MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  shftfn Structured version   Unicode version

Theorem shftfn 11888
Description: Functionality and domain of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
shftfn  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } )
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem shftfn
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5001 . . . . 5  |-  Rel  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
21a1i 11 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Rel  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
3 fnfun 5542 . . . . . 6  |-  ( F  Fn  B  ->  Fun  F )
43adantr 452 . . . . 5  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  F )
5 funmo 5470 . . . . . . 7  |-  ( Fun 
F  ->  E* w
( z  -  A
) F w )
6 vex 2959 . . . . . . . . . 10  |-  z  e. 
_V
7 vex 2959 . . . . . . . . . 10  |-  w  e. 
_V
8 eleq1 2496 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  e.  CC  <->  z  e.  CC ) )
9 oveq1 6088 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
x  -  A )  =  ( z  -  A ) )
109breq1d 4222 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( x  -  A
) F y  <->  ( z  -  A ) F y ) )
118, 10anbi12d 692 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F y ) ) )
12 breq2 4216 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
( z  -  A
) F y  <->  ( z  -  A ) F w ) )
1312anbi2d 685 . . . . . . . . . 10  |-  ( y  =  w  ->  (
( z  e.  CC  /\  ( z  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F w ) ) )
14 eqid 2436 . . . . . . . . . 10  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
156, 7, 11, 13, 14brab 4477 . . . . . . . . 9  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  <->  ( z  e.  CC  /\  ( z  -  A
) F w ) )
1615simprbi 451 . . . . . . . 8  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  ->  ( z  -  A
) F w )
1716moimi 2328 . . . . . . 7  |-  ( E* w ( z  -  A ) F w  ->  E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
185, 17syl 16 . . . . . 6  |-  ( Fun 
F  ->  E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
1918alrimiv 1641 . . . . 5  |-  ( Fun 
F  ->  A. z E* w  z { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
204, 19syl 16 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  A. z E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
21 dffun6 5469 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  <->  ( Rel  {
<. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  /\  A. z E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
) )
222, 20, 21sylanbrc 646 . . 3  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
23 shftfval.1 . . . . . 6  |-  F  e. 
_V
2423shftfval 11885 . . . . 5  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
2524adantl 453 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
2625funeqd 5475 . . 3  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( Fun  ( F 
shift  A )  <->  Fun  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } ) )
2722, 26mpbird 224 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  ( F  shift  A ) )
2823shftdm 11886 . . 3  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
29 fndm 5544 . . . . 5  |-  ( F  Fn  B  ->  dom  F  =  B )
3029eleq2d 2503 . . . 4  |-  ( F  Fn  B  ->  (
( x  -  A
)  e.  dom  F  <->  ( x  -  A )  e.  B ) )
3130rabbidv 2948 . . 3  |-  ( F  Fn  B  ->  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
3228, 31sylan9eqr 2490 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
33 df-fn 5457 . 2  |-  ( ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } 
<->  ( Fun  ( F 
shift  A )  /\  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } ) )
3427, 32, 33sylanbrc 646 1  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   E*wmo 2282   {crab 2709   _Vcvv 2956   class class class wbr 4212   {copab 4265   dom cdm 4878   Rel wrel 4883   Fun wfun 5448    Fn wfn 5449  (class class class)co 6081   CCcc 8988    - cmin 9291    shift cshi 11881
This theorem is referenced by:  shftf  11894  seqshft  11900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-shft 11882
  Copyright terms: Public domain W3C validator