Theorem opprsllem 14086

Description: Lemma for opprbasg 14087 and oppraddg 14088. (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 13214	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
2	1	slotex 13108	. . . 4 |- ( R e. V -> ( .r ` R ) e. _V )
3		tposexg 6423	. . . 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 13133	. . 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 2231	. . . 4 |- ( Base ` R ) = ( Base ` R )
11		eqid 2231	. . . 4 |- ( .r ` R ) = ( .r ` R )
12		opprbas.1	. . . 4 |- O = (oppr ` R )
13	10, 11, 12	opprvalg 14081	. . 3 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) )
14	13	fveq2d 5643	. 2 |- ( R e. V -> ( E ` O ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
15	9, 14	eqtr4d 2267	1 |- ( R e. V -> ( E ` R ) = ( E ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   =/= wne 2402   _Vcvv 2802   <.cop 3672   ` cfv 5326  (class class class)co 6017  tpos ctpos 6409   NNcn 9142   ndxcnx 13078   sSet csts 13079  Slot cslot 13080   Basecbs 13081   .rcmulr 13160  opp_rcoppr 14079

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-sets 13088  df-mulr 13173  df-oppr 14080

This theorem is referenced by:  opprbasg  14087  oppraddg  14088