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Theorem ovolsslem 21109
Description: Lemma for ovolss 21110. (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 3474 . . . . . . . . 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 2933 . . . . . 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 3544 . . . . 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 3508 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  N  C_  M )
9 sstr 3475 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A  C_  RR )
107ovolval 21099 . . . . . . . 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 3548 . . . . . . . . . 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 3497 . . . . . . . . 9  |-  M  C_  RR*
14 infmxrlb 11411 . . . . . . . . 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 4423 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  <_  x )
1817ralrimiva 2830 . . . . 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 3527 . . . 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 3548 . . . . 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 3497 . . . 4  |-  N  C_  RR*
24 ovolcl 21103 . . . . 5  |-  ( A 
C_  RR  ->  ( vol* `  A )  e.  RR* )
259, 24syl 16 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  e.  RR* )
26 infmxrgelb 11412 . . . 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 21099 . . 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 4429 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 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   {crab 2803    i^i cin 3438    C_ wss 3439   U.cuni 4202   class class class wbr 4403    X. cxp 4949   `'ccnv 4950   ran crn 4952    o. ccom 4955   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   supcsup 7805   RRcr 9396   1c1 9398    + caddc 9400   RR*cxr 9532    < clt 9533    <_ cle 9534    - cmin 9710   NNcn 10437   (,)cioo 11415    seqcseq 11927   abscabs 12845   vol*covol 21088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-ovol 21090
This theorem is referenced by:  ovolss  21110
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