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Theorem ovollb2 20977
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 20967). (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 20958 . . . . . . . 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 3371 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  A  C_  RR )
5 ovolcl 20966 . . . . . 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 11115 . . . 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 4323 . 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 2443 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
13 ovollb2.1 . . . . . . . . 9  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
1412, 13ovolsf 20961 . . . . . . . 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 5570 . . . . . . 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 9391 . . . . . . 7  |-  0  e.  RR
19 pnfxr 11097 . . . . . . 7  |- +oo  e.  RR*
20 icossre 11381 . . . . . . 7  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
2118, 19, 20mp2an 672 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
2217, 21syl6ss 3373 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  C_  RR )
23 1nn 10338 . . . . . . . 8  |-  1  e.  NN
24 fdm 5568 . . . . . . . . 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 2530 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  -> 
1  e.  dom  S
)
27 ne0i 3648 . . . . . . 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 5061 . . . . . . 7  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
3029necon3bii 2645 . . . . . 6  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
3128, 30sylib 196 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( [,]  o.  F ) )  ->  ran  S  =/=  (/) )
32 supxrre2 11299 . . . . 5  |-  ( ( ran  S  C_  RR  /\ 
ran  S  =/=  (/) )  -> 
( sup ( ran 
S ,  RR* ,  <  )  e.  RR  <->  sup ( ran  S ,  RR* ,  <  )  =/= +oo ) )
3322, 31, 32syl2anc 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 ) )
3433biimpar 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 )
35 fveq2 5696 . . . . . . . . . 10  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3635fveq2d 5700 . . . . . . . . 9  |-  ( m  =  n  ->  ( 1st `  ( F `  m ) )  =  ( 1st `  ( F `  n )
) )
37 oveq2 6104 . . . . . . . . . 10  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
3837oveq2d 6112 . . . . . . . . 9  |-  ( m  =  n  ->  (
( x  /  2
)  /  ( 2 ^ m ) )  =  ( ( x  /  2 )  / 
( 2 ^ n
) ) )
3936, 38oveq12d 6114 . . . . . . . 8  |-  ( m  =  n  ->  (
( 1st `  ( F `  m )
)  -  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 1st `  ( F `  n
) )  -  (
( x  /  2
)  /  ( 2 ^ n ) ) ) )
4035fveq2d 5700 . . . . . . . . 9  |-  ( m  =  n  ->  ( 2nd `  ( F `  m ) )  =  ( 2nd `  ( F `  n )
) )
4140, 38oveq12d 6114 . . . . . . . 8  |-  ( m  =  n  ->  (
( 2nd `  ( F `  m )
)  +  ( ( x  /  2 )  /  ( 2 ^ m ) ) )  =  ( ( 2nd `  ( F `  n
) )  +  ( ( x  /  2
)  /  ( 2 ^ n ) ) ) )
4239, 41opeq12d 4072 . . . . . . 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 4388 . . . . . 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 2443 . . . . . 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 757 . . . . . 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 758 . . . . . 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 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+ )
48 simplr 754 . . . . . 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 20976 . . . . 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 2804 . . . 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 11180 . . . . 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 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 ) ) )
5350, 52mpbird 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* ,  <  ) )
5434, 53syldan 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* ,  <  ) )
5511, 54pm2.61dane 2694 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 1369    e. wcel 1756    =/= wne 2611   A.wral 2720    i^i cin 3332    C_ wss 3333   (/)c0 3642   <.cop 3888   U.cuni 4096   class class class wbr 4297    e. cmpt 4355    X. cxp 4843   dom cdm 4845   ran crn 4846    o. ccom 4849   -->wf 5419   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   supcsup 7695   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290   +oocpnf 9420   RR*cxr 9422    < clt 9423    <_ cle 9424    - cmin 9600    / cdiv 9998   NNcn 10327   2c2 10376   RR+crp 10996   [,)cico 11307   [,]cicc 11308    seqcseq 11811   ^cexp 11870   abscabs 12728   vol*covol 20951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-ioo 11309  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-ovol 20953
This theorem is referenced by:  ovolctb  20978  ovolicc1  21004  ioombl1lem4  21047  uniiccvol  21065
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