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Theorem shft2rab 18867
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 3179 . . . . 5  |-  ( ph  ->  ( y  e.  A  ->  y  e.  RR ) )
32pm4.71rd 616 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  y  e.  A )
) )
4 recn 8827 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 ovolshft.2 . . . . . . . . 9  |-  ( ph  ->  C  e.  RR )
65recnd 8861 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
7 subneg 9096 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( y  -  -u C
)  =  ( y  +  C ) )
84, 6, 7syl2anr 464 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  -  -u C )  =  ( y  +  C
) )
9 ovolshft.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
109adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
118, 10eleq12d 2351 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  -  -u C
)  e.  B  <->  ( y  +  C )  e.  {
x  e.  RR  | 
( x  -  C
)  e.  A }
) )
12 id 19 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  RR )
13 readdcl 8820 . . . . . . . 8  |-  ( ( y  e.  RR  /\  C  e.  RR )  ->  ( y  +  C
)  e.  RR )
1412, 5, 13syl2anr 464 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  +  C )  e.  RR )
15 oveq1 5865 . . . . . . . . 9  |-  ( x  =  ( y  +  C )  ->  (
x  -  C )  =  ( ( y  +  C )  -  C ) )
1615eleq1d 2349 . . . . . . . 8  |-  ( x  =  ( y  +  C )  ->  (
( x  -  C
)  e.  A  <->  ( (
y  +  C )  -  C )  e.  A ) )
1716elrab3 2924 . . . . . . 7  |-  ( ( y  +  C )  e.  RR  ->  (
( y  +  C
)  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( (
y  +  C )  -  C )  e.  A ) )
1814, 17syl 15 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  +  C )  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( (
y  +  C )  -  C )  e.  A ) )
19 pncan 9057 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( ( y  +  C )  -  C
)  =  y )
204, 6, 19syl2anr 464 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  +  C )  -  C )  =  y )
2120eleq1d 2349 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( y  +  C
)  -  C )  e.  A  <->  y  e.  A ) )
2211, 18, 213bitrd 270 . . . . 5  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  -  -u C
)  e.  B  <->  y  e.  A ) )
2322pm5.32da 622 . . . 4  |-  ( ph  ->  ( ( y  e.  RR  /\  ( y  -  -u C )  e.  B )  <->  ( y  e.  RR  /\  y  e.  A ) ) )
243, 23bitr4d 247 . . 3  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) ) )
2524abbi2dv 2398 . 2  |-  ( ph  ->  A  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C )  e.  B ) } )
26 df-rab 2552 . 2  |-  { y  e.  RR  |  ( y  -  -u C
)  e.  B }  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) }
2725, 26syl6eqr 2333 1  |-  ( ph  ->  A  =  { y  e.  RR  |  ( y  -  -u C
)  e.  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547    C_ wss 3152  (class class class)co 5858   CCcc 8735   RRcr 8736    + caddc 8740    - cmin 9037   -ucneg 9038
This theorem is referenced by:  ovolshft  18870  shftmbl  18896
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-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-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-neg 9040
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