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

Theorem ovolsslem 21761
Description: Lemma for ovolss 21762. (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 3493 . . . . . . . . 9  |-  ( A 
C_  B  ->  ( B  C_  U. ran  ( (,)  o.  f )  ->  A  C_  U. ran  ( (,)  o.  f ) ) )
21ad2antrr 725 . . . . . . . 8  |-  ( ( ( A  C_  B  /\  B  C_  RR )  /\  y  e.  RR* )  ->  ( B  C_  U.
ran  ( (,)  o.  f )  ->  A  C_ 
U. ran  ( (,)  o.  f ) ) )
32anim1d 564 . . . . . . 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 2915 . . . . . 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 3563 . . . . 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 3527 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  N  C_  M )
9 sstr 3494 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A  C_  RR )
107ovolval 21751 . . . . . . . 8  |-  ( A 
C_  RR  ->  ( vol* `  A )  =  sup ( M ,  RR* ,  `'  <  )
)
1110adantr 465 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  =  sup ( M ,  RR* ,  `'  <  ) )
12 ssrab2 3567 . . . . . . . . . 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 3516 . . . . . . . . 9  |-  M  C_  RR*
14 infmxrlb 11529 . . . . . . . . 9  |-  ( ( M  C_  RR*  /\  x  e.  M )  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1513, 14mpan 670 . . . . . . . 8  |-  ( x  e.  M  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1615adantl 466 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1711, 16eqbrtrd 4453 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  <_  x )
1817ralrimiva 2855 . . . . 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 3546 . . . 4  |-  ( N 
C_  M  ->  ( A. x  e.  M  ( vol* `  A
)  <_  x  ->  A. x  e.  N  ( vol* `  A
)  <_  x )
)
218, 19, 20sylc 60 . . 3  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A. x  e.  N  ( vol* `  A
)  <_  x )
22 ssrab2 3567 . . . . 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 3516 . . . 4  |-  N  C_  RR*
24 ovolcl 21755 . . . . 5  |-  ( A 
C_  RR  ->  ( vol* `  A )  e.  RR* )
259, 24syl 16 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  e.  RR* )
26 infmxrgelb 11530 . . . 4  |-  ( ( N  C_  RR*  /\  ( vol* `  A )  e.  RR* )  ->  (
( vol* `  A )  <_  sup ( N ,  RR* ,  `'  <  )  <->  A. x  e.  N  ( vol* `  A
)  <_  x )
)
2723, 25, 26sylancr 663 . . 3  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( ( vol* `  A )  <_  sup ( N ,  RR* ,  `'  <  )  <->  A. x  e.  N  ( vol* `  A
)  <_  x )
)
2821, 27mpbird 232 . 2  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  <_  sup ( N ,  RR* ,  `'  <  ) )
296ovolval 21751 . . 3  |-  ( B 
C_  RR  ->  ( vol* `  B )  =  sup ( N ,  RR* ,  `'  <  )
)
3029adantl 466 . 2  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  B )  =  sup ( N ,  RR* ,  `'  <  ) )
3128, 30breqtrrd 4459 1  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  <_  ( vol* `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   {crab 2795    i^i cin 3457    C_ wss 3458   U.cuni 4230   class class class wbr 4433    X. cxp 4983   `'ccnv 4984   ran crn 4986    o. ccom 4989   ` cfv 5574  (class class class)co 6277    ^m cmap 7418   supcsup 7898   RRcr 9489   1c1 9491    + caddc 9493   RR*cxr 9625    < clt 9626    <_ cle 9627    - cmin 9805   NNcn 10537   (,)cioo 11533    seqcseq 12081   abscabs 13041   vol*covol 21740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-ovol 21742
This theorem is referenced by:  ovolss  21762
  Copyright terms: Public domain W3C validator