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Theorem ovollb2 19338
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 19328). (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 448 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  U. ran  ( [,]  o.  F ) )
2 ovolficcss 19319 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
32adantr 452 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  U. ran  ( [,]  o.  F )  C_  RR )
41, 3sstrd 3318 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  RR )
5 ovolcl 19327 . . . . . 6  |-  ( A 
C_  RR  ->  ( vol
* `  A )  e.  RR* )
64, 5syl 16 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
( vol * `  A )  e.  RR* )
76adantr 452 . . . 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 10683 . . . 4  |-  ( ( vol * `  A
)  e.  RR*  ->  ( vol * `  A
)  <_  +oo )
97, 8syl 16 . . 3  |-  ( ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U. ran  ( [,]  o.  F ) )  /\  sup ( ran 
S ,  RR* ,  <  )  =  +oo )  -> 
( vol * `  A )  <_  +oo )
10 simpr 448 . . 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 4198 . 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 2404 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
13 ovollb2.1 . . . . . . . . 9  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
1412, 13ovolsf 19322 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) 
+oo ) )
1514adantr 452 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  S : NN --> ( 0 [,)  +oo ) )
16 frn 5556 . . . . . . 7  |-  ( S : NN --> ( 0 [,)  +oo )  ->  ran  S 
C_  ( 0 [,) 
+oo ) )
1715, 16syl 16 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  C_  ( 0 [,)  +oo ) )
18 0re 9047 . . . . . . 7  |-  0  e.  RR
19 pnfxr 10669 . . . . . . 7  |-  +oo  e.  RR*
20 icossre 10947 . . . . . . 7  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
2118, 19, 20mp2an 654 . . . . . 6  |-  ( 0 [,)  +oo )  C_  RR
2217, 21syl6ss 3320 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  C_  RR )
23 1nn 9967 . . . . . . . 8  |-  1  e.  NN
24 fdm 5554 . . . . . . . . 9  |-  ( S : NN --> ( 0 [,)  +oo )  ->  dom  S  =  NN )
2515, 24syl 16 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  dom  S  =  NN )
2623, 25syl5eleqr 2491 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
1  e.  dom  S
)
27 ne0i 3594 . . . . . . 7  |-  ( 1  e.  dom  S  ->  dom  S  =/=  (/) )
2826, 27syl 16 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  dom  S  =/=  (/) )
29 dm0rn0 5045 . . . . . . 7  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
3029necon3bii 2599 . . . . . 6  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
3128, 30sylib 189 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  =/=  (/) )
32 supxrre2 10866 . . . . 5  |-  ( ( ran  S  C_  RR  /\ 
ran  S  =/=  (/) )  -> 
( sup ( ran 
S ,  RR* ,  <  )  e.  RR  <->  sup ( ran  S ,  RR* ,  <  )  =/=  +oo ) )
3322, 31, 32syl2anc 643 . . . 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 472 . . 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 5687 . . . . . . . . . 10  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3635fveq2d 5691 . . . . . . . . 9  |-  ( m  =  n  ->  ( 1st `  ( F `  m ) )  =  ( 1st `  ( F `  n )
) )
37 oveq2 6048 . . . . . . . . . 10  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
3837oveq2d 6056 . . . . . . . . 9  |-  ( m  =  n  ->  (
( x  /  2
)  /  ( 2 ^ m ) )  =  ( ( x  /  2 )  / 
( 2 ^ n
) ) )
3936, 38oveq12d 6058 . . . . . . . 8  |-  ( m  =  n  ->  (
( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 1st `  ( F `  n
) )  -  (
( x  /  2
)  /  ( 2 ^ n ) ) ) )
4035fveq2d 5691 . . . . . . . . 9  |-  ( m  =  n  ->  ( 2nd `  ( F `  m ) )  =  ( 2nd `  ( F `  n )
) )
4140, 38oveq12d 6058 . . . . . . . 8  |-  ( m  =  n  ->  (
( 2nd `  ( F `  m )
)  +  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 2nd `  ( F `  n
) )  +  ( ( x  /  2
)  /  ( 2 ^ n ) ) ) )
4239, 41opeq12d 3952 . . . . . . 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 4260 . . . . . 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 2404 . . . . . 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 735 . . . . . 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 736 . . . . . 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 448 . . . . . 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 732 . . . . . 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 19337 . . . . 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 2749 . . . 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 10747 . . . . 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 458 . . . 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 224 . . 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 457 . 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 2645 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    i^i cin 3279    C_ wss 3280   (/)c0 3588   <.cop 3777   U.cuni 3975   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   dom cdm 4837   ran crn 4838    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   supcsup 7403   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   RR+crp 10568   [,)cico 10874   [,]cicc 10875    seq cseq 11278   ^cexp 11337   abscabs 11994   vol *covol 19312
This theorem is referenced by:  ovolctb  19339  ovolicc1  19365  ioombl1lem4  19408  uniiccvol  19425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-ovol 19314
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