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Theorem 2shfti 11575
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 11565 . . . . . . . 8  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A ) F w ) } )
32breqd 4034 . . . . . . 7  |-  ( A  e.  CC  ->  (
( x  -  B
) ( F  shift  A ) y  <->  ( x  -  B ) { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) } y ) )
4 ovex 5883 . . . . . . . 8  |-  ( x  -  B )  e. 
_V
5 vex 2791 . . . . . . . 8  |-  y  e. 
_V
6 eleq1 2343 . . . . . . . . 9  |-  ( z  =  ( x  -  B )  ->  (
z  e.  CC  <->  ( x  -  B )  e.  CC ) )
7 oveq1 5865 . . . . . . . . . 10  |-  ( z  =  ( x  -  B )  ->  (
z  -  A )  =  ( ( x  -  B )  -  A ) )
87breq1d 4033 . . . . . . . . 9  |-  ( z  =  ( x  -  B )  ->  (
( z  -  A
) F w  <->  ( (
x  -  B )  -  A ) F w ) )
96, 8anbi12d 691 . . . . . . . 8  |-  ( z  =  ( x  -  B )  ->  (
( z  e.  CC  /\  ( z  -  A
) F w )  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F w ) ) )
10 breq2 4027 . . . . . . . . 9  |-  ( w  =  y  ->  (
( ( x  -  B )  -  A
) F w  <->  ( (
x  -  B )  -  A ) F y ) )
1110anbi2d 684 . . . . . . . 8  |-  ( w  =  y  ->  (
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F w )  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
12 eqid 2283 . . . . . . . 8  |-  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) }  =  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A
) F w ) }
134, 5, 9, 11, 12brab 4287 . . . . . . 7  |-  ( ( x  -  B ) { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A ) F w ) } y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) )
143, 13syl6bb 252 . . . . . 6  |-  ( A  e.  CC  ->  (
( x  -  B
) ( F  shift  A ) y  <->  ( (
x  -  B )  e.  CC  /\  (
( x  -  B
)  -  A ) F y ) ) )
1514ad2antrr 706 . . . . 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 9051 . . . . . . . 8  |-  ( ( x  e.  CC  /\  B  e.  CC )  ->  ( x  -  B
)  e.  CC )
1716biantrurd 494 . . . . . . 7  |-  ( ( x  e.  CC  /\  B  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
1817ancoms 439 . . . . . 6  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <-> 
( ( x  -  B )  e.  CC  /\  ( ( x  -  B )  -  A
) F y ) ) )
1918adantll 694 . . . . 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 9081 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  A
)  -  B )  =  ( ( x  -  B )  -  A ) )
21 subsub4 9080 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  A
)  -  B )  =  ( x  -  ( A  +  B
) ) )
2220, 21eqtr3d 2317 . . . . . . . 8  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( x  -  B
)  -  A )  =  ( x  -  ( A  +  B
) ) )
23223expb 1152 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
x  -  B )  -  A )  =  ( x  -  ( A  +  B )
) )
2423ancoms 439 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( x  -  B )  -  A )  =  ( x  -  ( A  +  B ) ) )
2524breq1d 4033 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( ( x  -  B )  -  A ) F y  <->  ( x  -  ( A  +  B
) ) F y ) )
2615, 19, 253bitr2d 272 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  x  e.  CC )  ->  ( ( x  -  B ) ( F  shift  A )
y  <->  ( x  -  ( A  +  B
) ) F y ) )
2726pm5.32da 622 . . 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 4082 . 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 5883 . . . 4  |-  ( F 
shift  A )  e.  _V
3029shftfval 11565 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A ) 
shift  B )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  B ) ( F  shift  A )
y ) } )
3130adantl 452 . 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 8819 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
331shftfval 11565 . . 3  |-  ( ( A  +  B )  e.  CC  ->  ( F  shift  ( A  +  B ) )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  ( A  +  B ) ) F y ) } )
3432, 33syl 15 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( F  shift  ( A  +  B ) )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  ( A  +  B )
) F y ) } )
3528, 31, 343eqtr4d 2325 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   {copab 4076  (class class class)co 5858   CCcc 8735    + caddc 8740    - cmin 9037    shift cshi 11561
This theorem is referenced by:  shftcan1  11578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-shft 11562
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