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Theorem ovollb2 19385
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 19375). (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 19366 . . . . . . . 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 3358 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  RR )
5 ovolcl 19374 . . . . . 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 10727 . . . 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 4238 . 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 2436 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
13 ovollb2.1 . . . . . . . . 9  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
1412, 13ovolsf 19369 . . . . . . . 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 5597 . . . . . . 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 9091 . . . . . . 7  |-  0  e.  RR
19 pnfxr 10713 . . . . . . 7  |-  +oo  e.  RR*
20 icossre 10991 . . . . . . 7  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
2118, 19, 20mp2an 654 . . . . . 6  |-  ( 0 [,)  +oo )  C_  RR
2217, 21syl6ss 3360 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  C_  RR )
23 1nn 10011 . . . . . . . 8  |-  1  e.  NN
24 fdm 5595 . . . . . . . . 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 2523 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
1  e.  dom  S
)
27 ne0i 3634 . . . . . . 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 5086 . . . . . . 7  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
3029necon3bii 2633 . . . . . 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 10910 . . . . 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 5728 . . . . . . . . . 10  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3635fveq2d 5732 . . . . . . . . 9  |-  ( m  =  n  ->  ( 1st `  ( F `  m ) )  =  ( 1st `  ( F `  n )
) )
37 oveq2 6089 . . . . . . . . . 10  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
3837oveq2d 6097 . . . . . . . . 9  |-  ( m  =  n  ->  (
( x  /  2
)  /  ( 2 ^ m ) )  =  ( ( x  /  2 )  / 
( 2 ^ n
) ) )
3936, 38oveq12d 6099 . . . . . . . 8  |-  ( m  =  n  ->  (
( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 1st `  ( F `  n
) )  -  (
( x  /  2
)  /  ( 2 ^ n ) ) ) )
4035fveq2d 5732 . . . . . . . . 9  |-  ( m  =  n  ->  ( 2nd `  ( F `  m ) )  =  ( 2nd `  ( F `  n )
) )
4140, 38oveq12d 6099 . . . . . . . 8  |-  ( m  =  n  ->  (
( 2nd `  ( F `  m )
)  +  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 2nd `  ( F `  n
) )  +  ( ( x  /  2
)  /  ( 2 ^ n ) ) ) )
4239, 41opeq12d 3992 . . . . . . 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 4300 . . . . . 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 2436 . . . . . 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 19384 . . . . 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 2789 . . . 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 10791 . . . . 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 2682 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    i^i cin 3319    C_ wss 3320   (/)c0 3628   <.cop 3817   U.cuni 4015   class class class wbr 4212    e. cmpt 4266    X. cxp 4876   dom cdm 4878   ran crn 4879    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   supcsup 7445   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   RR+crp 10612   [,)cico 10918   [,]cicc 10919    seq cseq 11323   ^cexp 11382   abscabs 12039   vol *covol 19359
This theorem is referenced by:  ovolctb  19386  ovolicc1  19412  ioombl1lem4  19455  uniiccvol  19472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-ovol 19361
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