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Theorem shftfval 10830
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 simplr 528 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  CC )
2 simpll 527 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  A  e.  CC )
31, 2subcld 8268 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  CC )
4 vex 2741 . . . . . . . . . . 11  |-  y  e. 
_V
5 breldmg 4834 . . . . . . . . . . 11  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
( x  -  A
)  e.  dom  F
)
64, 5mp3an2 1325 . . . . . . . . . 10  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  ( x  -  A )  e.  dom  F )
73, 6sylancom 420 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  dom  F )
8 npcan 8166 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
98eqcomd 2183 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
109ancoms 268 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
1110adantr 276 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  =  ( ( x  -  A )  +  A
) )
12 oveq1 5882 . . . . . . . . . . 11  |-  ( w  =  ( x  -  A )  ->  (
w  +  A )  =  ( ( x  -  A )  +  A ) )
1312eqeq2d 2189 . . . . . . . . . 10  |-  ( w  =  ( x  -  A )  ->  (
x  =  ( w  +  A )  <->  x  =  ( ( x  -  A )  +  A
) ) )
1413rspcev 2842 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  dom  F  /\  x  =  (
( x  -  A
)  +  A ) )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
157, 11, 14syl2anc 411 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
16 vex 2741 . . . . . . . . 9  |-  x  e. 
_V
17 eqeq1 2184 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  ( w  +  A )  <->  x  =  ( w  +  A
) ) )
1817rexbidv 2478 . . . . . . . . 9  |-  ( z  =  x  ->  ( E. w  e.  dom  F  z  =  ( w  +  A )  <->  E. w  e.  dom  F  x  =  ( w  +  A
) ) )
1916, 18elab 2882 . . . . . . . 8  |-  ( x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  <->  E. w  e.  dom  F  x  =  ( w  +  A ) )
2015, 19sylibr 134 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) } )
21 brelrng 4859 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
y  e.  ran  F
)
224, 21mp3an2 1325 . . . . . . . 8  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  y  e.  ran  F )
233, 22sylancom 420 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  y  e.  ran  F )
2420, 23jca 306 . . . . . 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 ) )
2524expl 378 . . . . 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 ) ) )
2625ssopab2dv 4279 . . . 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 ) } )
27 df-xp 4633 . . . 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 ) }
2826, 27sseqtrrdi 3205 . . 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 ) )
29 shftfval.1 . . . . . 6  |-  F  e. 
_V
3029dmex 4894 . . . . 5  |-  dom  F  e.  _V
3130abrexex 6118 . . . 4  |-  { z  |  E. w  e. 
dom  F  z  =  ( w  +  A
) }  e.  _V
3229rnex 4895 . . . 4  |-  ran  F  e.  _V
3331, 32xpex 4742 . . 3  |-  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) }  X.  ran  F )  e.  _V
34 ssexg 4143 . . 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 )
3528, 33, 34sylancl 413 . 2  |-  ( A  e.  CC  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  e.  _V )
36 breq 4006 . . . . . 6  |-  ( z  =  F  ->  (
( x  -  w
) z y  <->  ( x  -  w ) F y ) )
3736anbi2d 464 . . . . 5  |-  ( z  =  F  ->  (
( x  e.  CC  /\  ( x  -  w
) z y )  <-> 
( x  e.  CC  /\  ( x  -  w
) F y ) ) )
3837opabbidv 4070 . . . 4  |-  ( z  =  F  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) } )
39 oveq2 5883 . . . . . . 7  |-  ( w  =  A  ->  (
x  -  w )  =  ( x  -  A ) )
4039breq1d 4014 . . . . . 6  |-  ( w  =  A  ->  (
( x  -  w
) F y  <->  ( x  -  A ) F y ) )
4140anbi2d 464 . . . . 5  |-  ( w  =  A  ->  (
( x  e.  CC  /\  ( x  -  w
) F y )  <-> 
( x  e.  CC  /\  ( x  -  A
) F y ) ) )
4241opabbidv 4070 . . . 4  |-  ( w  =  A  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
43 df-shft 10824 . . . 4  |-  shift  =  ( z  e.  _V ,  w  e.  CC  |->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) } )
4438, 42, 43ovmpog 6009 . . 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 ) } )
4529, 44mp3an1 1324 . 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 ) } )
4635, 45mpdan 421 1  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2738    C_ wss 3130   class class class wbr 4004   {copab 4064    X. cxp 4625   dom cdm 4627   ran crn 4628  (class class class)co 5875   CCcc 7809    + caddc 7814    - cmin 8128    shift cshi 10823
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-resscn 7903  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-sub 8130  df-shft 10824
This theorem is referenced by:  shftdm  10831  shftfib  10832  shftfn  10833  2shfti  10840  shftidt2  10841
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