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Theorem ovollb2 18848
Description: It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 18838). (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypothesis
Ref Expression
ovollb2.1  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
Assertion
Ref Expression
ovollb2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )

Proof of Theorem ovollb2
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  U. ran  ( [,]  o.  F ) )
2 ovolficcss 18829 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
32adantr 451 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  U. ran  ( [,]  o.  F )  C_  RR )
41, 3sstrd 3189 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  RR )
5 ovolcl 18837 . . . . . 6  |-  ( A 
C_  RR  ->  ( vol
* `  A )  e.  RR* )
64, 5syl 15 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
( vol * `  A )  e.  RR* )
76adantr 451 . . . 4  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  =  +oo )  -> 
( vol * `  A )  e.  RR* )
8 pnfge 10469 . . . 4  |-  ( ( vol * `  A
)  e.  RR*  ->  ( vol * `  A
)  <_  +oo )
97, 8syl 15 . . 3  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  =  +oo )  -> 
( vol * `  A )  <_  +oo )
10 simpr 447 . . 3  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  =  +oo )  ->  sup ( ran  S ,  RR* ,  <  )  = 
+oo )
119, 10breqtrrd 4049 . 2  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  =  +oo )  -> 
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
12 eqid 2283 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
13 ovollb2.1 . . . . . . . . 9  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
1412, 13ovolsf 18832 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) 
+oo ) )
1514adantr 451 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  S : NN --> ( 0 [,)  +oo ) )
16 frn 5395 . . . . . . 7  |-  ( S : NN --> ( 0 [,)  +oo )  ->  ran  S 
C_  ( 0 [,) 
+oo ) )
1715, 16syl 15 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  C_  ( 0 [,)  +oo ) )
18 0re 8838 . . . . . . 7  |-  0  e.  RR
19 pnfxr 10455 . . . . . . 7  |-  +oo  e.  RR*
20 icossre 10730 . . . . . . 7  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
2118, 19, 20mp2an 653 . . . . . 6  |-  ( 0 [,)  +oo )  C_  RR
2217, 21syl6ss 3191 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  C_  RR )
23 1nn 9757 . . . . . . . 8  |-  1  e.  NN
24 fdm 5393 . . . . . . . . 9  |-  ( S : NN --> ( 0 [,)  +oo )  ->  dom  S  =  NN )
2515, 24syl 15 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  dom  S  =  NN )
2623, 25syl5eleqr 2370 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
1  e.  dom  S
)
27 ne0i 3461 . . . . . . 7  |-  ( 1  e.  dom  S  ->  dom  S  =/=  (/) )
2826, 27syl 15 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  dom  S  =/=  (/) )
29 dm0rn0 4895 . . . . . . 7  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
3029necon3bii 2478 . . . . . 6  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
3128, 30sylib 188 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  =/=  (/) )
32 supxrre2 10650 . . . . 5  |-  ( ( ran  S  C_  RR  /\ 
ran  S  =/=  (/) )  -> 
( sup ( ran 
S ,  RR* ,  <  )  e.  RR  <->  sup ( ran  S ,  RR* ,  <  )  =/=  +oo ) )
3322, 31, 32syl2anc 642 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
( sup ( ran 
S ,  RR* ,  <  )  e.  RR  <->  sup ( ran  S ,  RR* ,  <  )  =/=  +oo ) )
3433biimpar 471 . . 3  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  =/=  +oo )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR )
35 fveq2 5525 . . . . . . . . . 10  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3635fveq2d 5529 . . . . . . . . 9  |-  ( m  =  n  ->  ( 1st `  ( F `  m ) )  =  ( 1st `  ( F `  n )
) )
37 oveq2 5866 . . . . . . . . . 10  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
3837oveq2d 5874 . . . . . . . . 9  |-  ( m  =  n  ->  (
( x  /  2
)  /  ( 2 ^ m ) )  =  ( ( x  /  2 )  / 
( 2 ^ n
) ) )
3936, 38oveq12d 5876 . . . . . . . 8  |-  ( m  =  n  ->  (
( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 1st `  ( F `  n
) )  -  (
( x  /  2
)  /  ( 2 ^ n ) ) ) )
4035fveq2d 5529 . . . . . . . . 9  |-  ( m  =  n  ->  ( 2nd `  ( F `  m ) )  =  ( 2nd `  ( F `  n )
) )
4140, 38oveq12d 5876 . . . . . . . 8  |-  ( m  =  n  ->  (
( 2nd `  ( F `  m )
)  +  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 2nd `  ( F `  n
) )  +  ( ( x  /  2
)  /  ( 2 ^ n ) ) ) )
4239, 41opeq12d 3804 . . . . . . 7  |-  ( m  =  n  ->  <. (
( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) ) ,  ( ( 2nd `  ( F `  m
) )  +  ( ( x  /  2
)  /  ( 2 ^ m ) ) ) >.  =  <. ( ( 1st `  ( F `  n )
)  -  ( ( x  /  2 )  /  ( 2 ^ n ) ) ) ,  ( ( 2nd `  ( F `  n
) )  +  ( ( x  /  2
)  /  ( 2 ^ n ) ) ) >. )
4342cbvmptv 4111 . . . . . 6  |-  ( m  e.  NN  |->  <. (
( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) ) ,  ( ( 2nd `  ( F `  m
) )  +  ( ( x  /  2
)  /  ( 2 ^ m ) ) ) >. )  =  ( n  e.  NN  |->  <.
