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Theorem ovolmge0 18836
Description: The set  M is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypothesis
Ref Expression
ovolval.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* ,  <  ) ) }
Assertion
Ref Expression
ovolmge0  |-  ( B  e.  M  ->  0  <_  B )
Distinct variable groups:    y, f, A    B, f, y
Allowed substitution hints:    M( y, f)

Proof of Theorem ovolmge0
StepHypRef Expression
1 ovolval.1 . . 3  |-  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* ,  <  ) ) }
21elovolm 18834 . 2  |-  ( B  e.  M  <->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) )
3 reex 8828 . . . . . . . . 9  |-  RR  e.  _V
43, 3xpex 4801 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
54inex2 4156 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
6 nnex 9752 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 6796 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
8 eqid 2283 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  o.  f )  =  ( ( abs  o.  -  )  o.  f )
9 eqid 2283 . . . . . . . . . 10  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
108, 9ovolsf 18832 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,)  +oo ) )
11 1nn 9757 . . . . . . . . 9  |-  1  e.  NN
12 ffvelrn 5663 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) : NN --> ( 0 [,) 
+oo )  /\  1  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  ( 0 [,)  +oo ) )
1310, 11, 12sylancl 643 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  ( 0 [,)  +oo ) )
14 elrege0 10746 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) ` 
1 )  e.  ( 0 [,)  +oo )  <->  ( (  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) ` 
1 )  e.  RR  /\  0  <_  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
) ) )
1514simprbi 450 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) ` 
1 )  e.  ( 0 [,)  +oo )  ->  0  <_  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
) )
1613, 15syl 15 . . . . . . 7  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  0  <_  (  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) ` 
1 ) )
17 frn 5395 . . . . . . . . . 10  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,)  +oo )  ->  ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  C_  ( 0 [,)  +oo ) )
1810, 17syl 15 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  (
0 [,)  +oo ) )
19 icossxr 10734 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  RR*
2018, 19syl6ss 3191 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  RR* )
21 ffn 5389 . . . . . . . . . 10  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,)  +oo )  ->  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) )  Fn  NN )
2210, 21syl 15 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  Fn  NN )
23 fnfvelrn 5662 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  Fn  NN  /\  1  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) )
2422, 11, 23sylancl 643 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) )
25 supxrub 10643 . . . . . . . 8  |-  ( ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) )  C_  RR* 
/\  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) `  1 )  e.  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) )  ->  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) `  1 )  <_  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )
2620, 24, 25syl2anc 642 . . . . . . 7  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  <_  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) )
2719, 13sseldi 3178 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  RR* )
28 supxrcl 10633 . . . . . . . . 9  |-  ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  C_  RR* 
->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )
2920, 28syl 15 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  e. 
RR* )
30 0xr 8878 . . . . . . . . 9  |-  0  e.  RR*
31 xrletr 10489 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  RR*  /\  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  e. 
RR* )  ->  (
( 0  <_  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  /\  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  <_  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) )  ->  0  <_  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) )
3230, 31mp3an1 1264 . . . . . . . 8  |-  ( ( (  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) ` 
1 )  e.  RR*  /\ 
sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )  ->  ( ( 0  <_  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) `  1 )  /\  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) `  1 )  <_  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )  -> 
0  <_  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) )
3327, 29, 32syl2anc 642 . . . . . . 7  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( 0  <_  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  /\  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  <_  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) )  ->  0  <_  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) )
3416, 26, 33mp2and 660 . . . . . 6  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  0  <_  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )
357, 34sylbi 187 . . . . 5  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  0  <_  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )
36 breq2 4027 . . . . 5  |-  ( B  =  sup ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  )  -> 
( 0  <_  B  <->  0  <_  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) )
3735, 36syl5ibrcom 213 . . . 4  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  ( B  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  ->  0  <_  B ) )
3837adantld 453 . . 3  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  ( ( A 
C_  U. ran  ( (,) 
o.  f )  /\  B  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) )  ->  0  <_  B
) )
3938rexlimiv 2661 . 2  |-  ( E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )  -> 
0  <_  B )
402, 39sylbi 187 1  |-  ( B  e.  M  ->  0  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   U.cuni 3827   class class class wbr 4023    X. cxp 4687   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   (,)cioo 10656   [,)cico 10658    seq cseq 11046   abscabs 11719
This theorem is referenced by:  ovolge0  18840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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