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

Theorem ovolsslem 22379
Description: Lemma for ovolss 22380. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shofrtened by AV, 17-Sep-2020.)
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 3414 . . . . . . . . 9  |-  ( A 
C_  B  ->  ( B  C_  U. ran  ( (,)  o.  f )  ->  A  C_  U. ran  ( (,)  o.  f ) ) )
21ad2antrr 730 . . . . . . . 8  |-  ( ( ( A  C_  B  /\  B  C_  RR )  /\  y  e.  RR* )  ->  ( B  C_  U.
ran  ( (,)  o.  f )  ->  A  C_ 
U. ran  ( (,)  o.  f ) ) )
32anim1d 566 . . . . . . 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 2838 . . . . . 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 3485 . . . . 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 3448 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  N  C_  M )
9 sstr 3415 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A  C_  RR )
107ovolval 22368 . . . . . . . 8  |-  ( A 
C_  RR  ->  ( vol* `  A )  = inf ( M ,  RR* ,  <  ) )
1110adantr 466 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  = inf ( M ,  RR* ,  <  ) )
12 ssrab2 3489 . . . . . . . . . 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 3437 . . . . . . . . 9  |-  M  C_  RR*
14 infxrlb 11571 . . . . . . . . 9  |-  ( ( M  C_  RR*  /\  x  e.  M )  -> inf ( M ,  RR* ,  <  )  <_  x )
1513, 14mpan 674 . . . . . . . 8  |-  ( x  e.  M  -> inf ( M ,  RR* ,  <  )  <_  x )
1615adantl 467 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  M )  -> inf ( M ,  RR* ,  <  )  <_  x )
1711, 16eqbrtrd 4387 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  M )  ->  ( vol* `  A )  <_  x )
1817ralrimiva 2779 . . . . 5  |-  ( A 
C_  RR  ->  A. x  e.  M  ( vol* `  A )  <_  x )
199, 18syl 17 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A. x  e.  M  ( vol* `  A
)  <_  x )
20 ssralv 3468 . . . 4  |-  ( N 
C_  M  ->  ( A. x  e.  M  ( vol* `  A
)  <_  x  ->  A. x  e.  N  ( vol* `  A
)  <_  x )
)
218, 19, 20sylc 62 . . 3  |-  ( ( A  C_  B  /\  B  C_  RR )  ->  A. x  e.  N  ( vol* `  A
)  <_  x )
22 ssrab2 3489 . . . . 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 3437 . . . 4  |-  N  C_  RR*
24 ovolcl 22373 . . . . 5  |-  ( A 
C_  RR  ->  ( vol* `  A )  e.  RR* )
259, 24syl 17 . . . 4  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  e.  RR* )
26 infxrgelb 11572 . . . 4  |-  ( ( N  C_  RR*  /\  ( vol* `  A )  e.  RR* )  ->  (
( vol* `  A )  <_ inf ( N ,  RR* ,  <  )  <->  A. x  e.  N  ( vol* `  A
)  <_  x )
)
2723, 25, 26sylancr 667 . . 3  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( ( vol* `  A )  <_ inf ( N ,  RR* ,  <  )  <->  A. x  e.  N  ( vol* `  A
)  <_  x )
)
2821, 27mpbird 235 . 2  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  <_ inf ( N ,  RR* ,  <  )
)
296ovolval 22368 . . 3  |-  ( B 
C_  RR  ->  ( vol* `  B )  = inf ( N ,  RR* ,  <  ) )
3029adantl 467 . 2  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  B )  = inf ( N ,  RR* ,  <  ) )
3128, 30breqtrrd 4393 1  |-  ( ( A  C_  B  /\  B  C_  RR )  -> 
( vol* `  A )  <_  ( vol* `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2714   E.wrex 2715   {crab 2718    i^i cin 3378    C_ wss 3379   U.cuni 4162   class class class wbr 4366    X. cxp 4794   ran crn 4797    o. ccom 4800   ` cfv 5544  (class class class)co 6249    ^m cmap 7427   supcsup 7907  infcinf 7908   RRcr 9489   1c1 9491    + caddc 9493   RR*cxr 9625    < clt 9626    <_ cle 9627    - cmin 9811   NNcn 10560   (,)cioo 11586    seqcseq 12163   abscabs 13241   vol*covol 22355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  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 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-ovol 22358
This theorem is referenced by:  ovolss  22380
  Copyright terms: Public domain W3C validator