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Theorem ovollb2 22025
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 22015). (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 461 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  U. ran  ( [,]  o.  F ) )
2 ovolficcss 22006 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
32adantr 465 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  U. ran  ( [,]  o.  F )  C_  RR )
41, 3sstrd 3509 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  RR )
5 ovolcl 22014 . . . . . 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 465 . . . 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 11364 . . . 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 461 . . 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 4482 . 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 2457 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
13 ovollb2.1 . . . . . . . . 9  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
1412, 13ovolsf 22009 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) +oo ) )
1514adantr 465 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  S : NN --> ( 0 [,) +oo ) )
16 frn 5743 . . . . . . 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 rge0ssre 11653 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
1917, 18syl6ss 3511 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  C_  RR )
20 1nn 10567 . . . . . . . 8  |-  1  e.  NN
21 fdm 5741 . . . . . . . . 9  |-  ( S : NN --> ( 0 [,) +oo )  ->  dom  S  =  NN )
2215, 21syl 16 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  dom  S  =  NN )
2320, 22syl5eleqr 2552 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
1  e.  dom  S
)
24 ne0i 3799 . . . . . . 7  |-  ( 1  e.  dom  S  ->  dom  S  =/=  (/) )
2523, 24syl 16 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  dom  S  =/=  (/) )
26 dm0rn0 5229 . . . . . . 7  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
2726necon3bii 2725 . . . . . 6  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
2825, 27sylib 196 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  =/=  (/) )
29 supxrre2 11548 . . . . 5  |-  ( ( ran  S  C_  RR  /\ 
ran  S  =/=  (/) )  -> 
( sup ( ran 
S ,  RR* ,  <  )  e.  RR  <->  sup ( ran  S ,  RR* ,  <  )  =/= +oo ) )
3019, 28, 29syl2anc 661 . . . 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 ) )
3130biimpar 485 . . 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 )
32 fveq2 5872 . . . . . . . . . 10  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3332fveq2d 5876 . . . . . . . . 9  |-  ( m  =  n  ->  ( 1st `  ( F `  m ) )  =  ( 1st `  ( F `  n )
) )
34 oveq2 6304 . . . . . . . . . 10  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
3534oveq2d 6312 . . . . . . . . 9  |-  ( m  =  n  ->  (
( x  /  2
)  /  ( 2 ^ m ) )  =  ( ( x  /  2 )  / 
( 2 ^ n
) ) )
3633, 35oveq12d 6314 . . . . . . . 8  |-  ( m  =  n  ->  (
( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 1st `  ( F `  n
) )  -  (
( x  /  2
)  /  ( 2 ^ n ) ) ) )
3732fveq2d 5876 . . . . . . . . 9  |-  ( m  =  n  ->  ( 2nd `  ( F `  m ) )  =  ( 2nd `  ( F `  n )
) )
3837, 35oveq12d 6314 . . . . . . . 8  |-  ( m  =  n  ->  (
( 2nd `  ( F `  m )
)  +  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 2nd `  ( F `  n
) )  +  ( ( x  /  2
)  /  ( 2 ^ n ) ) ) )
3936, 38opeq12d 4227 . . . . . . 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 ) ) ) >. )
4039cbvmptv 4548 . . . . . 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 ) ) ) >. )
41 eqid 2457 . . . . . 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 ) ) ) >. ) ) )
42 simplll 759 . . . . . 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 ) ) )
43 simpllr 760 . . . . . 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 ) )
44 simpr 461 . . . . . 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+ )
45 simplr 755 . . . . . 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 )
4613, 40, 41, 42, 43, 44, 45ovollb2lem 22024 . . . . 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 ) )
4746ralrimiva 2871 . . . 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 ) )
48 xralrple 11429 . . . . 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 ) ) )
496, 48sylan 471 . . . 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 ) ) )
5047, 49mpbird 232 . . 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* ,  <  ) )
5131, 50syldan 470 . 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* ,  <  ) )
5211, 51pm2.61dane 2775 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
( vol* `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    i^i cin 3470    C_ wss 3471   (/)c0 3793   <.cop 4038   U.cuni 4251   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   supcsup 7918   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   RR+crp 11245   [,)cico 11556   [,]cicc 11557    seqcseq 12109   ^cexp 12168   abscabs 13078   vol*covol 21999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-ovol 22001
This theorem is referenced by:  ovolctb  22026  ovolicc1  22052  ioombl1lem4  22096  uniiccvol  22114
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