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Theorem ovolmge0 18852
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 18850 . 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 8844 . . . . . . . . 9  |-  RR  e.  _V
43, 3xpex 4817 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
54inex2 4172 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
6 nnex 9768 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 6812 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
8 eqid 2296 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  o.  f )  =  ( ( abs  o.  -  )  o.  f )
9 eqid 2296 . . . . . . . . . 10  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
108, 9ovolsf 18848 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,)  +oo ) )
11 1nn 9773 . . . . . . . . 9  |-  1  e.  NN
12 ffvelrn 5679 . . . . . . . . 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 10762 . . . . . . . . 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 5411 . . . . . . . . . 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 10750 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  RR*
2018, 19syl6ss 3204 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  RR* )
21 ffn 5405 . . . . . . . . . 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 5678 . . . . . . . . 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 10659 . . . . . . . 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 3191 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  RR* )
28 supxrcl 10649 . . . . . . . . 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 8894 . . . . . . . . 9  |-  0  e.  RR*
31 xrletr 10505 . . . . . . . . 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 4043 . . . . 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 2674 . 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   U.cuni 3843   class class class wbr 4039    X. cxp 4703   ran crn 4706    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   (,)cioo 10672   [,)cico 10674    seq cseq 11062   abscabs 11735
This theorem is referenced by:  ovolge0  18856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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