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Theorem uniiccvol 19472
Description: An almost-disjoint union of closed intervals (disjoint interiors) has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 19448.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Hypotheses
Ref Expression
uniioombl.1  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
uniioombl.2  |-  ( ph  -> Disj  x  e.  NN ( (,) `  ( F `  x ) ) )
uniioombl.3  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
Assertion
Ref Expression
uniiccvol  |-  ( ph  ->  ( vol * `  U. ran  ( [,]  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
Distinct variable groups:    x, F    ph, x
Allowed substitution hint:    S( x)

Proof of Theorem uniiccvol
StepHypRef Expression
1 uniioombl.1 . . 3  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2 ssid 3367 . . 3  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
3 uniioombl.3 . . . 4  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
43ovollb2 19385 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  U. ran  ( [,]  o.  F
)  C_  U. ran  ( [,]  o.  F ) )  ->  ( vol * `  U. ran  ( [,] 
o.  F ) )  <_  sup ( ran  S ,  RR* ,  <  )
)
51, 2, 4sylancl 644 . 2  |-  ( ph  ->  ( vol * `  U. ran  ( [,]  o.  F ) )  <_  sup ( ran  S ,  RR* ,  <  ) )
6 uniioombl.2 . . . 4  |-  ( ph  -> Disj  x  e.  NN ( (,) `  ( F `  x ) ) )
71, 6, 3uniioovol 19471 . . 3  |-  ( ph  ->  ( vol * `  U. ran  ( (,)  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
8 ioossicc 10996 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  C_  (
( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) )
9 df-ov 6084 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
10 df-ov 6084 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
118, 9, 103sstr3i 3386 . . . . . . . . . . 11  |-  ( (,) `  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )  C_  ( [,] `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. )
1211a1i 11 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )  C_  ( [,] `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
13 inss2 3562 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
14 ffvelrn 5868 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3346 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6386 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  ( RR  X.  RR )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1715, 16syl 16 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  =  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1817fveq2d 5732 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  =  ( (,) `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
1917fveq2d 5732 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( [,] `  ( F `  x ) )  =  ( [,] `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
2012, 18, 193sstr4d 3391 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  C_  ( [,] `  ( F `
 x ) ) )
21 fvco3 5800 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
22 fvco3 5800 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
2320, 21, 223sstr4d 3391 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  C_  ( ( [,]  o.  F ) `  x
) )
241, 23sylan 458 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  C_  ( ( [,]  o.  F ) `  x
) )
2524ralrimiva 2789 . . . . . 6  |-  ( ph  ->  A. x  e.  NN  ( ( (,)  o.  F ) `  x
)  C_  ( ( [,]  o.  F ) `  x ) )
26 ss2iun 4108 . . . . . 6  |-  ( A. x  e.  NN  (
( (,)  o.  F
) `  x )  C_  ( ( [,]  o.  F ) `  x
)  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  C_  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x ) )
2725, 26syl 16 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  C_  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x ) )
28 ioof 11002 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
29 ffn 5591 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3028, 29ax-mp 8 . . . . . . 7  |-  (,)  Fn  ( RR*  X.  RR* )
31 ressxr 9129 . . . . . . . . . 10  |-  RR  C_  RR*
32 xpss12 4981 . . . . . . . . . 10  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
3331, 31, 32mp2an 654 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
3413, 33sstri 3357 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
35 fss 5599 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
