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Theorem oppr1g 13064
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
oppr1g  |-  ( R  e.  V  ->  .1.  =  ( 1r `  O ) )

Proof of Theorem oppr1g
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2177 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
3 opprbas.1 . . . . . . . . . . 11  |-  O  =  (oppr
`  R )
4 eqid 2177 . . . . . . . . . . 11  |-  ( .r
`  O )  =  ( .r `  O
)
51, 2, 3, 4opprmulg 13055 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( .r `  O ) y )  =  ( y ( .r `  R ) x ) )
653expa 1203 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  ( x
( .r `  O
) y )  =  ( y ( .r
`  R ) x ) )
76eqeq1d 2186 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  ( (
x ( .r `  O ) y )  =  y  <->  ( y
( .r `  R
) x )  =  y ) )
8 simpll 527 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  R  e.  V )
9 simpr 110 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  y  e.  ( Base `  R )
)
10 simplr 528 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  x  e.  ( Base `  R )
)
111, 2, 3, 4opprmulg 13055 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  (
y ( .r `  O ) x )  =  ( x ( .r `  R ) y ) )
128, 9, 10, 11syl3anc 1238 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  ( y
( .r `  O
) x )  =  ( x ( .r
`  R ) y ) )
1312eqeq1d 2186 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  ( (
y ( .r `  O ) x )  =  y  <->  ( x
( .r `  R
) y )  =  y ) )
147, 13anbi12d 473 . . . . . . 7  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  ( (
( x ( .r
`  O ) y )  =  y  /\  ( y ( .r
`  O ) x )  =  y )  <-> 
( ( y ( .r `  R ) x )  =  y  /\  ( x ( .r `  R ) y )  =  y ) ) )
1514biancomd 271 . . . . . 6  |-  ( ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  /\  y  e.  (
Base `  R )
)  ->  ( (
( x ( .r
`  O ) y )  =  y  /\  ( y ( .r
`  O ) x )  =  y )  <-> 
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) ) )
1615ralbidva 2473 . . . . 5  |-  ( ( R  e.  V  /\  x  e.  ( Base `  R ) )  -> 
( 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 ) ) )
1716riotabidva 5840 . . . 4  |-  ( R  e.  V  ->  ( iota_ x  e.  ( Base `  R ) A. y  e.  ( Base `  R
) ( ( x ( .r `  O
) y )  =  y  /\  ( y ( .r `  O
) x )  =  y ) )  =  ( iota_ x  e.  (
Base `  R ) A. y  e.  ( Base `  R ) ( ( x ( .r
`  R ) y )  =  y  /\  ( y ( .r
`  R ) x )  =  y ) ) )
18 df-riota 5824 . . . 4  |-  ( iota_ 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 `  O ) y )  =  y  /\  (
y ( .r `  O ) x )  =  y ) ) )
19 df-riota 5824 . . . 4  |-  ( iota_ x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) )  =  ( iota x ( x  e.  ( Base `  R )  /\  A. y  e.  ( Base `  R ) ( ( x ( .r `  R ) y )  =  y  /\  (
y ( .r `  R ) x )  =  y ) ) )
2017, 18, 193eqtr3g 2233 . . 3  |-  ( R  e.  V  ->  ( 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 ) ) ) )
213opprex 13057 . . . . 5  |-  ( R  e.  V  ->  O  e.  _V )
22 eqid 2177 . . . . . 6  |-  (mulGrp `  O )  =  (mulGrp `  O )
2322mgpex 12949 . . . . 5  |-  ( O  e.  _V  ->  (mulGrp `  O )  e.  _V )
24 eqid 2177 . . . . . 6  |-  ( Base `  (mulGrp `  O )
)  =  ( Base `  (mulGrp `  O )
)
25 eqid 2177 . . . . . 6  |-  ( +g  `  (mulGrp `  O )
)  =  ( +g  `  (mulGrp `  O )
)
26 eqid 2177 . . . . . 6  |-  ( 0g
`  (mulGrp `  O )
)  =  ( 0g
`  (mulGrp `  O )
)
2724, 25, 26grpidvalg 12671 . . . . 5  |-  ( (mulGrp `  O )  e.  _V  ->  ( 0g `  (mulGrp `  O ) )  =  ( iota x ( x  e.  ( Base `  (mulGrp `  O )
)  /\  A. y  e.  ( Base `  (mulGrp `  O ) ) ( ( x ( +g  `  (mulGrp `  O )
