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Theorem opprlem 15660
Description: Lemma for opprbas 15661 and oppradd 15662. (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 13417 . . 3  |-  E  = Slot  ( E `  ndx )
42nnrei 9941 . . . . 5  |-  N  e.  RR
5 opprlem.4 . . . . 5  |-  N  <  3
64, 5ltneii 9117 . . . 4  |-  N  =/=  3
71, 2ndxarg 13416 . . . . 5  |-  ( E `
 ndx )  =  N
8 mulrndx 13501 . . . . 5  |-  ( .r
`  ndx )  =  3
97, 8neeq12i 2562 . . . 4  |-  ( ( E `  ndx )  =/=  ( .r `  ndx ) 
<->  N  =/=  3 )
106, 9mpbir 201 . . 3  |-  ( E `
 ndx )  =/=  ( .r `  ndx )
113, 10setsnid 13436 . 2  |-  ( E `
 R )  =  ( E `  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r
`  R ) >.
) )
12 eqid 2387 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2387 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
14 opprbas.1 . . . 4  |-  O  =  (oppr
`  R )
1512, 13, 14opprval 15656 . . 3  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r `  R
) >. )
1615fveq2i 5671 . 2  |-  ( E `
 O )  =  ( E `  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r
`  R ) >.
) )
1711, 16eqtr4i 2410 1  |-  ( E `
 R )  =  ( E `  O
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717    =/= wne 2550   <.cop 3760   class class class wbr 4153   ` cfv 5394  (class class class)co 6020  tpos ctpos 6414    < clt 9053   NNcn 9932   3c3 9982   ndxcnx 13393   sSet csts 13394  Slot cslot 13395   Basecbs 13396   .rcmulr 13457  opprcoppr 15654
This theorem is referenced by:  opprbas  15661  oppradd  15662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-i2m1 8991  ax-1ne0 8992  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-tpos 6415  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-nn 9933  df-2 9990  df-3 9991  df-ndx 13399  df-slot 13400  df-sets 13402  df-mulr 13470  df-oppr 15655
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