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Theorem shftfn 11658
Description: Functionality and domain of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
shftfn  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } )
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem shftfn
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4891 . . . . 5  |-  Rel  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
21a1i 10 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Rel  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
3 fnfun 5420 . . . . . 6  |-  ( F  Fn  B  ->  Fun  F )
43adantr 451 . . . . 5  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  F )
5 funmo 5350 . . . . . . 7  |-  ( Fun 
F  ->  E* w
( z  -  A
) F w )
6 vex 2867 . . . . . . . . . 10  |-  z  e. 
_V
7 vex 2867 . . . . . . . . . 10  |-  w  e. 
_V
8 eleq1 2418 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  e.  CC  <->  z  e.  CC ) )
9 oveq1 5949 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
x  -  A )  =  ( z  -  A ) )
109breq1d 4112 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( x  -  A
) F y  <->  ( z  -  A ) F y ) )
118, 10anbi12d 691 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F y ) ) )
12 breq2 4106 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
( z  -  A
) F y  <->  ( z  -  A ) F w ) )
1312anbi2d 684 . . . . . . . . . 10  |-  ( y  =  w  ->  (
( z  e.  CC  /\  ( z  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F w ) ) )
14 eqid 2358 . . . . . . . . . 10  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
156, 7, 11, 13, 14brab 4366 . . . . . . . . 9  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  <->  ( z  e.  CC  /\  ( z  -  A
) F w ) )
1615simprbi 450 . . . . . . . 8  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  ->  ( z  -  A
) F w )
1716moimi 2256 . . . . . . 7  |-  ( E* w ( z  -  A ) F w  ->  E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
185, 17syl 15 . . . . . 6  |-  ( Fun 
F  ->  E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
1918alrimiv 1631 . . . . 5  |-  ( Fun 
F  ->  A. z E* w  z { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
204, 19syl 15 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  A. z E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
21 dffun6 5349 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  <->  ( Rel  {
<. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  /\  A. z E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
) )
222, 20, 21sylanbrc 645 . . 3  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
23 shftfval.1 . . . . . 6  |-  F  e. 
_V
2423shftfval 11655 . . . . 5  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
2524adantl 452 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
2625funeqd 5355 . . 3  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( Fun  ( F 
shift  A )  <->  Fun  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } ) )
2722, 26mpbird 223 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  ( F  shift  A ) )
2823shftdm 11656 . . 3  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
29 fndm 5422 . . . . 5  |-  ( F  Fn  B  ->  dom  F  =  B )
3029eleq2d 2425 . . . 4  |-  ( F  Fn  B  ->  (
( x  -  A
)  e.  dom  F  <->  ( x  -  A )  e.  B ) )
3130rabbidv 2856 . . 3  |-  ( F  Fn  B  ->  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
3228, 31sylan9eqr 2412 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
33 df-fn 5337 . 2  |-  ( ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } 
<->  ( Fun  ( F 
shift  A )  /\  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } ) )
3427, 32, 33sylanbrc 645 1  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1540    = wceq 1642    e. wcel 1710   E*wmo 2210   {crab 2623   _Vcvv 2864   class class class wbr 4102   {copab 4155   dom cdm 4768   Rel wrel 4773   Fun wfun 5328    Fn wfn 5329  (class class class)co 5942   CCcc 8822    - cmin 9124    shift cshi 11651
This theorem is referenced by:  shftf  11664  seqshft  11670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-ltxr 8959  df-sub 9126  df-shft 11652
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