) y )  =  y  /\  ( y ( +g  `  (mulGrp `  O ) ) x )  =  y ) ) ) )
2821, 23, 273syl 17 . . . 4  |-  ( R  e.  V  ->  ( 0g `  (mulGrp `  O
) )  =  ( iota x ( x  e.  ( Base `  (mulGrp `  O ) )  /\  A. y  e.  ( Base `  (mulGrp `  O )
) ( ( x ( +g  `  (mulGrp `  O ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  O )
) x )  =  y ) ) ) )
293, 1opprbasg 13059 . . . . . . . 8  |-  ( R  e.  V  ->  ( Base `  R )  =  ( Base `  O
) )
30 eqid 2177 . . . . . . . . . 10  |-  ( Base `  O )  =  (
Base `  O )
3122, 30mgpbasg 12950 . . . . . . . . 9  |-  ( O  e.  _V  ->  ( Base `  O )  =  ( Base `  (mulGrp `  O ) ) )
3221, 31syl 14 . . . . . . . 8  |-  ( R  e.  V  ->  ( Base `  O )  =  ( Base `  (mulGrp `  O ) ) )
3329, 32eqtrd 2210 . . . . . . 7  |-  ( R  e.  V  ->  ( Base `  R )  =  ( Base `  (mulGrp `  O ) ) )
3433eleq2d 2247 . . . . . 6  |-  ( R  e.  V  ->  (
x  e.  ( Base `  R )  <->  x  e.  ( Base `  (mulGrp `  O
) ) ) )
3522, 4mgpplusgg 12948 . . . . . . . . . . 11  |-  ( O  e.  _V  ->  ( .r `  O )  =  ( +g  `  (mulGrp `  O ) ) )
3621, 35syl 14 . . . . . . . . . 10  |-  ( R  e.  V  ->  ( .r `  O )  =  ( +g  `  (mulGrp `  O ) ) )
3736oveqd 5885 . . . . . . . . 9  |-  ( R  e.  V  ->  (
x ( .r `  O ) y )  =  ( x ( +g  `  (mulGrp `  O ) ) y ) )
3837eqeq1d 2186 . . . . . . . 8  |-  ( R  e.  V  ->  (
( x ( .r
`  O ) y )  =  y  <->  ( x
( +g  `  (mulGrp `  O ) ) y )  =  y ) )
3936oveqd 5885 . . . . . . . . 9  |-  ( R  e.  V  ->  (
y ( .r `  O ) x )  =  ( y ( +g  `  (mulGrp `  O ) ) x ) )
4039eqeq1d 2186 . . . . . . . 8  |-  ( R  e.  V  ->  (
( y ( .r
`  O ) x )  =  y  <->  ( y
( +g  `  (mulGrp `  O ) ) x )  =  y ) )
4138, 40anbi12d 473 . . . . . . 7  |-  ( R  e.  V  ->  (
( ( x ( .r `  O ) y )  =  y  /\  ( y ( .r `  O ) x )  =  y )  <->  ( ( x ( +g  `  (mulGrp `  O ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  O )
) x )  =  y ) ) )
4233, 41raleqbidv 2684 . . . . . 6  |-  ( R  e.  V  ->  ( A. y  e.  ( Base `  R ) ( ( x ( .r
`  O ) y )  =  y  /\  ( y ( .r
`  O ) x )  =  y )  <->  A. y  e.  ( Base `  (mulGrp `  O
) ) ( ( x ( +g  `  (mulGrp `  O ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  O )
) x )  =  y ) ) )
4334, 42anbi12d 473 . . . . 5  |-  ( R  e.  V  ->  (
( x  e.  (
Base `  R )  /\  A. y  e.  (
Base `  R )
( ( x ( .r `  O ) y )  =  y  /\  ( y ( .r `  O ) x )  =  y ) )  <->  ( x  e.  ( Base `  (mulGrp `  O ) )  /\  A. y  e.  ( Base `  (mulGrp `  O )
) ( ( x ( +g  `  (mulGrp `  O ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  O )
) x )  =  y ) ) ) )
4443iotabidv 5194 . . . 4  |-  ( R  e.  V  ->  ( 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 `  (mulGrp `  O
) )  /\  A. y  e.  ( Base `  (mulGrp `  O )
) ( ( x ( +g  `  (mulGrp `  O ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  O )
) x )  =  y ) ) ) )
4528, 44eqtr4d 2213 . . 3  |-  ( R  e.  V  ->  ( 0g `  (mulGrp `  O
) )  =  ( iota x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  O
) y )  =  y  /\  ( y ( .r `  O
) x )  =  y ) ) ) )
46 eqid 2177 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
4746mgpex 12949 . . . . 5  |-  ( R  e.  V  ->  (mulGrp `  R )  e.  _V )
48 eqid 2177 . . . . . 6  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
49 eqid 2177 . . . . . 6  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
50 eqid 2177 . . . . . 6  |-  ( 0g
`  (mulGrp `  R )
)  =  ( 0g
`  (mulGrp `  R )
)
5148, 49, 50grpidvalg 12671 . . . . 5  |-  ( (mulGrp `  R )  e.  _V  ->  ( 0g `  (mulGrp `  R ) )  =  ( iota x ( x  e.  ( Base `  (mulGrp `  R )
