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Theorem shftfvalg 9647
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 107 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  F  e.  V )
2 simpl 106 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  A  e.  CC )
3 simplr 490 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  CC )
4 simpll 489 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  A  e.  CC )
53, 4subcld 7385 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  CC )
6 vex 2577 . . . . . . . . . . 11  |-  y  e. 
_V
7 breldmg 4569 . . . . . . . . . . 11  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
( x  -  A
)  e.  dom  F
)
86, 7mp3an2 1231 . . . . . . . . . 10  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  ( x  -  A )  e.  dom  F )
95, 8sylancom 405 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  dom  F )
10 npcan 7283 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
1110eqcomd 2061 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
1211ancoms 259 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
1312adantr 265 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  =  ( ( x  -  A )  +  A
) )
14 oveq1 5547 . . . . . . . . . . 11  |-  ( w  =  ( x  -  A )  ->  (
w  +  A )  =  ( ( x  -  A )  +  A ) )
1514eqeq2d 2067 . . . . . . . . . 10  |-  ( w  =  ( x  -  A )  ->  (
x  =  ( w  +  A )  <->  x  =  ( ( x  -  A )  +  A
) ) )
1615rspcev 2673 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  dom  F  /\  x  =  (
( x  -  A
)  +  A ) )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
179, 13, 16syl2anc 397 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
18 vex 2577 . . . . . . . . 9  |-  x  e. 
_V
19 eqeq1 2062 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  ( w  +  A )  <->  x  =  ( w  +  A
) ) )
2019rexbidv 2344 . . . . . . . . 9  |-  ( z  =  x  ->  ( E. w  e.  dom  F  z  =  ( w  +  A )  <->  E. w  e.  dom  F  x  =  ( w  +  A
) ) )
2118, 20elab 2710 . . . . . . . 8  |-  ( x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  <->  E. w  e.  dom  F  x  =  ( w  +  A ) )
2217, 21sylibr 141 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) } )
23 brelrng 4593 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
y  e.  ran  F
)
246, 23mp3an2 1231 . . . . . . . 8  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  y  e.  ran  F )
255, 24sylancom 405 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  y  e.  ran  F )
2622, 25jca 294 . . . . . 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 364 . . . . 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 4043 . . . 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 4379 . . . 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, 29syl6sseqr 3020 . . 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 4624 . . . . 5  |-  ( F  e.  V  ->  dom  F  e.  _V )
32 abrexexg 5773 . . . . 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 4625 . . . 4  |-  ( F  e.  V  ->  ran  F  e.  _V )
35 xpexg 4480 . . . 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 397 . . 3  |-  ( F  e.  V  ->  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  X.  ran  F )  e.  _V )
37 ssexg 3924 . . 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 277 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  e.  _V )
39 elex 2583 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
40 breq 3794 . . . . . 6  |-  ( z  =  F  ->  (
( x  -  w
) z y  <->  ( x  -  w ) F y ) )
4140anbi2d 445 . . . . 5  |-  ( z  =  F  ->  (
( x  e.  CC  /\  ( x  -  w
) z y )  <-> 
( x  e.  CC  /\  ( x  -  w
) F y ) ) )
4241opabbidv 3851 . . . 4  |-  ( z  =  F  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) } )
43 oveq2 5548 . . . . . . 7  |-  ( w  =  A  ->  (
x  -  w )  =  ( x  -  A ) )
4443breq1d 3802 . . . . . 6  |-  ( w  =  A  ->  (
( x  -  w
) F y  <->  ( x  -  A ) F y ) )
4544anbi2d 445 . . . . 5  |-  ( w  =  A  ->  (
( x  e.  CC  /\  ( x  -  w
) F y )  <-> 
( x  e.  CC  /\  ( x  -  A
) F y ) ) )
4645opabbidv 3851 . . . 4  |-  ( w  =  A  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
47 df-shft 9644 . . . 4  |-  shift  =  ( z  e.  _V ,  w  e.  CC  |->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) } )
4842, 46, 47ovmpt2g 5663 . . 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 1179 . 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 1146 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 101    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574    C_ wss 2945   class class class wbr 3792   {copab 3845    X. cxp 4371   dom cdm 4373   ran crn 4374  (class class class)co 5540   CCcc 6945    + caddc 6950    - cmin 7245    shift cshi 9643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-shft 9644
This theorem is referenced by:  ovshftex  9648  shftfibg  9649  2shfti  9660
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