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Theorem 2shfti 11850
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 11840 . . . . . . . 8  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A ) F w ) } )
32breqd 4183 . . . . . . 7  |-  ( A  e.  CC  ->  (
( x  -  B
) ( F  shift  A ) y  <->  ( x  -  B ) { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) } y ) )
4 ovex 6065 . . . . . . . 8  |-  ( x  -  B )  e. 
_V
5 vex 2919 . . . . . . . 8  |-  y  e. 
_V
6 eleq1 2464 . . . . . . . . 9  |-  ( z  =  ( x  -  B )  ->  (
z  e.  CC  <->  ( x  -  B )  e.  CC ) )
7 oveq1 6047 . . . . . . . . . 10  |-  ( z  =  ( x  -  B )  ->  (
z  -  A )  =  ( ( x  -  B )  -  A ) )
87breq1d 4182 . . . . . . . . 9  |-  ( z  =  ( x  -  B )  ->  (
( z  -  A
) F w  <->  ( (
x  -  B )  -  A ) F w ) )
96, 8anbi12d 692 . . . . . . . 8  |-  ( z  =  ( x  -  B )  ->  (
( z  e.  CC  /\  ( z  -  A
) F w )  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F w ) ) )
10 breq2 4176 . . . . . . . . 9  |-  ( w  =  y  ->  (
( ( x  -  B )  -  A
) F w  <->  ( (
x  -  B )  -  A ) F y ) )
1110anbi2d 685 . . . . . . . 8  |-  ( w  =  y  ->  (
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F w )  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
12 eqid 2404 . . . . . . . 8  |-  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) }  =  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) }
134, 5, 9, 11, 12brab 4437 . . . . . . 7  |-  ( ( x  -  B ) { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A ) F w ) } y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) )
143, 13syl6bb 253 . . . . . 6  |-  ( A  e.  CC  ->  (
( x  -  B
) ( F  shift  A ) y  <->  ( (
x  -  B )  e.  CC  /\  (
( x  -  B
)  -  A ) F y ) ) )
1514ad2antrr 707 . . . . 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 9261 . . . . . . . 8  |-  ( ( x  e.  CC  /\  B  e.  CC )  ->  ( x  -  B
)  e.  CC )
1716biantrurd 495 . . . . . . 7  |-  ( ( x  e.  CC  /\  B  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
1817ancoms 440 . . . . . 6  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
1918adantll 695 . . . . 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 9291 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  A
)  -  B )  =  ( ( x  -  B )  -  A ) )
21 subsub4 9290 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  A
)  -  B )  =  ( x  -  ( A  +  B
) ) )
2220, 21eqtr3d 2438 . . . . . . . 8  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  B
)  -  A )  =  ( x  -  ( A  +  B
) ) )
23223expb 1154 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
x  -  B )  -  A )  =  ( x  -  ( A  +  B )
) )
2423ancoms 440 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( x  -  B )  -  A )  =  ( x  -  ( A  +  B ) ) )
2524breq1d 4182 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <->  ( x  -  ( A  +  B
) ) F y ) )
2615, 19, 253bitr2d 273 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( x  -  B ) ( F  shift  A )
y  <->  ( x  -  ( A  +  B
) ) F y ) )
2726pm5.32da 623 . . 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 4231 . 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 6065 . . . 4  |-  ( F 
shift  A )  e.  _V
3029shftfval 11840 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A ) 
shift  B )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  B ) ( F  shift  A )
y ) } )
3130adantl 453 . 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 9028 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
331shftfval 11840 . . 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 2446 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A )  shift  B )  =  ( F  shift  ( A  +  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   class class class wbr 4172   {copab 4225  (class class class)co 6040   CCcc 8944    + caddc 8949    - cmin 9247    shift cshi 11836
This theorem is referenced by:  shftcan1  11853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-shft 11837
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