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Theorem ovolscalem2 21019
Description: Lemma for ovolshft 21016. (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 465 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
3 ovolsca.4 . . . . . 6  |-  ( ph  ->  ( vol* `  A )  e.  RR )
43adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol* `  A )  e.  RR )
5 ovolsca.2 . . . . . 6  |-  ( ph  ->  C  e.  RR+ )
6 rpmulcl 11033 . . . . . 6  |-  ( ( C  e.  RR+  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
75, 6sylan 471 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
8 eqid 2443 . . . . . 6  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 20985 . . . . 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 1218 . . . 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 725 . . . . 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 725 . . . . 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 725 . . . . 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 725 . . . . 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 5712 . . . . . . . . 9  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
1716fveq2d 5716 . . . . . . . 8  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
1817oveq1d 6127 . . . . . . 7  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  /  C )  =  ( ( 1st `  ( f `  n
) )  /  C
) )
1916fveq2d 5716 . . . . . . . 8  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
2019oveq1d 6127 . . . . . . 7  |-  ( m  =  n  ->  (
( 2nd `  (
f `  m )
)  /  C )  =  ( ( 2nd `  ( f `  n
) )  /  C
) )
2118, 20opeq12d 4088 . . . . . 6  |-  ( m  =  n  ->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >.  =  <. (
( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
2221cbvmptv 4404 . . . . 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 7255 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2423ad2antrl 727 . . . . 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 763 . . . . 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 754 . . . . 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 764 . . . . 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 21018 . . . 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 2866 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol* `  B )  <_ 
( ( ( vol* `  A )  /  C )  +  y ) )
3029ralrimiva 2820 . 2  |-  ( ph  ->  A. y  e.  RR+  ( vol* `  B
)  <_  ( (
( vol* `  A )  /  C
)  +  y ) )
31 ssrab2 3458 . . . . 5  |-  { x  e.  RR  |  ( C  x.  x )  e.  A }  C_  RR
3213, 31syl6eqss 3427 . . . 4  |-  ( ph  ->  B  C_  RR )
33 ovolcl 20983 . . . 4  |-  ( B 
C_  RR  ->  ( vol* `  B )  e.  RR* )
3432, 33syl 16 . . 3  |-  ( ph  ->  ( vol* `  B )  e.  RR* )
353, 5rerpdivcld 11075 . . 3  |-  ( ph  ->  ( ( vol* `  A )  /  C
)  e.  RR )
36 xralrple 11196 . . 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 661 . 2  |-  ( ph  ->  ( ( vol* `  B )  <_  (
( vol* `  A )  /  C
)  <->  A. y  e.  RR+  ( vol* `  B
)  <_  ( (
( vol* `  A )  /  C
)  +  y ) ) )
3830, 37mpbird 232 1  |-  ( ph  ->  ( vol* `  B )  <_  (
( vol* `  A )  /  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740    i^i cin 3348    C_ wss 3349   <.cop 3904   U.cuni 4112   class class class wbr 4313    e. cmpt 4371    X. cxp 4859   ran crn 4862    o. ccom 4865   -->wf 5435   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597    ^m cmap 7235   supcsup 7711   RRcr 9302   1c1 9304    + caddc 9306    x. cmul 9308   RR*cxr 9438    < clt 9439    <_ cle 9440    - cmin 9616    / cdiv 10014   NNcn 10343   RR+crp 11012   (,)cioo 11321    seqcseq 11827   abscabs 12744   vol*covol 20968
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-ioo 11325  df-ico 11327  df-fz 11459  df-fzo 11570  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185  df-ovol 20970
This theorem is referenced by:  ovolsca  21020
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