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Theorem shft2rab 22044
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 3498 . . . . 5  |-  ( ph  ->  ( y  e.  A  ->  y  e.  RR ) )
32pm4.71rd 635 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  y  e.  A )
) )
4 recn 9599 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 ovolshft.2 . . . . . . . . 9  |-  ( ph  ->  C  e.  RR )
65recnd 9639 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
7 subneg 9887 . . . . . . . 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 2539 . . . . . 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 9592 . . . . . . . 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 6303 . . . . . . . . 9  |-  ( x  =  ( y  +  C )  ->  (
x  -  C )  =  ( ( y  +  C )  -  C ) )
1615eleq1d 2526 . . . . . . . 8  |-  ( x  =  ( y  +  C )  ->  (
( x  -  C
)  e.  A  <->  ( (
y  +  C )  -  C )  e.  A ) )
1716elrab3 3258 . . . . . . 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 9845 . . . . . . . 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 2526 . . . . . 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 2594 . 2  |-  ( ph  ->  A  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C )  e.  B ) } )
26 df-rab 2816 . 2  |-  { y  e.  RR  |  ( y  -  -u C
)  e.  B }  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) }
2725, 26syl6eqr 2516 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 1395    e. wcel 1819   {cab 2442   {crab 2811    C_ wss 3471  (class class class)co 6296   CCcc 9507   RRcr 9508    + caddc 9512    - cmin 9824   -ucneg 9825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-sub 9826  df-neg 9827
This theorem is referenced by:  ovolshft  22047  shftmbl  22074
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