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Theorem opprlem 17151
Description: Lemma for opprbas 17152 and oppradd 17153. (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 14530 . . 3  |-  E  = Slot  ( E `  ndx )
42nnrei 10551 . . . . 5  |-  N  e.  RR
5 opprlem.4 . . . . 5  |-  N  <  3
64, 5ltneii 9700 . . . 4  |-  N  =/=  3
71, 2ndxarg 14529 . . . . 5  |-  ( E `
 ndx )  =  N
8 mulrndx 14619 . . . . 5  |-  ( .r
`  ndx )  =  3
97, 8neeq12i 2732 . . . 4  |-  ( ( E `  ndx )  =/=  ( .r `  ndx ) 
<->  N  =/=  3 )
106, 9mpbir 209 . . 3  |-  ( E `
 ndx )  =/=  ( .r `  ndx )
113, 10setsnid 14551 . 2  |-  ( E `
 R )  =  ( E `  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r
`  R ) >.
) )
12 eqid 2443 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2443 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
14 opprbas.1 . . . 4  |-  O  =  (oppr
`  R )
1512, 13, 14opprval 17147 . . 3  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r `  R
) >. )
1615fveq2i 5859 . 2  |-  ( E `
 O )  =  ( E `  ( R sSet  <. ( .r `  ndx ) , tpos  ( .r
`  R ) >.
) )
1711, 16eqtr4i 2475 1  |-  ( E `
 R )  =  ( E `  O
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    e. wcel 1804    =/= wne 2638   <.cop 4020   class class class wbr 4437   ` cfv 5578  (class class class)co 6281  tpos ctpos 6956    < clt 9631   NNcn 10542   3c3 10592   ndxcnx 14506   sSet csts 14507  Slot cslot 14508   Basecbs 14509   .rcmulr 14575  opprcoppr 17145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-i2m1 9563  ax-1ne0 9564  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636  df-nn 10543  df-2 10600  df-3 10601  df-ndx 14512  df-slot 14513  df-sets 14515  df-mulr 14588  df-oppr 17146
This theorem is referenced by:  opprbas  17152  oppradd  17153
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