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Theorem shft2rab 19404
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 3347 . . . . 5  |-  ( ph  ->  ( y  e.  A  ->  y  e.  RR ) )
32pm4.71rd 617 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  y  e.  A )
) )
4 recn 9080 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 ovolshft.2 . . . . . . . . 9  |-  ( ph  ->  C  e.  RR )
65recnd 9114 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
7 subneg 9350 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( y  -  -u C
)  =  ( y  +  C ) )
84, 6, 7syl2anr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  -  -u C )  =  ( y  +  C
) )
9 ovolshft.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
109adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
118, 10eleq12d 2504 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  -  -u C
)  e.  B  <->  ( y  +  C )  e.  {
x  e.  RR  | 
( x  -  C
)  e.  A }
) )
12 id 20 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  RR )
13 readdcl 9073 . . . . . . . 8  |-  ( ( y  e.  RR  /\  C  e.  RR )  ->  ( y  +  C
)  e.  RR )
1412, 5, 13syl2anr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  +  C )  e.  RR )
15 oveq1 6088 . . . . . . . . 9  |-  ( x  =  ( y  +  C )  ->  (
x  -  C )  =  ( ( y  +  C )  -  C ) )
1615eleq1d 2502 . . . . . . . 8  |-  ( x  =  ( y  +  C )  ->  (
( x  -  C
)  e.  A  <->  ( (
y  +  C )  -  C )  e.  A ) )
1716elrab3 3093 . . . . . . 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 9311 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( ( y  +  C )  -  C
)  =  y )
204, 6, 19syl2anr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  +  C )  -  C )  =  y )
2120eleq1d 2502 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( y  +  C
)  -  C )  e.  A  <->  y  e.  A ) )
2211, 18, 213bitrd 271 . . . . 5  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  -  -u C
)  e.  B  <->  y  e.  A ) )
2322pm5.32da 623 . . . 4  |-  ( ph  ->  ( ( y  e.  RR  /\  ( y  -  -u C )  e.  B )  <->  ( y  e.  RR  /\  y  e.  A ) ) )
243, 23bitr4d 248 . . 3  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) ) )
2524abbi2dv 2551 . 2  |-  ( ph  ->  A  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C )  e.  B ) } )
26 df-rab 2714 . 2  |-  { y  e.  RR  |  ( y  -  -u C
)  e.  B }  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) }
2725, 26syl6eqr 2486 1  |-  ( ph  ->  A  =  { y  e.  RR  |  ( y  -  -u C
)  e.  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   {crab 2709    C_ wss 3320  (class class class)co 6081   CCcc 8988   RRcr 8989    + caddc 8993    - cmin 9291   -ucneg 9292
This theorem is referenced by:  ovolshft  19407  shftmbl  19433
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-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-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-neg 9294
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