Theorem opprsllem 14379

Description: Lemma for opprbasg 14380 and oppraddg 14381. (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 13486	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
2	1	slotex 13379	. . . 4 |- ( R e. V -> ( .r ` R ) e. _V )
3		tposexg 6529	. . . 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 13404	. . 3 |- ( ( R e. V /\ tpos ( .r ` R ) e. _V ) -> ( E ` R ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
9	4, 8	mpdan 425	. 2 |- ( R e. V -> ( E ` R ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
10		eqid 2238	. . . 4 |- ( Base ` R ) = ( Base ` R )
11		eqid 2238	. . . 4 |- ( .r ` R ) = ( .r ` R )
12		opprbas.1	. . . 4 |- O = (oppr ` R )
13	10, 11, 12	opprvalg 14374	. . 3 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) )
14	13	fveq2d 5699	. 2 |- ( R e. V -> ( E ` O ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
15	9, 14	eqtr4d 2274	1 |- ( R e. V -> ( E ` R ) = ( E ` O ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   <.cop 3712   ` cfv 5377  (class class class)co 6085  tpos ctpos 6515   NNcn 9304   ndxcnx 13349   sSet csts 13350  Slot cslot 13351   Basecbs 13352   .rcmulr 13432  opp_rcoppr 14372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-sets 13359  df-mulr 13445  df-oppr 14373

This theorem is used by:  opprbasg  14380  oppraddg  14381