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Theorem ovolscalem2 19270
Description: Lemma for ovolshft 19267. (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 10558 . . . . . 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 2380 . . . . . 6  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 19236 . . . . 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 5661 . . . . . . . . 9  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
1716fveq2d 5665 . . . . . . . 8  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
1817oveq1d 6028 . . . . . . 7  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  /  C )  =  ( ( 1st `  ( f `  n
) )  /  C
) )
1916fveq2d 5665 . . . . . . . 8  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
2019oveq1d 6028 . . . . . . 7  |-  ( m  =  n  ->  (
( 2nd `  (
f `  m )
)  /  C )  =  ( ( 2nd `  ( f `  n
) )  /  C
) )
2118, 20opeq12d 3927 . . . . . 6  |-  ( m  =  n  ->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >.  =  <. (
( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
2221cbvmptv 4234 . . . . 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 6967 . . . . . 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 19269 . . . 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 2770 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol * `
 B )  <_ 
( ( ( vol
* `  A )  /  C )  +  y ) )
3029ralrimiva 2725 . 2  |-  ( ph  ->  A. y  e.  RR+  ( vol * `  B
)  <_  ( (
( vol * `  A )  /  C
)  +  y ) )
31 ssrab2 3364 . . . . 5  |-  { x  e.  RR  |  ( C  x.  x )  e.  A }  C_  RR
3213, 31syl6eqss 3334 . . . 4  |-  ( ph  ->  B  C_  RR )
33 ovolcl 19234 . . . 4  |-  ( B 
C_  RR  ->  ( vol
* `  B )  e.  RR* )
3432, 33syl 16 . . 3  |-  ( ph  ->  ( vol * `  B )  e.  RR* )
353, 5rerpdivcld 10600 . . 3  |-  ( ph  ->  ( ( vol * `  A )  /  C
)  e.  RR )
36 xralrple 10716 . . 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 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   {crab 2646    i^i cin 3255    C_ wss 3256   <.cop 3753   U.cuni 3950   class class class wbr 4146    e. cmpt 4200    X. cxp 4809   ran crn 4812    o. ccom 4815   -->wf 5383   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280    ^m cmap 6947   supcsup 7373   RRcr 8915   1c1 8917    + caddc 8919    x. cmul 8921   RR*cxr 9045    < clt 9046    <_ cle 9047    - cmin 9216    / cdiv 9602   NNcn 9925   RR+crp 10537   (,)cioo 10841    seq cseq 11243   abscabs 11959   vol
*covol 19219
This theorem is referenced by:  ovolsca  19271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-ioo 10845  df-ico 10847  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-ovol 19221
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