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Theorem shftfvalg 10602
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.)
Assertion
Ref Expression
shftfvalg  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
Distinct variable groups:    x, y, A   
x, F, y
Allowed substitution hints:    V( x, y)

Proof of Theorem shftfvalg
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  F  e.  V )
2 simpl 108 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  A  e.  CC )
3 simplr 519 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  CC )
4 simpll 518 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  A  e.  CC )
53, 4subcld 8085 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  CC )
6 vex 2689 . . . . . . . . . . 11  |-  y  e. 
_V
7 breldmg 4745 . . . . . . . . . . 11  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
( x  -  A
)  e.  dom  F
)
86, 7mp3an2 1303 . . . . . . . . . 10  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  ( x  -  A )  e.  dom  F )
95, 8sylancom 416 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  dom  F )
10 npcan 7983 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
1110eqcomd 2145 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
1211ancoms 266 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
1312adantr 274 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  =  ( ( x  -  A )  +  A
) )
14 oveq1 5781 . . . . . . . . . . 11  |-  ( w  =  ( x  -  A )  ->  (
w  +  A )  =  ( ( x  -  A )  +  A ) )
1514eqeq2d 2151 . . . . . . . . . 10  |-  ( w  =  ( x  -  A )  ->  (
x  =  ( w  +  A )  <->  x  =  ( ( x  -  A )  +  A
) ) )
1615rspcev 2789 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  dom  F  /\  x  =  (
( x  -  A
)  +  A ) )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
179, 13, 16syl2anc 408 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
18 vex 2689 . . . . . . . . 9  |-  x  e. 
_V
19 eqeq1 2146 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  ( w  +  A )  <->  x  =  ( w  +  A
) ) )
2019rexbidv 2438 . . . . . . . . 9  |-  ( z  =  x  ->  ( E. w  e.  dom  F  z  =  ( w  +  A )  <->  E. w  e.  dom  F  x  =  ( w  +  A
) ) )
2118, 20elab 2828 . . . . . . . 8  |-  ( x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  <->  E. w  e.  dom  F  x  =  ( w  +  A ) )
2217, 21sylibr 133 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) } )
23 brelrng 4770 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
y  e.  ran  F
)
246, 23mp3an2 1303 . . . . . . . 8  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  y  e.  ran  F )
255, 24sylancom 416 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  y  e.  ran  F )
2622, 25jca 304 . . . . . 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 ) )
2726expl 375 . . . . 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 ) ) )
2827ssopab2dv 4200 . . . 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 ) } )
29 df-xp 4545 . . . 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 ) }
3028, 29sseqtrrdi 3146 . . 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 ) )
31 dmexg 4803 . . . . 5  |-  ( F  e.  V  ->  dom  F  e.  _V )
32 abrexexg 6016 . . . . 5  |-  ( dom 
F  e.  _V  ->  { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) }  e.  _V )
3331, 32syl 14 . . . 4  |-  ( F  e.  V  ->  { z  |  E. w  e. 
dom  F  z  =  ( w  +  A
) }  e.  _V )
34 rnexg 4804 . . . 4  |-  ( F  e.  V  ->  ran  F  e.  _V )
35 xpexg 4653 . . . 4  |-  ( ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  e.  _V  /\  ran  F  e. 
_V )  ->  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  X.  ran  F )  e.  _V )
3633, 34, 35syl2anc 408 . . 3  |-  ( F  e.  V  ->  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  X.  ran  F )  e.  _V )
37 ssexg 4067 . . 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 )
3830, 36, 37syl2an 287 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  e.  _V )
39 elex 2697 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
40 breq 3931 . . . . . 6  |-  ( z  =  F  ->  (
( x  -  w
) z y  <->  ( x  -  w ) F y ) )
4140anbi2d 459 . . . . 5  |-  ( z  =  F  ->  (
( x  e.  CC  /\  ( x  -  w
) z y )  <-> 
( x  e.  CC  /\  ( x  -  w
) F y ) ) )
4241opabbidv 3994 . . . 4  |-  ( z  =  F  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) } )
43 oveq2 5782 . . . . . . 7  |-  ( w  =  A  ->  (
x  -  w )  =  ( x  -  A ) )
4443breq1d 3939 . . . . . 6  |-  ( w  =  A  ->  (
( x  -  w
) F y  <->  ( x  -  A ) F y ) )
4544anbi2d 459 . . . . 5  |-  ( w  =  A  ->  (
( x  e.  CC  /\  ( x  -  w
) F y )  <-> 
( x  e.  CC  /\  ( x  -  A
) F y ) ) )
4645opabbidv 3994 . . . 4  |-  ( w  =  A  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
47 df-shft 10599 . . . 4  |-  shift  =  ( z  e.  _V ,  w  e.  CC  |->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) } )
4842, 46, 47ovmpog 5905 . . 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 ) } )
4939, 48syl3an1 1249 . 2  |-  ( ( 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 ) } )
501, 2, 38, 49syl3anc 1216 1  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   {cab 2125   E.wrex 2417   _Vcvv 2686    C_ wss 3071   class class class wbr 3929   {copab 3988    X. cxp 4537   dom cdm 4539   ran crn 4540  (class class class)co 5774   CCcc 7630    + caddc 7635    - cmin 7945    shift cshi 10598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-resscn 7724  ax-1cn 7725  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7947  df-shft 10599
This theorem is referenced by:  ovshftex  10603  shftfibg  10604  2shfti  10615
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