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Theorem sca2rab 21128
Description: If  B is a scale of  A by  C, then  A is a scale of  B by  1  /  C. (Contributed by Mario Carneiro, 22-Mar-2014.)
Hypotheses
Ref Expression
ovolsca.1  |-  ( ph  ->  A  C_  RR )
ovolsca.2  |-  ( ph  ->  C  e.  RR+ )
ovolsca.3  |-  ( ph  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
Assertion
Ref Expression
sca2rab  |-  ( ph  ->  A  =  { y  e.  RR  |  ( ( 1  /  C
)  x.  y )  e.  B } )
Distinct variable groups:    x, y, A    y, B    x, C, y    ph, y
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem sca2rab
StepHypRef Expression
1 ovolsca.1 . . . . . 6  |-  ( ph  ->  A  C_  RR )
21sseld 3464 . . . . 5  |-  ( ph  ->  ( y  e.  A  ->  y  e.  RR ) )
32pm4.71rd 635 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  y  e.  A )
) )
4 ovolsca.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
54adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
65eleq2d 2524 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( 1  /  C
)  x.  y )  e.  B  <->  ( (
1  /  C )  x.  y )  e. 
{ x  e.  RR  |  ( C  x.  x )  e.  A } ) )
7 ovolsca.2 . . . . . . . . . 10  |-  ( ph  ->  C  e.  RR+ )
87adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR )  ->  C  e.  RR+ )
98rprecred 11150 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( 1  /  C )  e.  RR )
10 remulcl 9479 . . . . . . . 8  |-  ( ( ( 1  /  C
)  e.  RR  /\  y  e.  RR )  ->  ( ( 1  /  C )  x.  y
)  e.  RR )
119, 10sylancom 667 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( 1  /  C )  x.  y )  e.  RR )
12 oveq2 6209 . . . . . . . . 9  |-  ( x  =  ( ( 1  /  C )  x.  y )  ->  ( C  x.  x )  =  ( C  x.  ( ( 1  /  C )  x.  y
) ) )
1312eleq1d 2523 . . . . . . . 8  |-  ( x  =  ( ( 1  /  C )  x.  y )  ->  (
( C  x.  x
)  e.  A  <->  ( C  x.  ( ( 1  /  C )  x.  y
) )  e.  A
) )
1413elrab3 3225 . . . . . . 7  |-  ( ( ( 1  /  C
)  x.  y )  e.  RR  ->  (
( ( 1  /  C )  x.  y
)  e.  { x  e.  RR  |  ( C  x.  x )  e.  A }  <->  ( C  x.  ( ( 1  /  C )  x.  y
) )  e.  A
) )
1511, 14syl 16 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( 1  /  C
)  x.  y )  e.  { x  e.  RR  |  ( C  x.  x )  e.  A }  <->  ( C  x.  ( ( 1  /  C )  x.  y
) )  e.  A
) )
16 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR )  ->  y  e.  RR )
1716recnd 9524 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  y  e.  CC )
188rpcnd 11141 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  C  e.  CC )
198rpne0d 11144 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  C  =/=  0 )
2017, 18, 19divrec2d 10223 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  /  C )  =  ( ( 1  /  C )  x.  y
) )
2120oveq2d 6217 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( C  x.  ( y  /  C ) )  =  ( C  x.  (
( 1  /  C
)  x.  y ) ) )
2217, 18, 19divcan2d 10221 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( C  x.  ( y  /  C ) )  =  y )
2321, 22eqtr3d 2497 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( C  x.  ( ( 1  /  C )  x.  y ) )  =  y )
2423eleq1d 2523 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( C  x.  ( ( 1  /  C )  x.  y ) )  e.  A  <->  y  e.  A ) )
256, 15, 243bitrd 279 . . . . 5  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( 1  /  C
)  x.  y )  e.  B  <->  y  e.  A ) )
2625pm5.32da 641 . . . 4  |-  ( ph  ->  ( ( y  e.  RR  /\  ( ( 1  /  C )  x.  y )  e.  B )  <->  ( y  e.  RR  /\  y  e.  A ) ) )
273, 26bitr4d 256 . . 3  |-  ( ph  ->  ( y  e.  A  <->  ( y  e.  RR  /\  ( ( 1  /  C )  x.  y
)  e.  B ) ) )
2827abbi2dv 2591 . 2  |-  ( ph  ->  A  =  { y  |  ( y  e.  RR  /\  ( ( 1  /  C )  x.  y )  e.  B ) } )
29 df-rab 2808 . 2  |-  { y  e.  RR  |  ( ( 1  /  C
)  x.  y )  e.  B }  =  { y  |  ( y  e.  RR  /\  ( ( 1  /  C )  x.  y
)  e.  B ) }
3028, 29syl6eqr 2513 1  |-  ( ph  ->  A  =  { y  e.  RR  |  ( ( 1  /  C
)  x.  y )  e.  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   {crab 2803    C_ wss 3437  (class class class)co 6201   RRcr 9393   1c1 9395    x. cmul 9399    / cdiv 10105   RR+crp 11103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-rp 11104
This theorem is referenced by:  ovolsca  21131
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