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Theorem shftfvalg 10217
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 108 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  F  e.  V )
2 simpl 107 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  A  e.  CC )
3 simplr 497 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  CC )
4 simpll 496 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  A  e.  CC )
53, 4subcld 7772 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  CC )
6 vex 2622 . . . . . . . . . . 11  |-  y  e. 
_V
7 breldmg 4630 . . . . . . . . . . 11  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
( x  -  A
)  e.  dom  F
)
86, 7mp3an2 1261 . . . . . . . . . 10  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  ( x  -  A )  e.  dom  F )
95, 8sylancom 411 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  ( x  -  A )  e.  dom  F )
10 npcan 7670 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
1110eqcomd 2093 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
1211ancoms 264 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
1312adantr 270 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  =  ( ( x  -  A )  +  A
) )
14 oveq1 5641 . . . . . . . . . . 11  |-  ( w  =  ( x  -  A )  ->  (
w  +  A )  =  ( ( x  -  A )  +  A ) )
1514eqeq2d 2099 . . . . . . . . . 10  |-  ( w  =  ( x  -  A )  ->  (
x  =  ( w  +  A )  <->  x  =  ( ( x  -  A )  +  A
) ) )
1615rspcev 2722 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  dom  F  /\  x  =  (
( x  -  A
)  +  A ) )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
179, 13, 16syl2anc 403 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  E. w  e.  dom  F  x  =  ( w  +  A
) )
18 vex 2622 . . . . . . . . 9  |-  x  e. 
_V
19 eqeq1 2094 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  ( w  +  A )  <->  x  =  ( w  +  A
) ) )
2019rexbidv 2381 . . . . . . . . 9  |-  ( z  =  x  ->  ( E. w  e.  dom  F  z  =  ( w  +  A )  <->  E. w  e.  dom  F  x  =  ( w  +  A
) ) )
2118, 20elab 2758 . . . . . . . 8  |-  ( x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  <->  E. w  e.  dom  F  x  =  ( w  +  A ) )
2217, 21sylibr 132 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  x  e.  { z  |  E. w  e.  dom  F  z  =  ( w  +  A
) } )
23 brelrng 4654 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  CC  /\  y  e.  _V  /\  (
x  -  A ) F y )  -> 
y  e.  ran  F
)
246, 23mp3an2 1261 . . . . . . . 8  |-  ( ( ( x  -  A
)  e.  CC  /\  ( x  -  A
) F y )  ->  y  e.  ran  F )
255, 24sylancom 411 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  CC )  /\  ( x  -  A ) F y )  ->  y  e.  ran  F )
2622, 25jca 300 . . . . . 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 370 . . . . 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 4096 . . . 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 4434 . . . 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 3071 . . 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 4685 . . . . 5  |-  ( F  e.  V  ->  dom  F  e.  _V )
32 abrexexg 5871 . . . . 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 4686 . . . 4  |-  ( F  e.  V  ->  ran  F  e.  _V )
35 xpexg 4540 . . . 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 403 . . 3  |-  ( F  e.  V  ->  ( { z  |  E. w  e.  dom  F  z  =  ( w  +  A ) }  X.  ran  F )  e.  _V )
37 ssexg 3970 . . 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 283 . 2  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  e.  _V )
39 elex 2630 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
40 breq 3839 . . . . . 6  |-  ( z  =  F  ->  (
( x  -  w
) z y  <->  ( x  -  w ) F y ) )
4140anbi2d 452 . . . . 5  |-  ( z  =  F  ->  (
( x  e.  CC  /\  ( x  -  w
) z y )  <-> 
( x  e.  CC  /\  ( x  -  w
) F y ) ) )
4241opabbidv 3896 . . . 4  |-  ( z  =  F  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) } )
43 oveq2 5642 . . . . . . 7  |-  ( w  =  A  ->  (
x  -  w )  =  ( x  -  A ) )
4443breq1d 3847 . . . . . 6  |-  ( w  =  A  ->  (
( x  -  w
) F y  <->  ( x  -  A ) F y ) )
4544anbi2d 452 . . . . 5  |-  ( w  =  A  ->  (
( x  e.  CC  /\  ( x  -  w
) F y )  <-> 
( x  e.  CC  /\  ( x  -  A
) F y ) ) )
4645opabbidv 3896 . . . 4  |-  ( w  =  A  ->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
47 df-shft 10214 . . . 4  |-  shift  =  ( z  e.  _V ,  w  e.  CC  |->  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  w
) z y ) } )
4842, 46, 47ovmpt2g 5761 . . 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 1207 . 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 1174 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 102    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619    C_ wss 2997   class class class wbr 3837   {copab 3890    X. cxp 4426   dom cdm 4428   ran crn 4429  (class class class)co 5634   CCcc 7327    + caddc 7332    - cmin 7632    shift cshi 10213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-shft 10214
This theorem is referenced by:  ovshftex  10218  shftfibg  10219  2shfti  10230
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