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

Theorem ovolsslem 21763
Description: Lemma for ovolss 21764. (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 3516 . . . . . . . . 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 2941 . . . . . 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 3586 . . . . 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 3550 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  N  C_  M )
9 sstr 3517 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A  C_  RR )
107ovolval 21753 . . . . . . . 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 3590 . . . . . . . . . 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 3539 . . . . . . . . 9  |-  M  C_  RR*
14 infmxrlb 11537 . . . . . . . . 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 4473 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  <_  x )
1817ralrimiva 2881 . . . . 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 3569 . . . 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 3590 . . . . 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 3539 . . . 4  |-  N  C_  RR*
24 ovolcl 21757 . . . . 5  |-  ( A 
C_  RR  ->  ( vol* `  A )  e.  RR* )
259, 24syl 16 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  e.  RR* )
26 infmxrgelb 11538 . . . 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 21753 . . 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 4479 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 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821    i^i cin 3480    C_ wss 3481   U.cuni 4251   class class class wbr 4453    X. cxp 5003   `'ccnv 5004   ran crn 5006    o. ccom 5009   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   supcsup 7912   RRcr 9503   1c1 9505    + caddc 9507   RR*cxr 9639    < clt 9640    <_ cle 9641    - cmin 9817   NNcn 10548   (,)cioo 11541    seqcseq 12087   abscabs 13047   vol*covol 21742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-ovol 21744
This theorem is referenced by:  ovolss  21764
  Copyright terms: Public domain W3C validator