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Theorem shft2rab 21003
Description: If  B is a shift of  A by  C, then  A is a shift of  B by  -u C. (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
ovolshft.1  |-  ( ph  ->  A  C_  RR )
ovolshft.2  |-  ( ph  ->  C  e.  RR )
ovolshft.3  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
Assertion
Ref Expression
shft2rab  |-  ( ph  ->  A  =  { y  e.  RR  |  ( y  -  -u C
)  e.  B }
)
Distinct variable groups:    x, y, A    x, C, y    y, B    ph, y
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem shft2rab
StepHypRef Expression
1 ovolshft.1 . . . . . 6  |-  ( ph  ->  A  C_  RR )
21sseld 3367 . . . . 5  |-  ( ph  ->  ( y  e.  A  ->  y  e.  RR ) )
32pm4.71rd 635 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  y  e.  A )
) )
4 recn 9384 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 ovolshft.2 . . . . . . . . 9  |-  ( ph  ->  C  e.  RR )
65recnd 9424 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
7 subneg 9670 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( y  -  -u C
)  =  ( y  +  C ) )
84, 6, 7syl2anr 478 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  -  -u C )  =  ( y  +  C
) )
9 ovolshft.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
109adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
118, 10eleq12d 2511 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  -  -u C
)  e.  B  <->  ( y  +  C )  e.  {
x  e.  RR  | 
( x  -  C
)  e.  A }
) )
12 id 22 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  RR )
13 readdcl 9377 . . . . . . . 8  |-  ( ( y  e.  RR  /\  C  e.  RR )  ->  ( y  +  C
)  e.  RR )
1412, 5, 13syl2anr 478 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  +  C )  e.  RR )
15 oveq1 6110 . . . . . . . . 9  |-  ( x  =  ( y  +  C )  ->  (
x  -  C )  =  ( ( y  +  C )  -  C ) )
1615eleq1d 2509 . . . . . . . 8  |-  ( x  =  ( y  +  C )  ->  (
( x  -  C
)  e.  A  <->  ( (
y  +  C )  -  C )  e.  A ) )
1716elrab3 3130 . . . . . . 7  |-  ( ( y  +  C )  e.  RR  ->  (
( y  +  C
)  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( (
y  +  C )  -  C )  e.  A ) )
1814, 17syl 16 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  +  C )  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( (
y  +  C )  -  C )  e.  A ) )
19 pncan 9628 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( ( y  +  C )  -  C
)  =  y )
204, 6, 19syl2anr 478 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  +  C )  -  C )  =  y )
2120eleq1d 2509 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( y  +  C
)  -  C )  e.  A  <->  y  e.  A ) )
2211, 18, 213bitrd 279 . . . . 5  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  -  -u C
)  e.  B  <->  y  e.  A ) )
2322pm5.32da 641 . . . 4  |-  ( ph  ->  ( ( y  e.  RR  /\  ( y  -  -u C )  e.  B )  <->  ( y  e.  RR  /\  y  e.  A ) ) )
243, 23bitr4d 256 . . 3  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) ) )
2524abbi2dv 2564 . 2  |-  ( ph  ->  A  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C )  e.  B ) } )
26 df-rab 2736 . 2  |-  { y  e.  RR  |  ( y  -  -u C
)  e.  B }  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) }
2725, 26syl6eqr 2493 1  |-  ( ph  ->  A  =  { y  e.  RR  |  ( y  -  -u C
)  e.  B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   {crab 2731    C_ wss 3340  (class class class)co 6103   CCcc 9292   RRcr 9293    + caddc 9297    - cmin 9607   -ucneg 9608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-ltxr 9435  df-sub 9609  df-neg 9610
This theorem is referenced by:  ovolshft  21006  shftmbl  21032
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