Theorem opprsllem 14037

Description: Lemma for opprbasg 14038 and oppraddg 14039. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)

Hypotheses

Ref	Expression
opprbas.1	|- O = (oppr ` R )
opprsllem.2	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
opprlem.3	|- ( E ` ndx ) =/= ( .r ` ndx )

Assertion

Ref	Expression
opprsllem	|- ( R e. V -> ( E ` R ) = ( E ` O ) )

Proof of Theorem opprsllem

Step	Hyp	Ref	Expression
1		mulrslid 13165	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
2	1	slotex 13059	. . . 4 |- ( R e. V -> ( .r ` R ) e. _V )
3		tposexg 6404	. . . 4 |- ( ( .r ` R ) e. _V -> tpos ( .r ` R ) e. _V )
4	2, 3	syl 14	. . 3 |- ( R e. V -> tpos ( .r ` R ) e. _V )
5		opprsllem.2	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
6		opprlem.3	. . . 4 |- ( E ` ndx ) =/= ( .r ` ndx )
7	1	simpri 113	. . . 4 |- ( .r ` ndx ) e. NN
8	5, 6, 7	setsslnid 13084	. . 3 |- ( ( R e. V /\ tpos ( .r ` R ) e. _V ) -> ( E ` R ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
9	4, 8	mpdan 421	. 2 |- ( R e. V -> ( E ` R ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
10		eqid 2229	. . . 4 |- ( Base ` R ) = ( Base ` R )
11		eqid 2229	. . . 4 |- ( .r ` R ) = ( .r ` R )
12		opprbas.1	. . . 4 |- O = (oppr ` R )
13	10, 11, 12	opprvalg 14032	. . 3 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) )
14	13	fveq2d 5631	. 2 |- ( R e. V -> ( E ` O ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
15	9, 14	eqtr4d 2265	1 |- ( R e. V -> ( E ` R ) = ( E ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   <.cop 3669   ` cfv 5318  (class class class)co 6001  tpos ctpos 6390   NNcn 9110   ndxcnx 13029   sSet csts 13030  Slot cslot 13031   Basecbs 13032   .rcmulr 13111  opp_rcoppr 14030

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-tpos 6391  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-sets 13039  df-mulr 13124  df-oppr 14031

This theorem is referenced by:  opprbasg  14038  oppraddg  14039