MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolsslem Structured version   Unicode version

Theorem ovolsslem 19382
Description: Lemma for ovolss 19383. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypotheses
Ref Expression
ovolss.1  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
ovolss.2  |-  N  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
Assertion
Ref Expression
ovolsslem  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol * `  A )  <_  ( vol * `  B ) )
Distinct variable groups:    y, f, A    B, f, y
Allowed substitution hints:    M( y, f)    N( y, f)

Proof of Theorem ovolsslem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sstr2 3357 . . . . . . . . 9  |-  ( A 
C_  B  ->  ( B  C_  U. ran  ( (,)  o.  f )  ->  A  C_  U. ran  ( (,)  o.  f ) ) )
21ad2antrr 708 . . . . . . . 8  |-  ( ( ( A  C_  B  /\  B  C_  RR )  /\  y  e.  RR* )  ->  ( B  C_  U.
ran  ( (,)  o.  f )  ->  A  C_ 
U. ran  ( (,)  o.  f ) ) )
32anim1d 549 . . . . . . 7  |-  ( ( ( A  C_  B  /\  B  C_  RR )  /\  y  e.  RR* )  ->  ( ( B 
C_  U. ran  ( (,) 
o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) )  ->  ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) ) )
43reximdv 2819 . . . . . 6  |-  ( ( ( A  C_  B  /\  B  C_  RR )  /\  y  e.  RR* )  ->  ( E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )  ->  E. f  e.  (
(  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) ) )
54ss2rabdv 3426 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }  C_  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) } )
6 ovolss.2 . . . . 5  |-  N  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
7 ovolss.1 . . . . 5  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
85, 6, 73sstr4g 3391 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  N  C_  M )
9 sstr 3358 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A  C_  RR )
107ovolval 19372 . . . . . . . 8  |-  ( A 
C_  RR  ->  ( vol
* `  A )  =  sup ( M ,  RR* ,  `'  <  )
)
1110adantr 453 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol * `  A )  =  sup ( M ,  RR* ,  `'  <  ) )
12 ssrab2 3430 . . . . . . . . . 10  |-  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) } 
C_  RR*
137, 12eqsstri 3380 . . . . . . . . 9  |-  M  C_  RR*
14 infmxrlb 10914 . . . . . . . . 9  |-  ( ( M  C_  RR*  /\  x  e.  M )  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1513, 14mpan 653 . . . . . . . 8  |-  ( x  e.  M  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1615adantl 454 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1711, 16eqbrtrd 4234 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol * `  A )  <_  x )
1817ralrimiva 2791 . . . . 5  |-  ( A 
C_  RR  ->  A. x  e.  M  ( vol * `
 A )  <_  x )
199, 18syl 16 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A. x  e.  M  ( vol * `  A
)  <_  x )
20 ssralv 3409 . . . 4  |-  ( N 
C_  M  ->  ( A. x  e.  M  ( vol * `  A
)  <_  x  ->  A. x  e.  N  ( vol * `  A
)  <_  x )
)
218, 19, 20sylc 59 . . 3  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A. x  e.  N  ( vol * `  A
)  <_  x )
22 ssrab2 3430 . . . . 5  |-  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) } 
C_  RR*
236, 22eqsstri 3380 . . . 4  |-  N  C_  RR*
24 ovolcl 19376 . . . . 5  |-  ( A 
C_  RR  ->  ( vol
* `  A )  e.  RR* )
259, 24syl 16 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol * `  A )  e.  RR* )
26 infmxrgelb 10915 . . . 4  |-  ( ( N  C_  RR*  /\  ( vol * `  A )  e.  RR* )  ->  (
( vol * `  A )  <_  sup ( N ,  RR* ,  `'  <  )  <->  A. x  e.  N  ( vol * `  A
)  <_  x )
)
2723, 25, 26sylancr 646 . . 3  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( ( vol * `  A )  <_  sup ( N ,  RR* ,  `'  <  )  <->  A. x  e.  N  ( vol * `  A
)  <_  x )
)
2821, 27mpbird 225 . 2  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol * `  A )  <_  sup ( N ,  RR* ,  `'  <  ) )
296ovolval 19372 . . 3  |-  ( B 
C_  RR  ->  ( vol
* `  B )  =  sup ( N ,  RR* ,  `'  <  )
)
3029adantl 454 . 2  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol * `  B )  =  sup ( N ,  RR* ,  `'  <  ) )
3128, 30breqtrrd 4240 1  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol * `  A )  <_  ( vol * `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711    i^i cin 3321    C_ wss 3322   U.cuni 4017   class class class wbr 4214    X. cxp 4878   `'ccnv 4879   ran crn 4881    o. ccom 4884   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   supcsup 7447   RRcr 8991   1c1 8993    + caddc 8995   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   NNcn 10002   (,)cioo 10918    seq cseq 11325   abscabs 12041   vol
*covol 19361
This theorem is referenced by:  ovolss  19383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-ovol 19363
  Copyright terms: Public domain W3C validator