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Theorem shftval 10865
Description: Value of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
shftval  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) `  B )  =  ( F `  ( B  -  A
) ) )

Proof of Theorem shftval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 shftfval.1 . . . . 5  |-  F  e. 
_V
21shftfib 10863 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  ( F " { ( B  -  A ) } ) )
32eleq2d 2259 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( x  e.  ( ( F  shift  A )
" { B }
)  <->  x  e.  ( F " { ( B  -  A ) } ) ) )
43iotabidv 5218 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( iota x x  e.  ( ( F 
shift  A ) " { B } ) )  =  ( iota x x  e.  ( F " { ( B  -  A ) } ) ) )
5 simpr 110 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
6 dffv3g 5530 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A ) `
 B )  =  ( iota x x  e.  ( ( F 
shift  A ) " { B } ) ) )
75, 6syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) `  B )  =  ( iota x x  e.  ( ( F  shift  A ) " { B } ) ) )
8 simpl 109 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
95, 8subcld 8297 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  -  A
)  e.  CC )
10 dffv3g 5530 . . 3  |-  ( ( B  -  A )  e.  CC  ->  ( F `  ( B  -  A ) )  =  ( iota x x  e.  ( F " { ( B  -  A ) } ) ) )
119, 10syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( F `  ( B  -  A )
)  =  ( iota
x x  e.  ( F " { ( B  -  A ) } ) ) )
124, 7, 113eqtr4d 2232 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) `  B )  =  ( F `  ( B  -  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   {csn 3607   "cima 4647   iotacio 5194   ` cfv 5235  (class class class)co 5895   CCcc 7838    - cmin 8157    shift cshi 10854
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-resscn 7932  ax-1cn 7933  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-distr 7944  ax-i2m1 7945  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-sub 8159  df-shft 10855
This theorem is referenced by:  shftval2  10866  shftval4  10868  shftval5  10869  shftf  10870  shftvalg  10876  isumshft  11529
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