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Theorem ovolmge0 22308
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 22306 . 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 9629 . . . . . . . . 9  |-  RR  e.  _V
43, 3xpex 6609 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
54inex2 4567 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
6 nnex 10615 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 7508 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
8 eqid 2429 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  o.  f )  =  ( ( abs  o.  -  )  o.  f )
9 eqid 2429 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
108, 9ovolsf 22304 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo ) )
11 1nn 10620 . . . . . . . . 9  |-  1  e.  NN
12 ffvelrn 6035 . . . . . . . . 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 666 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  ( 0 [,) +oo ) )
14 elrege0 11737 . . . . . . . . 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 465 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) ` 
1 )  e.  ( 0 [,) +oo )  ->  0  <_  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
) )
1613, 15syl 17 . . . . . . 7  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  0  <_  (  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) ` 
1 ) )
17 frn 5752 . . . . . . . . . 10  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo )  ->  ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  C_  ( 0 [,) +oo ) )
1810, 17syl 17 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  (
0 [,) +oo )
)
19 icossxr 11719 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  RR*
2018, 19syl6ss 3482 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  RR* )
21 ffn 5746 . . . . . . . . . 10  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  Fn  NN )
2210, 21syl 17 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  Fn  NN )
23 fnfvelrn 6034 . . . . . . . . 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 666 . . . . . . . 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 11610 . . . . . . . 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 665 . . . . . . 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 3468 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  RR* )
28 supxrcl 11600 . . . . . . . . 9  |-  ( ran 
seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  C_  RR* 
->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )
2920, 28syl 17 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  e. 
RR* )
30 0xr 9686 . . . . . . . . 9  |-  0  e.  RR*
31 xrletr 11455 . . . . . . . . 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 1347 . . . . . . . 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 665 . . . . . . 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 683 . . . . . 6  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  0  <_  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )
357, 34sylbi 198 . . . . 5  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  0  <_  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )
36 breq2 4430 . . . . 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 225 . . . 4  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  ( B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  ->  0  <_  B ) )
3837adantld 468 . . 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 2918 . 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 198 1  |-  ( B  e.  M  ->  0  <_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   {crab 2786    i^i cin 3441    C_ wss 3442   U.cuni 4222   class class class wbr 4426    X. cxp 4852   ran crn 4855    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7480   supcsup 7960   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859   NNcn 10609   (,)cioo 11635   [,)cico 11637    seqcseq 12210   abscabs 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278
This theorem is referenced by:  ovolge0  22312
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