)  /\  A. y  e.  ( Base `  (mulGrp `  R ) ) ( ( x ( +g  `  (mulGrp `  R )
) y )  =  y  /\  ( y ( +g  `  (mulGrp `  R ) ) x )  =  y ) ) ) )
5247, 51syl 14 . . . 4  |-  ( R  e.  V  ->  ( 0g `  (mulGrp `  R
) )  =  ( iota x ( x  e.  ( Base `  (mulGrp `  R ) )  /\  A. y  e.  ( Base `  (mulGrp `  R )
) ( ( x ( +g  `  (mulGrp `  R ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  R )
) x )  =  y ) ) ) )
5346, 1mgpbasg 12950 . . . . . . 7  |-  ( R  e.  V  ->  ( Base `  R )  =  ( Base `  (mulGrp `  R ) ) )
5453eleq2d 2247 . . . . . 6  |-  ( R  e.  V  ->  (
x  e.  ( Base `  R )  <->  x  e.  ( Base `  (mulGrp `  R
) ) ) )
5546, 2mgpplusgg 12948 . . . . . . . . . 10  |-  ( R  e.  V  ->  ( .r `  R )  =  ( +g  `  (mulGrp `  R ) ) )
5655oveqd 5885 . . . . . . . . 9  |-  ( R  e.  V  ->  (
x ( .r `  R ) y )  =  ( x ( +g  `  (mulGrp `  R ) ) y ) )
5756eqeq1d 2186 . . . . . . . 8  |-  ( R  e.  V  ->  (
( x ( .r
`  R ) y )  =  y  <->  ( x
( +g  `  (mulGrp `  R ) ) y )  =  y ) )
5855oveqd 5885 . . . . . . . . 9  |-  ( R  e.  V  ->  (
y ( .r `  R ) x )  =  ( y ( +g  `  (mulGrp `  R ) ) x ) )
5958eqeq1d 2186 . . . . . . . 8  |-  ( R  e.  V  ->  (
( y ( .r
`  R ) x )  =  y  <->  ( y
( +g  `  (mulGrp `  R ) ) x )  =  y ) )
6057, 59anbi12d 473 . . . . . . 7  |-  ( R  e.  V  ->  (
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y )  <->  ( ( x ( +g  `  (mulGrp `  R ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  R )
) x )  =  y ) ) )
6153, 60raleqbidv 2684 . . . . . 6  |-  ( R  e.  V  ->  ( A. y  e.  ( Base `  R ) ( ( x ( .r
`  R ) y )  =  y  /\  ( y ( .r
`  R ) x )  =  y )  <->  A. y  e.  ( Base `  (mulGrp `  R
) ) ( ( x ( +g  `  (mulGrp `  R ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  R )
) x )  =  y ) ) )
6254, 61anbi12d 473 . . . . 5  |-  ( R  e.  V  ->  (
( x  e.  (
Base `  R )  /\  A. y  e.  (
Base `  R )
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) )  <->  ( x  e.  ( Base `  (mulGrp `  R ) )  /\  A. y  e.  ( Base `  (mulGrp `  R )
) ( ( x ( +g  `  (mulGrp `  R ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  R )
) x )  =  y ) ) ) )
6362iotabidv 5194 . . . 4  |-  ( R  e.  V  ->  ( iota x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) ) )  =  ( iota x
( x  e.  (
Base `  (mulGrp `  R
) )  /\  A. y  e.  ( Base `  (mulGrp `  R )
) ( ( x ( +g  `  (mulGrp `  R ) ) y )  =  y  /\  ( y ( +g  `  (mulGrp `  R )
) x )  =  y ) ) ) )
6452, 63eqtr4d 2213 . . 3  |-  ( R  e.  V  ->  ( 0g `  (mulGrp `  R
) )  =  ( iota x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) ) ) )
6520, 45, 643eqtr4d 2220 . 2  |-  ( R  e.  V  ->  ( 0g `  (mulGrp `  O
) )  =  ( 0g `  (mulGrp `  R ) ) )
66 eqid 2177 . . . 4  |-  ( 1r
`  O )  =  ( 1r `  O
)
6722, 66ringidvalg 12957 . . 3  |-  ( O  e.  _V  ->  ( 1r `  O )  =  ( 0g `  (mulGrp `  O ) ) )
6821, 67syl 14 . 2  |-  ( R  e.  V  ->  ( 1r `  O )  =  ( 0g `  (mulGrp `  O ) ) )
69 oppr1.2 . . 3  |-  .1.  =  ( 1r `  R )
7046, 69ringidvalg 12957 . 2  |-  ( R  e.  V  ->  .1.  =  ( 0g `  (mulGrp `  R ) ) )
7165, 68, 703eqtr4rd 2221 1  |-  ( R  e.  V  ->  .1.  =  ( 1r `  O ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737   iotacio 5171   ` cfv 5211   iota_crio 5823  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   .rcmulr 12506   0gc0g 12640  mulGrpcmgp 12944   1rcur 12955  opprcoppr 13051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-tpos 6239  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgp 12945  df-ur 12956  df-oppr 13052
This theorem is referenced by:  opprunitd  13091
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