MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2shfti Structured version   Unicode version

Theorem 2shfti 12565
Description: Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
2shfti  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A )  shift  B )  =  ( F  shift  ( A  +  B ) ) )

Proof of Theorem 2shfti
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 shftfval.1 . . . . . . . . 9  |-  F  e. 
_V
21shftfval 12555 . . . . . . . 8  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A ) F w ) } )
32breqd 4300 . . . . . . 7  |-  ( A  e.  CC  ->  (
( x  -  B
) ( F  shift  A ) y  <->  ( x  -  B ) { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) } y ) )
4 ovex 6115 . . . . . . . 8  |-  ( x  -  B )  e. 
_V
5 vex 2973 . . . . . . . 8  |-  y  e. 
_V
6 eleq1 2501 . . . . . . . . 9  |-  ( z  =  ( x  -  B )  ->  (
z  e.  CC  <->  ( x  -  B )  e.  CC ) )
7 oveq1 6097 . . . . . . . . . 10  |-  ( z  =  ( x  -  B )  ->  (
z  -  A )  =  ( ( x  -  B )  -  A ) )
87breq1d 4299 . . . . . . . . 9  |-  ( z  =  ( x  -  B )  ->  (
( z  -  A
) F w  <->  ( (
x  -  B )  -  A ) F w ) )
96, 8anbi12d 705 . . . . . . . 8  |-  ( z  =  ( x  -  B )  ->  (
( z  e.  CC  /\  ( z  -  A
) F w )  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F w ) ) )
10 breq2 4293 . . . . . . . . 9  |-  ( w  =  y  ->  (
( ( x  -  B )  -  A
) F w  <->  ( (
x  -  B )  -  A ) F y ) )
1110anbi2d 698 . . . . . . . 8  |-  ( w  =  y  ->  (
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F w )  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
12 eqid 2441 . . . . . . . 8  |-  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) }  =  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) }
134, 5, 9, 11, 12brab 4609 . . . . . . 7  |-  ( ( x  -  B ) { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A ) F w ) } y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) )
143, 13syl6bb 261 . . . . . 6  |-  ( A  e.  CC  ->  (
( x  -  B
) ( F  shift  A ) y  <->  ( (
x  -  B )  e.  CC  /\  (
( x  -  B
)  -  A ) F y ) ) )
1514ad2antrr 720 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( x  -  B ) ( F  shift  A )
y  <->  ( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A ) F y ) ) )
16 subcl 9605 . . . . . . . 8  |-  ( ( x  e.  CC  /\  B  e.  CC )  ->  ( x  -  B
)  e.  CC )
1716biantrurd 505 . . . . . . 7  |-  ( ( x  e.  CC  /\  B  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
1817ancoms 450 . . . . . 6  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
1918adantll 708 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <->  ( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A ) F y ) ) )
20 sub32 9639 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  A
)  -  B )  =  ( ( x  -  B )  -  A ) )
21 subsub4 9638 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  A
)  -  B )  =  ( x  -  ( A  +  B
) ) )
2220, 21eqtr3d 2475 . . . . . . . 8  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  B
)  -  A )  =  ( x  -  ( A  +  B
) ) )
23223expb 1183 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
x  -  B )  -  A )  =  ( x  -  ( A  +  B )
) )
2423ancoms 450 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( x  -  B )  -  A )  =  ( x  -  ( A  +  B ) ) )
2524breq1d 4299 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <->  ( x  -  ( A  +  B
) ) F y ) )
2615, 19, 253bitr2d 281 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( x  -  B ) ( F  shift  A )
y  <->  ( x  -  ( A  +  B
) ) F y ) )
2726pm5.32da 636 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( x  e.  CC  /\  ( x  -  B ) ( F  shift  A )
y )  <->  ( x  e.  CC  /\  ( x  -  ( A  +  B ) ) F y ) ) )
2827opabbidv 4352 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  B ) ( F  shift  A )
y ) }  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  ( A  +  B ) ) F y ) } )
29 ovex 6115 . . . 4  |-  ( F 
shift  A )  e.  _V
3029shftfval 12555 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A ) 
shift  B )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  B ) ( F  shift  A )
y ) } )
3130adantl 463 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A )  shift  B )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  B ) ( F  shift  A )
y ) } )
32 addcl 9360 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
331shftfval 12555 . . 3  |-  ( ( A  +  B )  e.  CC  ->  ( F  shift  ( A  +  B ) )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  ( A  +  B ) ) F y ) } )
3432, 33syl 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( F  shift  ( A  +  B ) )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  ( A  +  B )
) F y ) } )
3528, 31, 343eqtr4d 2483 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A )  shift  B )  =  ( F  shift  ( A  +  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970   class class class wbr 4289   {copab 4346  (class class class)co 6090   CCcc 9276    + caddc 9281    - cmin 9591    shift cshi 12551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-shft 12552
This theorem is referenced by:  shftcan1  12568
  Copyright terms: Public domain W3C validator