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Theorem ovolscalem2 19402
Description: Lemma for ovolshft 19399. (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 452 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
3 ovolsca.4 . . . . . 6  |-  ( ph  ->  ( vol * `  A )  e.  RR )
43adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol * `
 A )  e.  RR )
5 ovolsca.2 . . . . . 6  |-  ( ph  ->  C  e.  RR+ )
6 rpmulcl 10625 . . . . . 6  |-  ( ( C  e.  RR+  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
75, 6sylan 458 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
8 eqid 2435 . . . . . 6  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 19368 . . . . 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 1184 . . . 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 707 . . . . 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 ) ) ) ) )  ->  A  C_  RR )
125ad2antrr 707 . . . . 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 ) ) ) ) )  ->  C  e.  RR+ )
13 ovolsca.3 . . . . . 6  |-  ( ph  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
1413ad2antrr 707 . . . . 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 ) ) ) ) )  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
153ad2antrr 707 . . . . 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 * `  A )  e.  RR )
16 fveq2 5720 . . . . . . . . 9  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
1716fveq2d 5724 . . . . . . . 8  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
1817oveq1d 6088 . . . . . . 7  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  /  C )  =  ( ( 1st `  ( f `  n
) )  /  C
) )
1916fveq2d 5724 . . . . . . . 8  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
2019oveq1d 6088 . . . . . . 7  |-  ( m  =  n  ->  (
( 2nd `  (
f `  m )
)  /  C )  =  ( ( 2nd `  ( f `  n
) )  /  C
) )
2118, 20opeq12d 3984 . . . . . 6  |-  ( m  =  n  ->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >.  =  <. (
( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
2221cbvmptv 4292 . . . . 5  |-  ( m  e.  NN  |->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >. )  =  ( n  e.  NN  |->  <.
( ( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
23 elmapi 7030 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2423ad2antrl 709 . . . . 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 ) ) ) ) )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
25 simprrl 741 . . . . 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 ) ) ) ) )  ->  A  C_ 
U. ran  ( (,)  o.  f ) )
26 simplr 732 . . . . 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 ) ) ) ) )  ->  y  e.  RR+ )
27 simprrr 742 . . . . 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 ) ) ) ) )  ->  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol * `  A )  +  ( C  x.  y ) ) )
2811, 12, 14, 15, 8, 22, 24, 25, 26, 27ovolscalem1 19401 . . . 4  |-  ( ( ( 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 ) )
2910, 28rexlimddv 2826 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol * `
 B )  <_ 
( ( ( vol
* `  A )  /  C )  +  y ) )
3029ralrimiva 2781 . 2  |-  ( ph  ->  A. y  e.  RR+  ( vol * `  B
)  <_  ( (
( vol * `  A )  /  C
)  +  y ) )
31 ssrab2 3420 . . . . 5  |-  { x  e.  RR  |  ( C  x.  x )  e.  A }  C_  RR
3213, 31syl6eqss 3390 . . . 4  |-  ( ph  ->  B  C_  RR )
33 ovolcl 19366 . . . 4  |-  ( B 
C_  RR  ->  ( vol
* `  B )  e.  RR* )
3432, 33syl 16 . . 3  |-  ( ph  ->  ( vol * `  B )  e.  RR* )
353, 5rerpdivcld 10667 . . 3  |-  ( ph  ->  ( ( vol * `  A )  /  C
)  e.  RR )
36 xralrple 10783 . . 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 ) ) )
3734, 35, 36syl2anc 643 . 2  |-  ( ph  ->  ( ( vol * `  B )  <_  (
( vol * `  A )  /  C
)  <->  A. y  e.  RR+  ( vol * `  B
)  <_  ( (
( vol * `  A )  /  C
)  +  y ) ) )
3830, 37mpbird 224 1  |-  ( ph  ->  ( vol * `  B )  <_  (
( vol * `  A )  /  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701    i^i cin 3311    C_ wss 3312   <.cop 3809   U.cuni 4007   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   ran crn 4871    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340    ^m cmap 7010   supcsup 7437   RRcr 8981   1c1 8983    + caddc 8985    x. cmul 8987   RR*cxr 9111    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   RR+crp 10604   (,)cioo 10908    seq cseq 11315   abscabs 12031   vol
*covol 19351
This theorem is referenced by:  ovolsca  19403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-ioo 10912  df-ico 10914  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ovol 19353
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