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Theorem ovolscalem2 18873
Description: Lemma for ovolshft 18870. (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 } )
ovolsca.4  |-  ( ph  ->  ( vol * `  A )  e.  RR )
Assertion
Ref Expression
ovolscalem2  |-  ( ph  ->  ( vol * `  B )  <_  (
( vol * `  A )  /  C
) )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem ovolscalem2
Dummy variables  f  n  y  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolsca.1 . . . . . 6  |-  ( ph  ->  A  C_  RR )
21adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
3 ovolsca.4 . . . . . 6  |-  ( ph  ->  ( vol * `  A )  e.  RR )
43adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol * `
 A )  e.  RR )
5 ovolsca.2 . . . . . 6  |-  ( ph  ->  C  e.  RR+ )
6 rpmulcl 10375 . . . . . 6  |-  ( ( C  e.  RR+  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
75, 6sylan 457 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
8 eqid 2283 . . . . . 6  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 18839 . . . . 5  |-  ( ( A  C_  RR  /\  ( vol * `  A )  e.  RR  /\  ( C  x.  y )  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol * `  A )  +  ( C  x.  y ) ) ) )
102, 4, 7, 9syl3anc 1182 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol * `  A )  +  ( C  x.  y ) ) ) )
111ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  A  C_  RR )
125ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  C  e.  RR+ )
13 ovolsca.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
1413ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
153ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  ( vol * `  A )  e.  RR )
16 fveq2 5525 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
1716fveq2d 5529 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
1817oveq1d 5873 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  /  C )  =  ( ( 1st `  ( f `  n
) )  /  C
) )
1916fveq2d 5529 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
2019oveq1d 5873 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 2nd `  (
f `  m )
)  /  C )  =  ( ( 2nd `  ( f `  n
) )  /  C
) )
2118, 20opeq12d 3804 . . . . . . . 8  |-  ( m  =  n  ->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >.  =  <. (
( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
2221cbvmptv 4111 . . . . . . 7  |-  ( m  e.  NN  |->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >. )  =  ( n  e.  NN  |->  <.
( ( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
23 elmapi 6792 . . . . . . . 8  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2423ad2antrl 708 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
25 simprrl 740 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  A  C_ 
U. ran  ( (,)  o.  f ) )
26 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  y  e.  RR+ )
27 simprrr 741 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol * `  A )  +  ( C  x.  y ) ) )
2811, 12, 14, 15, 8, 22, 24, 25, 26, 27ovolscalem1 18872 . . . . . 6  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( C  x.  y ) ) ) ) )  ->  ( vol * `  B )  <_  ( ( ( vol * `  A
)  /  C )  +  y ) )
2928expr 598 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  ->  (
( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol * `  A )  +  ( C  x.  y ) ) )  ->  ( vol * `  B )  <_  ( ( ( vol * `  A
)  /  C )  +  y ) ) )
3029rexlimdva 2667 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol * `  A )  +  ( C  x.  y ) ) )  ->  ( vol * `  B )  <_  ( ( ( vol * `  A
)  /  C )  +  y ) ) )
3110, 30mpd 14 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol * `
 B )  <_ 
( ( ( vol
* `  A )  /  C )  +  y ) )
3231ralrimiva 2626 . 2  |-  ( ph  ->  A. y  e.  RR+  ( vol * `  B
)  <_  ( (
( vol * `  A )  /  C
)  +  y ) )
33 ssrab2 3258 . . . . . 6  |-  { x  e.  RR  |  ( C  x.  x )  e.  A }  C_  RR
3433a1i 10 . . . . 5  |-  ( ph  ->  { x  e.  RR  |  ( C  x.  x )  e.  A }  C_  RR )
3513, 34eqsstrd 3212 . . . 4  |-  ( ph  ->  B  C_  RR )
36 ovolcl 18837 . . . 4  |-  ( B 
C_  RR  ->  ( vol
* `  B )  e.  RR* )
3735, 36syl 15 . . 3  |-  ( ph  ->  ( vol * `  B )  e.  RR* )
383, 5rerpdivcld 10417 . . 3  |-  ( ph  ->  ( ( vol * `  A )  /  C
)  e.  RR )
39 xralrple 10532 . . 3  |-  ( ( ( vol * `  B )  e.  RR*  /\  ( ( vol * `  A )  /  C
)  e.  RR )  ->  ( ( vol
* `  B )  <_  ( ( vol * `  A )  /  C
)  <->  A. y  e.  RR+  ( vol * `  B
)  <_  ( (
( vol * `  A )  /  C
)  +  y ) ) )
4037, 38, 39syl2anc 642 . 2  |-  ( ph  ->  ( ( vol * `  B )  <_  (
( vol * `  A )  /  C
)  <->  A. y  e.  RR+  ( vol * `  B
)  <_  ( (
( vol * `  A )  /  C
)  +  y ) ) )
4132, 40mpbird 223 1  |-  ( ph  ->  ( vol * `  B )  <_  (
( vol * `  A )  /  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   ran crn 4690    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   supcsup 7193   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   RR+crp 10354   (,)cioo 10656    seq cseq 11046   abscabs 11719   vol
*covol 18822
This theorem is referenced by:  ovolsca  18874
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  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  ax-pre-mulgt0 8814  ax-pre-sup 8815
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-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ovol 18824
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