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Theorem sca2rab 22089
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 3488 . . . . 5  |-  ( ph  ->  ( y  e.  A  ->  y  e.  RR ) )
32pm4.71rd 633 . . . 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 463 . . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR )  ->  C  e.  RR+ )
98rprecred 11270 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( 1  /  C )  e.  RR )
10 remulcl 9566 . . . . . . . 8  |-  ( ( ( 1  /  C
)  e.  RR  /\  y  e.  RR )  ->  ( ( 1  /  C )  x.  y
)  e.  RR )
119, 10sylancom 665 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( 1  /  C )  x.  y )  e.  RR )
12 oveq2 6278 . . . . . . . . 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 3255 . . . . . . 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 459 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR )  ->  y  e.  RR )
1716recnd 9611 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  y  e.  CC )
188rpcnd 11261 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  C  e.  CC )
198rpne0d 11264 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR )  ->  C  =/=  0 )
2017, 18, 19divrec2d 10320 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  /  C )  =  ( ( 1  /  C )  x.  y
) )
2120oveq2d 6286 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  ( C  x.  ( y  /  C ) )  =  ( C  x.  (
( 1  /  C
)  x.  y ) ) )
2217, 18, 19divcan2d 10318 . . . . . . . 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 639 . . . 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 2813 . 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 367    = wceq 1398    e. wcel 1823   {cab 2439   {crab 2808    C_ wss 3461  (class class class)co 6270   RRcr 9480   1c1 9482    x. cmul 9486    / cdiv 10202   RR+crp 11221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-rp 11222
This theorem is referenced by:  ovolsca  22092
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