( ( 1st `  ( F `  n )
)  -  ( ( x  /  2 )  /  ( 2 ^ n ) ) ) ,  ( ( 2nd `  ( F `  n
) )  +  ( ( x  /  2
)  /  ( 2 ^ n ) ) ) >. )
44 eqid 2283 . . . . . 6  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  ( m  e.  NN  |->  <. ( ( 1st `  ( F `  m
) )  -  (
( x  /  2
)  /  ( 2 ^ m ) ) ) ,  ( ( 2nd `  ( F `
 m ) )  +  ( ( x  /  2 )  / 
( 2 ^ m
) ) ) >.
) ) )  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  ( m  e.  NN  |->  <. ( ( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) ) ,  ( ( 2nd `  ( F `  m
) )  +  ( ( x  /  2
)  /  ( 2 ^ m ) ) ) >. ) ) )
45 simplll 734 . . . . . 6  |-  ( ( ( ( F : NN
--> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran  S ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
46 simpllr 735 . . . . . 6  |-  ( ( ( ( F : NN
--> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran  S ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A  C_  U. ran  ( [,]  o.  F ) )
47 simpr 447 . . . . . 6  |-  ( ( ( ( F : NN
--> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran  S ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
48 simplr 731 . . . . . 6  |-  ( ( ( ( F : NN
--> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran  S ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR )
4913, 43, 44, 45, 46, 47, 48ovollb2lem 18847 . . . . 5  |-  ( ( ( ( F : NN
--> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran  S ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  -> 
( vol * `  A )  <_  ( sup ( ran  S ,  RR* ,  <  )  +  x ) )
5049ralrimiva 2626 . . . 4  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  e.  RR )  ->  A. x  e.  RR+  ( vol * `  A )  <_  ( sup ( ran  S ,  RR* ,  <  )  +  x ) )
51 xralrple 10532 . . . . 5  |-  ( ( ( vol * `  A )  e.  RR*  /\ 
sup ( ran  S ,  RR* ,  <  )  e.  RR )  ->  (
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  )  <->  A. x  e.  RR+  ( vol * `  A )  <_  ( sup ( ran  S ,  RR* ,  <  )  +  x ) ) )
526, 51sylan 457 . . . 4  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  e.  RR )  -> 
( ( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  )  <->  A. x  e.  RR+  ( vol * `  A )  <_  ( sup ( ran  S ,  RR* ,  <  )  +  x ) ) )
5350, 52mpbird 223 . . 3  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  e.  RR )  -> 
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
5434, 53syldan 456 . 2  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  =/=  +oo )  ->  ( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  )
)
5511, 54pm2.61dane 2524 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151    C_ wss 3152   (/)c0 3455   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   RR+crp 10354   [,)cico 10658   [,]cicc 10659    seq cseq 11046   ^cexp 11104   abscabs 11719   vol *covol 18822
This theorem is referenced by:  ovolctb  18849  ovolicc1  18875  ioombl1lem4  18918  uniiccvol  18935
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  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-int 3863  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-se 4353  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-isom 5264  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-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-q 10317  df-rp 10355  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ovol 18824
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