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Theorem opprex 13176
Description: Existence of the opposite ring. If you know that  R is a ring, see opprring 13180. (Contributed by Jim Kingdon, 10-Jan-2025.)
Hypothesis
Ref Expression
opprex.o  |-  O  =  (oppr
`  R )
Assertion
Ref Expression
opprex  |-  ( R  e.  V  ->  O  e.  _V )

Proof of Theorem opprex
StepHypRef Expression
1 eqid 2177 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2177 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 opprex.o . . 3  |-  O  =  (oppr
`  R )
41, 2, 3opprvalg 13172 . 2  |-  ( R  e.  V  ->  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r `  R
) >. ) )
5 mulrslid 12582 . . . . 5  |-  ( .r  = Slot  ( .r `  ndx )  /\  ( .r `  ndx )  e.  NN )
65simpri 113 . . . 4  |-  ( .r
`  ndx )  e.  NN
76a1i 9 . . 3  |-  ( R  e.  V  ->  ( .r `  ndx )  e.  NN )
85slotex 12481 . . . 4  |-  ( R  e.  V  ->  ( .r `  R )  e. 
_V )
9 tposexg 6256 . . . 4  |-  ( ( .r `  R )  e.  _V  -> tpos  ( .r
`  R )  e. 
_V )
108, 9syl 14 . . 3  |-  ( R  e.  V  -> tpos  ( .r
`  R )  e. 
_V )
11 setsex 12486 . . 3  |-  ( ( R  e.  V  /\  ( .r `  ndx )  e.  NN  /\ tpos  ( .r `  R )  e.  _V )  ->  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r `  R
) >. )  e.  _V )
127, 10, 11mpd3an23 1339 . 2  |-  ( R  e.  V  ->  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r
`  R ) >.
)  e.  _V )
134, 12eqeltrd 2254 1  |-  ( R  e.  V  ->  O  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737   <.cop 3595   ` cfv 5215  (class class class)co 5872  tpos ctpos 6242   NNcn 8915   ndxcnx 12451   sSet csts 12452  Slot cslot 12453   Basecbs 12454   .rcmulr 12529  opprcoppr 13170
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 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 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-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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-sets 12461  df-mulr 12542  df-oppr 13171
This theorem is referenced by:  oppr0g  13182  oppr1g  13183  opprnegg  13184
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