Theorem opprex 14316

Description: Existence of the opposite ring. If you know that R is a ring, see opprring 14322. (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

Step	Hyp	Ref	Expression
1		eqid 2234	. . 3 |- ( Base ` R ) = ( Base ` R )
2		eqid 2234	. . 3 |- ( .r ` R ) = ( .r ` R )
3		opprex.o	. . 3 |- O = (oppr ` R )
4	1, 2, 3	opprvalg 14312	. 2 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) )
5		mulrslid 13429	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
6	5	simpri 113	. . . 4 |- ( .r ` ndx ) e. NN
7	6	a1i 9	. . 3 |- ( R e. V -> ( .r ` ndx ) e. NN )
8	5	slotex 13323	. . . 4 |- ( R e. V -> ( .r ` R ) e. _V )
9		tposexg 6502	. . . 4 |- ( ( .r ` R ) e. _V -> tpos ( .r ` R ) e. _V )
10	8, 9	syl 14	. . 3 |- ( R e. V -> tpos ( .r ` R ) e. _V )
11		setsex 13328	. . 3 |- ( ( R e. V /\ ( .r ` ndx ) e. NN /\ tpos ( .r ` R ) e. _V ) -> ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) e. _V )
12	7, 10, 11	mpd3an23 1376	. 2 |- ( R e. V -> ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) e. _V )
13	4, 12	eqeltrd 2311	1 |- ( R e. V -> O e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   <.cop 3697   ` cfv 5357  (class class class)co 6058  tpos ctpos 6488   NNcn 9254   ndxcnx 13293   sSet csts 13294  Slot cslot 13295   Basecbs 13296   .rcmulr 13375  opp_rcoppr 14310

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-sets 13303  df-mulr 13388  df-oppr 14311

This theorem is referenced by:  opprrngbg  14321  oppr0g  14325  oppr1g  14326  opprnegg  14327  opprsubgg  14328  crngridl  14804