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Theorem ovolmge0 22367
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 22365 . 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 9576 . . . . . . . . 9  |-  RR  e.  _V
43, 3xpex 6548 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
54inex2 4504 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
6 nnex 10561 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 7450 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
8 eqid 2423 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  o.  f )  =  ( ( abs  o.  -  )  o.  f )
9 eqid 2423 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
108, 9ovolsf 22362 . . . . . . . . 9  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo ) )
11 1nn 10566 . . . . . . . . 9  |-  1  e.  NN
12 ffvelrn 5974 . . . . . . . . 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 11684 . . . . . . . . 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 5690 . . . . . . . . . 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 11665 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  RR*
2018, 19syl6ss 3414 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) )  C_  RR* )
21 ffn 5684 . . . . . . . . . 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 5973 . . . . . . . . 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 11556 . . . . . . . 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 3400 . . . . . . . 8  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) `  1
)  e.  RR* )
28 supxrcl 11546 . . . . . . . . 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 9633 . . . . . . . . 9  |-  0  e.  RR*
31 xrletr 11401 . . . . . . . . 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 4365 . . . . 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 2845 . 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 1872   E.wrex 2710   {crab 2713    i^i cin 3373    C_ wss 3374   U.cuni 4157   class class class wbr 4361    X. cxp 4789   ran crn 4792    o. ccom 4795    Fn wfn 5534   -->wf 5535   ` cfv 5539  (class class class)co 6244    ^m cmap 7422   supcsup 7902   RRcr 9484   0cc0 9485   1c1 9486    + caddc 9488   +oocpnf 9618   RR*cxr 9620    < clt 9621    <_ cle 9622    - cmin 9806   NNcn 10555   (,)cioo 11581   [,)cico 11583    seqcseq 12158   abscabs 13236
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 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563
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 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-map 7424  df-en 7520  df-dom 7521  df-sdom 7522  df-sup 7904  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-ico 11587  df-fz 11731  df-seq 12159  df-exp 12218  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238
This theorem is referenced by:  ovolge0  22371
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