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Theorem ovolmge0 21095
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 21093 . 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 9487 . . . . . . . . 9  |-  RR  e.  _V
43, 3xpex 6621 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
54inex2 4545 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
6 nnex 10442 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 7354 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
8 eqid 2454 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  o.  f )  =  ( ( abs  o.  -  )  o.  f )
9 eqid 2454 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
108, 9ovolsf 21091 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo ) )
11 1nn 10447 . . . . . . . . 9  |-  1  e.  NN
12 ffvelrn 5953 . . . . . . . . 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 662 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  ( 0 [,) +oo ) )
14 elrege0 11512 . . . . . . . . 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 464 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) ` 
1 )  e.  ( 0 [,) +oo )  ->  0  <_  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
) )
1613, 15syl 16 . . . . . . 7  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  0  <_  (  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) ` 
1 ) )
17 frn 5676 . . . . . . . . . 10  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo )  ->  ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  C_  ( 0 [,) +oo ) )
1810, 17syl 16 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  (
0 [,) +oo )
)
19 icossxr 11494 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  RR*
2018, 19syl6ss 3479 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  RR* )
21 ffn 5670 . . . . . . . . . 10  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  Fn  NN )
2210, 21syl 16 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  Fn  NN )
23 fnfvelrn 5952 . . . . . . . . 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 662 . . . . . . . 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 11401 . . . . . . . 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 661 . . . . . . 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 3465 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  RR* )
28 supxrcl 11391 . . . . . . . . 9  |-  ( ran 
seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  C_  RR* 
->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )
2920, 28syl 16 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  e. 
RR* )
30 0xr 9544 . . . . . . . . 9  |-  0  e.  RR*
31 xrletr 11246 . . . . . . . . 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 1302 . . . . . . . 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 661 . . . . . . 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 679 . . . . . 6  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  0  <_  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )
357, 34sylbi 195 . . . . 5  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  0  <_  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )
36 breq2 4407 . . . . 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 222 . . . 4  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  ( B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  ->  0  <_  B ) )
3837adantld 467 . . 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 2941 . 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 195 1  |-  ( B  e.  M  ->  0  <_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800   {crab 2803    i^i cin 3438    C_ wss 3439   U.cuni 4202   class class class wbr 4403    X. cxp 4949   ran crn 4952    o. ccom 4955    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   supcsup 7804   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399   +oocpnf 9529   RR*cxr 9531    < clt 9532    <_ cle 9533    - cmin 9709   NNcn 10436   (,)cioo 11414   [,)cico 11416    seqcseq 11926   abscabs 12844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-ico 11420  df-fz 11558  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846
This theorem is referenced by:  ovolge0  21099
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