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Theorem shftfval 13052
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 6306 . . . . . . . . . 10  |-  ( x  -  A )  e. 
_V
2 vex 3062 . . . . . . . . . 10  |-  y  e. 
_V
31, 2breldm 5028 . . . . . . . . 9  |-  ( ( x  -  A ) F y  ->  (
x  -  A )  e.  dom  F )
4 npcan 9865 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
54eqcomd 2410 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
65ancoms 451 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
7 oveq1 6285 . . . . . . . . . . 11  |-  ( w  =  ( x  -  A )  ->  (
w  +  A )  =  ( ( x  -  A )  +  A ) )
87eqeq2d 2416 . . . . . . . . . 10  |-  ( w  =  ( x  -  A )  ->  (
x  =  ( w  +  A )  <->  x  =  ( ( x  -  A )  +  A
) ) )
98rspcev 3160 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  dom  F  /\  x  =  (
( x  -  A
)  +  A ) )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
103, 6, 9syl2anr 476 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
11 vex 3062 . . . . . . . . 9  |-  x  e. 
_V
12 eqeq1 2406 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  ( w  +  A )  <->  x  =  ( w  +  A
) ) )
1312rexbidv 2918 . . . . . . . . 9  |-  ( z  =  x  ->  ( E. w  e.  dom  F  z  =  ( w  +  A )  <->  E. w  e.  dom  F  x  =  ( w  +  A
) ) )
1411, 13elab 3196 . . . . . . . 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 5054 . . . . . . . 8  |-  ( ( x  -  A ) F y  ->  y  e.  ran  F )
1716adantl 464 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  y  e.  ran  F )
1815, 17jca 530 . . . . . 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 616 . . . . 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 4719 . . . 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 4829 . . . 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 3489 . . 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 6717 . . . . 5  |-  dom  F  e.  _V
2524abrexex 6758 . . . 4  |-  { z  |  E. w  e. 
dom  F  z  =  ( w  +  A
) }  e.  _V
2623rnex 6718 . . . 4  |-  ran  F  e.  _V
2725, 26xpex 6586 . . 3  |-  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) }  X.  ran  F )  e.  _V
28 ssexg 4540 . . 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 660 . 2  |-  ( A  e.  CC  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  e.  _V )
30 breq 4397 . . . . . 6  |-  ( z  =  F  ->  (
( x  -  w
) z y  <->  ( x  -  w ) F y ) )
3130anbi2d 702 . . . . 5  |-  ( z  =  F  ->  (
( x  e.  CC  /\  ( x  -  w
) z y )  <-> 
( x  e.  CC  /\  ( x  -  w
) F y ) ) )
3231opabbidv 4458 . . . 4  |-  ( z  =  F  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) } )
33 oveq2 6286 . . . . . . 7  |-  ( w  =  A  ->  (
x  -  w )  =  ( x  -  A ) )
3433breq1d 4405 . . . . . 6  |-  ( w  =  A  ->  (
( x  -  w
) F y  <->  ( x  -  A ) F y ) )
3534anbi2d 702 . . . . 5  |-  ( w  =  A  ->  (
( x  e.  CC  /\  ( x  -  w
) F y )  <-> 
( x  e.  CC  /\  ( x  -  A
) F y ) ) )
3635opabbidv 4458 . . . 4  |-  ( w  =  A  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
37 df-shft 13049 . . . 4  |-  shift  =  ( z  e.  _V ,  w  e.  CC  |->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) } )
3832, 36, 37ovmpt2g 6418 . . 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 1313 . 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 666 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 367    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2755   _Vcvv 3059    C_ wss 3414   class class class wbr 4395   {copab 4452    X. cxp 4821   dom cdm 4823   ran crn 4824  (class class class)co 6278   CCcc 9520    + caddc 9525    - cmin 9841    shift cshi 13048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-sub 9843  df-shft 13049
This theorem is referenced by:  shftdm  13053  shftfib  13054  shftfn  13055  2shfti  13062  shftidt2  13063
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