.  (
Base `  R )  /\  A. y  e.  (
Base `  R )
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) ) )
13 eqid 2454 . . . . 5  |-  (mulGrp `  O )  =  (mulGrp `  O )
143, 1opprbas 16847 . . . . 5  |-  ( Base `  R )  =  (
Base `  O )
1513, 14mgpbas 16722 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  O
) )
1613, 4mgpplusg 16720 . . . 4  |-  ( .r
`  O )  =  ( +g  `  (mulGrp `  O ) )
17 eqid 2454 . . . 4  |-  ( 0g
`  (mulGrp `  O )
)  =  ( 0g
`  (mulGrp `  O )
)
1815, 16, 17grpidval 15554 . . 3  |-  ( 0g
`  (mulGrp `  O )
)  =  ( iota
x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  O
) y )  =  y  /\  ( y ( .r `  O
) x )  =  y ) ) )
19 eqid 2454 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
2019, 1mgpbas 16722 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
2119, 2mgpplusg 16720 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
22 eqid 2454 . . . 4  |-  ( 0g
`  (mulGrp `  R )
)  =  ( 0g
`  (mulGrp `  R )
)
2320, 21, 22grpidval 15554 . . 3  |-  ( 0g
`  (mulGrp `  R )
)  =  ( iota
x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) ) )
2412, 18, 233eqtr4i 2493 . 2  |-  ( 0g
`  (mulGrp `  O )
)  =  ( 0g
`  (mulGrp `  R )
)
25 eqid 2454 . . 3  |-  ( 1r
`  O )  =  ( 1r `  O
)
2613, 25rngidval 16730 . 2  |-  ( 1r
`  O )  =  ( 0g `  (mulGrp `  O ) )
27 oppr1.2 . . 3  |-  .1.  =  ( 1r `  R )
2819, 27rngidval 16730 . 2  |-  .1.  =  ( 0g `  (mulGrp `  R ) )
2924, 26, 283eqtr4ri 2494 1  |-  .1.  =  ( 1r `  O )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   iotacio 5490   ` cfv 5529  (class class class)co 6203   Basecbs 14295   .rcmulr 14361   0gc0g 14500  mulGrpcmgp 16716   1rcur 16728  opprcoppr 16840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-tpos 6858  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-plusg 14373  df-mulr 14374  df-0g 14502  df-mgp 16717  df-ur 16729  df-oppr 16841
This theorem is referenced by:  opprunit  16879  isdrngrd  16984  opprsubrg  17012  srng1  17070  issrngd  17072  fidomndrng  17505  rhmopp  26452  ldual1  33151  lduallmodlem  33155  ldualvsub  33158  lcd1  35612  lcdvsub  35620
  Copyright terms: Public domain W3C validator