Theorem opprsllem 13869

Description: Lemma for opprbasg 13870 and oppraddg 13871. (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 12997	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
2	1	slotex 12892	. . . 4 |- ( R e. V -> ( .r ` R ) e. _V )
3		tposexg 6346	. . . 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 12917	. . 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 2205	. . . 4 |- ( Base ` R ) = ( Base ` R )
11		eqid 2205	. . . 4 |- ( .r ` R ) = ( .r ` R )
12		opprbas.1	. . . 4 |- O = (oppr ` R )
13	10, 11, 12	opprvalg 13864	. . 3 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) )
14	13	fveq2d 5582	. 2 |- ( R e. V -> ( E ` O ) = ( E ` ( R sSet <. ( .r ` ndx ) , tpos ( .r ` R ) >. ) ) )
15	9, 14	eqtr4d 2241	1 |- ( R e. V -> ( E ` R ) = ( E ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   =/= wne 2376   _Vcvv 2772   <.cop 3636   ` cfv 5272  (class class class)co 5946  tpos ctpos 6332   NNcn 9038   ndxcnx 12862   sSet csts 12863  Slot cslot 12864   Basecbs 12865   .rcmulr 12943  opp_rcoppr 13862

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-tpos 6333  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-sets 12872  df-mulr 12956  df-oppr 13863

This theorem is referenced by:  opprbasg  13870  oppraddg  13871