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Theorem shftfval 12862
Description: The value of the sequence shifter operation is a function on 
CC.  A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
shftfval  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
Distinct variable groups:    x, y, A    x, F, y

Proof of Theorem shftfval
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6307 . . . . . . . . . 10  |-  ( x  -  A )  e. 
_V
2 vex 3116 . . . . . . . . . 10  |-  y  e. 
_V
31, 2breldm 5205 . . . . . . . . 9  |-  ( ( x  -  A ) F y  ->  (
x  -  A )  e.  dom  F )
4 npcan 9825 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
54eqcomd 2475 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
65ancoms 453 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
7 oveq1 6289 . . . . . . . . . . 11  |-  ( w  =  ( x  -  A )  ->  (
w  +  A )  =  ( ( x  -  A )  +  A ) )
87eqeq2d 2481 . . . . . . . . . 10  |-  ( w  =  ( x  -  A )  ->  (
x  =  ( w  +  A )  <->  x  =  ( ( x  -  A )  +  A
) ) )
98rspcev 3214 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  dom  F  /\  x  =  (
( x  -  A
)  +  A ) )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
103, 6, 9syl2anr 478 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
11 vex 3116 . . . . . . . . 9  |-  x  e. 
_V
12 eqeq1 2471 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  ( w  +  A )  <->  x  =  ( w  +  A
) ) )
1312rexbidv 2973 . . . . . . . . 9  |-  ( z  =  x  ->  ( E. w  e.  dom  F  z  =  ( w  +  A )  <->  E. w  e.  dom  F  x  =  ( w  +  A
) ) )
1411, 13elab 3250 . . . . . . . 8  |-  ( x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  <->  E. w  e.  dom  F  x  =  ( w  +  A ) )
1510, 14sylibr 212 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) } )
161, 2brelrn 5231 . . . . . . . 8  |-  ( ( x  -  A ) F y  ->  y  e.  ran  F )
1716adantl 466 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  y  e.  ran  F )
1815, 17jca 532 . . . . . 6  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  /\  y  e.  ran  F ) )
1918expl 618 . . . . 5  |-  ( A  e.  CC  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  ->  ( x  e. 
{ z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  /\  y  e.  ran  F ) ) )
2019ssopab2dv 4776 . . . 4  |-  ( A  e.  CC  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  C_  { <. x ,  y >.  |  ( x  e.  { z  |  E. w  e. 
dom  F  z  =  ( w  +  A
) }  /\  y  e.  ran  F ) } )
21 df-xp 5005 . . . 4  |-  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) }  X.  ran  F )  =  { <. x ,  y >.  |  ( x  e.  { z  |  E. w  e. 
dom  F  z  =  ( w  +  A
) }  /\  y  e.  ran  F ) }
2220, 21syl6sseqr 3551 . . 3  |-  ( A  e.  CC  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  C_  ( {
z  |  E. w  e.  dom  F  z  =  ( w  +  A
) }  X.  ran  F ) )
23 shftfval.1 . . . . . 6  |-  F  e. 
_V
2423dmex 6714 . . . . 5  |-  dom  F  e.  _V
2524abrexex 6755 . . . 4  |-  { z  |  E. w  e. 
dom  F  z  =  ( w  +  A
) }  e.  _V
2623rnex 6715 . . . 4  |-  ran  F  e.  _V
2725, 26xpex 6711 . . 3  |-  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) }  X.  ran  F )  e.  _V
28 ssexg 4593 . . 3  |-  ( ( { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  C_  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  X.  ran  F )  /\  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  X.  ran  F )  e.  _V )  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  e.  _V )
2922, 27, 28sylancl 662 . 2  |-  ( A  e.  CC  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  e.  _V )
30 breq 4449 . . . . . 6  |-  ( z  =  F  ->  (
( x  -  w
) z y  <->  ( x  -  w ) F y ) )
3130anbi2d 703 . . . . 5  |-  ( z  =  F  ->  (
( x  e.  CC  /\  ( x  -  w
) z y )  <-> 
( x  e.  CC  /\  ( x  -  w
) F y ) ) )
3231opabbidv 4510 . . . 4  |-  ( z  =  F  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) } )
33 oveq2 6290 . . . . . . 7  |-  ( w  =  A  ->  (
x  -  w )  =  ( x  -  A ) )
3433breq1d 4457 . . . . . 6  |-  ( w  =  A  ->  (
( x  -  w
) F y  <->  ( x  -  A ) F y ) )
3534anbi2d 703 . . . . 5  |-  ( w  =  A  ->  (
( x  e.  CC  /\  ( x  -  w
) F y )  <-> 
( x  e.  CC  /\  ( x  -  A
) F y ) ) )
3635opabbidv 4510 . . . 4  |-  ( w  =  A  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
37 df-shft 12859 . . . 4  |-  shift  =  ( z  e.  _V ,  w  e.  CC  |->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) } )
3832, 36, 37ovmpt2g 6419 . . 3  |-  ( ( F  e.  _V  /\  A  e.  CC  /\  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  e.  _V )  ->  ( F 
shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
3923, 38mp3an1 1311 . 2  |-  ( ( A  e.  CC  /\  {
<. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  e.  _V )  ->  ( F 
shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
4029, 39mpdan 668 1  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   {copab 4504    X. cxp 4997   dom cdm 4999   ran crn 5000  (class class class)co 6282   CCcc 9486    + caddc 9491    - cmin 9801    shift cshi 12858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629  df-sub 9803  df-shft 12859
This theorem is referenced by:  shftdm  12863  shftfib  12864  shftfn  12865  2shfti  12872  shftidt2  12873
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