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Theorem ovolsslem 20809
Description: Lemma for ovolss 20810. (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 3351 . . . . . . . . 9  |-  ( A 
C_  B  ->  ( B  C_  U. ran  ( (,)  o.  f )  ->  A  C_  U. ran  ( (,)  o.  f ) ) )
21ad2antrr 718 . . . . . . . 8  |-  ( ( ( A  C_  B  /\  B  C_  RR )  /\  y  e.  RR* )  ->  ( B  C_  U.
ran  ( (,)  o.  f )  ->  A  C_ 
U. ran  ( (,)  o.  f ) ) )
32anim1d 559 . . . . . . 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 2817 . . . . . 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 3421 . . . . 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 3385 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  N  C_  M )
9 sstr 3352 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A  C_  RR )
107ovolval 20799 . . . . . . . 8  |-  ( A 
C_  RR  ->  ( vol* `  A )  =  sup ( M ,  RR* ,  `'  <  )
)
1110adantr 462 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  =  sup ( M ,  RR* ,  `'  <  ) )
12 ssrab2 3425 . . . . . . . . . 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 3374 . . . . . . . . 9  |-  M  C_  RR*
14 infmxrlb 11284 . . . . . . . . 9  |-  ( ( M  C_  RR*  /\  x  e.  M )  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1513, 14mpan 663 . . . . . . . 8  |-  ( x  e.  M  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1615adantl 463 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  sup ( M ,  RR* ,  `'  <  )  <_  x )
1711, 16eqbrtrd 4300 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  <_  x )
1817ralrimiva 2789 . . . . 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 3404 . . . 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 3425 . . . . 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 3374 . . . 4  |-  N  C_  RR*
24 ovolcl 20803 . . . . 5  |-  ( A 
C_  RR  ->  ( vol* `  A )  e.  RR* )
259, 24syl 16 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  e.  RR* )
26 infmxrgelb 11285 . . . 4  |-  ( ( N  C_  RR*  /\  ( vol* `  A )  e.  RR* )  ->  (
( vol* `  A )  <_  sup ( N ,  RR* ,  `'  <  )  <->  A. x  e.  N  ( vol* `  A
)  <_  x )
)
2723, 25, 26sylancr 656 . . 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 20799 . . 3  |-  ( B 
C_  RR  ->  ( vol* `  B )  =  sup ( N ,  RR* ,  `'  <  )
)
3029adantl 463 . 2  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  B )  =  sup ( N ,  RR* ,  `'  <  ) )
3128, 30breqtrrd 4306 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 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   {crab 2709    i^i cin 3315    C_ wss 3316   U.cuni 4079   class class class wbr 4280    X. cxp 4825   `'ccnv 4826   ran crn 4828    o. ccom 4831   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   supcsup 7678   RRcr 9269   1c1 9271    + caddc 9273   RR*cxr 9405    < clt 9406    <_ cle 9407    - cmin 9583   NNcn 10310   (,)cioo 11288    seqcseq 11790   abscabs 12707   vol*covol 20788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-ovol 20790
This theorem is referenced by:  ovolss  20810
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