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Theorem sca2rab 20970
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 3350 . . . . 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 2505 . . . . . 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 11030 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( 1  /  C )  e.  RR )
10 remulcl 9359 . . . . . . . 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 6094 . . . . . . . . 9  |-  ( x  =  ( ( 1  /  C )  x.  y )  ->  ( C  x.  x )  =  ( C  x.  ( ( 1  /  C )  x.  y
) ) )
1312eleq1d 2504 . . . . . . . 8  |-  ( x  =  ( ( 1  /  C )  x.  y )  ->  (
( C  x.  x
)  e.  A  <->  ( C  x.  ( ( 1  /  C )  x.  y
) )  e.  A
) )
1413elrab3 3113 . . . . . . 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 9404 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  y  e.  CC )
188rpcnd 11021 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  C  e.  CC )
198rpne0d 11024 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  C  =/=  0 )
2017, 18, 19divrec2d 10103 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  /  C )  =  ( ( 1  /  C )  x.  y
) )
2120oveq2d 6102 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( C  x.  ( y  /  C ) )  =  ( C  x.  (
( 1  /  C
)  x.  y ) ) )
2217, 18, 19divcan2d 10101 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( C  x.  ( y  /  C ) )  =  y )
2321, 22eqtr3d 2472 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( C  x.  ( ( 1  /  C )  x.  y ) )  =  y )
2423eleq1d 2504 . . . . . 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 2553 . 2  |-  ( ph  ->  A  =  { y  |  ( y  e.  RR  /\  ( ( 1  /  C )  x.  y )  e.  B ) } )
29 df-rab 2719 . 2  |-  { y  e.  RR  |  ( ( 1  /  C
)  x.  y )  e.  B }  =  { y  |  ( y  e.  RR  /\  ( ( 1  /  C )  x.  y
)  e.  B ) }
3028, 29syl6eqr 2488 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 1369    e. wcel 1756   {cab 2424   {crab 2714    C_ wss 3323  (class class class)co 6086   RRcr 9273   1c1 9275    x. cmul 9279    / cdiv 9985   RR+crp 10983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-rp 10984
This theorem is referenced by:  ovolsca  20973
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