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Theorem opprlem 15725
Description: Lemma for opprbas 15726 and oppradd 15727. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprbas.1  |-  O  =  (oppr
`  R )
opprlem.2  |-  E  = Slot 
N
opprlem.3  |-  N  e.  NN
opprlem.4  |-  N  <  3
Assertion
Ref Expression
opprlem  |-  ( E `
 R )  =  ( E `  O
)

Proof of Theorem opprlem
StepHypRef Expression
1 opprlem.2 . . . 4  |-  E  = Slot 
N
2 opprlem.3 . . . 4  |-  N  e.  NN
31, 2ndxid 13482 . . 3  |-  E  = Slot  ( E `  ndx )
42nnrei 10001 . . . . 5  |-  N  e.  RR
5 opprlem.4 . . . . 5  |-  N  <  3
64, 5ltneii 9178 . . . 4  |-  N  =/=  3
71, 2ndxarg 13481 . . . . 5  |-  ( E `
 ndx )  =  N
8 mulrndx 13566 . . . . 5  |-  ( .r
`  ndx )  =  3
97, 8neeq12i 2610 . . . 4  |-  ( ( E `  ndx )  =/=  ( .r `  ndx ) 
<->  N  =/=  3 )
106, 9mpbir 201 . . 3  |-  ( E `
 ndx )  =/=  ( .r `  ndx )
113, 10setsnid 13501 . 2  |-  ( E `
 R )  =  ( E `  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r
`  R ) >.
) )
12 eqid 2435 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2435 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
14 opprbas.1 . . . 4  |-  O  =  (oppr
`  R )
1512, 13, 14opprval 15721 . . 3  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r `  R
) >. )
1615fveq2i 5723 . 2  |-  ( E `
 O )  =  ( E `  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r
`  R ) >.
) )
1711, 16eqtr4i 2458 1  |-  ( E `
 R )  =  ( E `  O
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    =/= wne 2598   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073  tpos ctpos 6470    < clt 9112   NNcn 9992   3c3 10042   ndxcnx 13458   sSet csts 13459  Slot cslot 13460   Basecbs 13461   .rcmulr 13522  opprcoppr 15719
This theorem is referenced by:  opprbas  15726  oppradd  15727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-tpos 6471  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-sets 13467  df-mulr 13535  df-oppr 15720
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