361, 34, 35sylancl 644 . . . . . . 7  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
37 fnfco 5609 . . . . . . 7  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
3830, 36, 37sylancr 645 . . . . . 6  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
39 fniunfv 5994 . . . . . 6  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  =  U. ran  ( (,)  o.  F
) )
4038, 39syl 16 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  =  U. ran  ( (,)  o.  F ) )
41 iccf 11003 . . . . . . . 8  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
42 ffn 5591 . . . . . . . 8  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4341, 42ax-mp 8 . . . . . . 7  |-  [,]  Fn  ( RR*  X.  RR* )
44 fnfco 5609 . . . . . . 7  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4543, 36, 44sylancr 645 . . . . . 6  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
46 fniunfv 5994 . . . . . 6  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x )  =  U. ran  ( [,]  o.  F
) )
4745, 46syl 16 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U. ran  ( [,]  o.  F ) )
4827, 40, 473sstr3d 3390 . . . 4  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
49 ovolficcss 19366 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
501, 49syl 16 . . . 4  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  RR )
51 ovolss 19381 . . . 4  |-  ( ( U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F )  /\  U. ran  ( [,]  o.  F
)  C_  RR )  ->  ( vol * `  U. ran  ( (,)  o.  F ) )  <_ 
( vol * `  U. ran  ( [,]  o.  F ) ) )
5248, 50, 51syl2anc 643 . . 3  |-  ( ph  ->  ( vol * `  U. ran  ( (,)  o.  F ) )  <_ 
( vol * `  U. ran  ( [,]  o.  F ) ) )
537, 52eqbrtrrd 4234 . 2  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( vol * `  U. ran  ( [,]  o.  F ) ) )
54 ovolcl 19374 . . . 4  |-  ( U. ran  ( [,]  o.  F
)  C_  RR  ->  ( vol * `  U. ran  ( [,]  o.  F
) )  e.  RR* )
5550, 54syl 16 . . 3  |-  ( ph  ->  ( vol * `  U. ran  ( [,]  o.  F ) )  e. 
RR* )
56 eqid 2436 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
5756, 3ovolsf 19369 . . . . . . 7  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) 
+oo ) )
581, 57syl 16 . . . . . 6  |-  ( ph  ->  S : NN --> ( 0 [,)  +oo ) )
59 frn 5597 . . . . . 6  |-  ( S : NN --> ( 0 [,)  +oo )  ->  ran  S 
C_  ( 0 [,) 
+oo ) )
6058, 59syl 16 . . . . 5  |-  ( ph  ->  ran  S  C_  (
0 [,)  +oo ) )
61 icossxr 10995 . . . . 5  |-  ( 0 [,)  +oo )  C_  RR*
6260, 61syl6ss 3360 . . . 4  |-  ( ph  ->  ran  S  C_  RR* )
63 supxrcl 10893 . . . 4  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
6462, 63syl 16 . . 3  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
65 xrletri3 10745 . . 3  |-  ( ( ( vol * `  U. ran  ( [,]  o.  F ) )  e. 
RR*  /\  sup ( ran  S ,  RR* ,  <  )  e.  RR* )  ->  (
( vol * `  U. ran  ( [,]  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )  <->  ( ( vol * `  U. ran  ( [,]  o.  F ) )  <_  sup ( ran  S ,  RR* ,  <  )  /\  sup ( ran  S ,  RR* ,  <  )  <_ 
( vol * `  U. ran  ( [,]  o.  F ) ) ) ) )
6655, 64, 65syl2anc 643 . 2  |-  ( ph  ->  ( ( vol * `  U. ran  ( [,] 
o.  F ) )  =  sup ( ran 
S ,  RR* ,  <  )  <-> 
( ( vol * `  U. ran  ( [,] 
o.  F ) )  <_  sup ( ran  S ,  RR* ,  <  )  /\  sup ( ran  S ,  RR* ,  <  )  <_  ( vol * `  U. ran  ( [,]  o.  F ) ) ) ) )
675, 53, 66mpbir2and 889 1  |-  ( ph  ->  ( vol * `  U. ran  ( [,]  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   <.cop 3817   U.cuni 4015   U_ciun 4093  Disj wdisj 4182   class class class wbr 4212    X. cxp 4876   ran crn 4879    o. ccom 4882    Fn wfn 5449   -->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   NNcn 10000   (,)cioo 10916   [,)cico 10918   [,]cicc 10919    seq cseq 11323   abscabs 12039   vol
*covol 19359
This theorem is referenced by:  mblfinlem2  26244
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-disj 4183  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-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  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-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  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-rlim 12283  df-sum 12480  df-rest 13650  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cmp 17450  df-ovol 19361  df-vol 19362
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