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

Theorem oppr1 15416
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 2283 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2283 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
3 opprbas.1 . . . . . . . . 9  |-  O  =  (oppr
`  R )
4 eqid 2283 . . . . . . . . 9  |-  ( .r
`  O )  =  ( .r `  O
)
51, 2, 3, 4opprmul 15408 . . . . . . . 8  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
65eqeq1i 2290 . . . . . . 7  |-  ( ( x ( .r `  O ) y )  =  y  <->  ( y
( .r `  R
) x )  =  y )
71, 2, 3, 4opprmul 15408 . . . . . . . 8  |-  ( y ( .r `  O
) x )  =  ( x ( .r
`  R ) y )
87eqeq1i 2290 . . . . . . 7  |-  ( ( y ( .r `  O ) x )  =  y  <->  ( x
( .r `  R
) y )  =  y )
96, 8anbi12ci 679 . . . . . 6  |-  ( ( ( x ( .r
`  O ) y )  =  y  /\  ( y ( .r
`  O ) x )  =  y )  <-> 
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) )
109ralbii 2567 . . . . 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 675 . . . 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 5241 . . 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 2283 . . . . 5  |-  (mulGrp `  O )  =  (mulGrp `  O )
143, 1opprbas 15411 . . . . 5  |-  ( Base `  R )  =  (
Base `  O )
1513, 14mgpbas 15331 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  O
) )
1613, 4mgpplusg 15329 . . . 4  |-  ( .r
`  O )  =  ( +g  `  (mulGrp `  O ) )
17 eqid 2283 . . . 4  |-  ( 0g
`  (mulGrp `  O )
)  =  ( 0g
`  (mulGrp `  O )
)
1815, 16, 17grpidval 14384 . . 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 2283 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
2019, 1mgpbas 15331 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
2119, 2mgpplusg 15329 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
22 eqid 2283 . . . 4  |-  ( 0g
`  (mulGrp `  R )
)  =  ( 0g
`  (mulGrp `  R )
)
2320, 21, 22grpidval 14384 . . 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 2313 . 2  |-  ( 0g
`  (mulGrp `  O )
)  =  ( 0g
`  (mulGrp `  R )
)
25 eqid 2283 . . 3  |-  ( 1r
`  O )  =  ( 1r `  O
)
2613, 25rngidval 15343 . 2  |-  ( 1r
`  O )  =  ( 0g `  (mulGrp `  O ) )
27 oppr1.2 . . 3  |-  .1.  =  ( 1r `  R )
2819, 27rngidval 15343 . 2  |-  .1.  =  ( 0g `  (mulGrp `  R ) )
2924, 26, 283eqtr4ri 2314 1  |-  .1.  =  ( 1r `  O )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   iotacio 5217   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400  mulGrpcmgp 15325   1rcur 15339  opprcoppr 15404
This theorem is referenced by:  opprunit  15443  isdrngrd  15538  opprsubrg  15566  srng1  15624  issrngd  15626  fidomndrng  16048  ldual1  28711  lduallmodlem  28715  ldualvsub  28718  lcd1  31172  lcdvsub  31180
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-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
  Copyright terms: Public domain W3C validator