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Theorem shft2rab 22403
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 3406 . . . . 5  |-  ( ph  ->  ( y  e.  A  ->  y  e.  RR ) )
32pm4.71rd 639 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  y  e.  A )
) )
4 recn 9580 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 ovolshft.2 . . . . . . . . 9  |-  ( ph  ->  C  e.  RR )
65recnd 9620 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
7 subneg 9874 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( y  -  -u C
)  =  ( y  +  C ) )
84, 6, 7syl2anr 480 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  -  -u C )  =  ( y  +  C
) )
9 ovolshft.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
109adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
118, 10eleq12d 2500 . . . . . 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 9573 . . . . . . . 8  |-  ( ( y  e.  RR  /\  C  e.  RR )  ->  ( y  +  C
)  e.  RR )
1412, 5, 13syl2anr 480 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  +  C )  e.  RR )
15 oveq1 6256 . . . . . . . . 9  |-  ( x  =  ( y  +  C )  ->  (
x  -  C )  =  ( ( y  +  C )  -  C ) )
1615eleq1d 2490 . . . . . . . 8  |-  ( x  =  ( y  +  C )  ->  (
( x  -  C
)  e.  A  <->  ( (
y  +  C )  -  C )  e.  A ) )
1716elrab3 3172 . . . . . . 7  |-  ( ( y  +  C )  e.  RR  ->  (
( y  +  C
)  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( (
y  +  C )  -  C )  e.  A ) )
1814, 17syl 17 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  +  C )  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( (
y  +  C )  -  C )  e.  A ) )
19 pncan 9832 . . . . . . . 8  |-  ( ( y  e.  CC  /\  C  e.  CC )  ->  ( ( y  +  C )  -  C
)  =  y )
204, 6, 19syl2anr 480 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  +  C )  -  C )  =  y )
2120eleq1d 2490 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( y  +  C
)  -  C )  e.  A  <->  y  e.  A ) )
2211, 18, 213bitrd 282 . . . . 5  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( y  -  -u C
)  e.  B  <->  y  e.  A ) )
2322pm5.32da 645 . . . 4  |-  ( ph  ->  ( ( y  e.  RR  /\  ( y  -  -u C )  e.  B )  <->  ( y  e.  RR  /\  y  e.  A ) ) )
243, 23bitr4d 259 . . 3  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) ) )
2524abbi2dv 2547 . 2  |-  ( ph  ->  A  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C )  e.  B ) } )
26 df-rab 2723 . 2  |-  { y  e.  RR  |  ( y  -  -u C
)  e.  B }  =  { y  |  ( y  e.  RR  /\  ( y  -  -u C
)  e.  B ) }
2725, 26syl6eqr 2480 1  |-  ( ph  ->  A  =  { y  e.  RR  |  ( y  -  -u C
)  e.  B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2414   {crab 2718    C_ wss 3379  (class class class)co 6249   CCcc 9488   RRcr 9489    + caddc 9493    - cmin 9811   -ucneg 9812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-ltxr 9631  df-sub 9813  df-neg 9814
This theorem is referenced by:  ovolshft  22406  shftmbl  22434
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