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Theorem uniiccvol 21857
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 21832.) (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 3528 . . 3  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
3 uniioombl.3 . . . 4  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
43ovollb2 21768 . . 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 662 . 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 21856 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
8 ioossicc 11622 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  C_  (
( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) )
9 df-ov 6298 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
10 df-ov 6298 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
118, 9, 103sstr3i 3547 . . . . . . . . . . 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 3724 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
14 ffvelrn 6030 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3507 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6832 . . . . . . . . . . . 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 5876 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  =  ( (,) `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
1917fveq2d 5876 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( [,] `  ( F `  x ) )  =  ( [,] `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
2012, 18, 193sstr4d 3552 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  C_  ( [,] `  ( F `
 x ) ) )
21 fvco3 5951 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
22 fvco3 5951 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
2320, 21, 223sstr4d 3552 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  C_  ( ( [,]  o.  F ) `  x
) )
241, 23sylan 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  C_  ( ( [,]  o.  F ) `  x
) )
2524ralrimiva 2881 . . . . . 6  |-  ( ph  ->  A. x  e.  NN  ( ( (,)  o.  F ) `  x
)  C_  ( ( [,]  o.  F ) `  x ) )
26 ss2iun 4347 . . . . . 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 11634 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
29 ffn 5737 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3028, 29ax-mp 5 . . . . . . 7  |-  (,)  Fn  ( RR*  X.  RR* )
31 rexpssxrxp 9650 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
3213, 31sstri 3518 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
33 fss 5745 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
341, 32, 33sylancl 662 . . . . . . 7  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
35 fnfco 5756 . . . . . . 7  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
3630, 34, 35sylancr 663 . . . . . 6  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
37 fniunfv 6158 . . . . . 6  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  =  U. ran  ( (,)  o.  F
) )
3836, 37syl 16 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  =  U. ran  ( (,)  o.  F ) )
39 iccf 11635 . . . . . . . 8  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
40 ffn 5737 . . . . . . . 8  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4139, 40ax-mp 5 . . . . . . 7  |-  [,]  Fn  ( RR*  X.  RR* )
42 fnfco 5756 . . . . . . 7  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4341, 34, 42sylancr 663 . . . . . 6  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
44 fniunfv 6158 . . . . . 6  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x )  =  U. ran  ( [,]  o.  F
) )
4543, 44syl 16 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U. ran  ( [,]  o.  F ) )
4627, 38, 453sstr3d 3551 . . . 4  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
47 ovolficcss 21749 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
481, 47syl 16 . . . 4  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  RR )
49 ovolss 21764 . . . 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 ) ) )
5046, 48, 49syl2anc 661 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  F ) )  <_ 
( vol* `  U. ran  ( [,]  o.  F ) ) )
517, 50eqbrtrrd 4475 . 2  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( vol* `  U. ran  ( [,]  o.  F ) ) )
52 ovolcl 21757 . . . 4  |-  ( U. ran  ( [,]  o.  F
)  C_  RR  ->  ( vol* `  U. ran  ( [,]  o.  F
) )  e.  RR* )
5348, 52syl 16 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( [,]  o.  F ) )  e. 
RR* )
54 eqid 2467 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
5554, 3ovolsf 21752 . . . . . . 7  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) +oo ) )
561, 55syl 16 . . . . . 6  |-  ( ph  ->  S : NN --> ( 0 [,) +oo ) )
57 frn 5743 . . . . . 6  |-  ( S : NN --> ( 0 [,) +oo )  ->  ran  S  C_  ( 0 [,) +oo ) )
5856, 57syl 16 . . . . 5  |-  ( ph  ->  ran  S  C_  (
0 [,) +oo )
)
59 icossxr 11621 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
6058, 59syl6ss 3521 . . . 4  |-  ( ph  ->  ran  S  C_  RR* )
61 supxrcl 11518 . . . 4  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
6260, 61syl 16 . . 3  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
63 xrletri3 11370 . . 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 ) ) ) ) )
6453, 62, 63syl2anc 661 . 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 ) ) ) ) )
655, 51, 64mpbir2and 920 1  |-  ( ph  ->  ( vol* `  U. ran  ( [,]  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    i^i cin 3480    C_ wss 3481   ~Pcpw 4016   <.cop 4039   U.cuni 4251   U_ciun 4331  Disj wdisj 4423   class class class wbr 4453    X. cxp 5003   ran crn 5006    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   supcsup 7912   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641    - cmin 9817   NNcn 10548   (,)cioo 11541   [,)cico 11543   [,]cicc 11544    seqcseq 12087   abscabs 13047   vol*covol 21742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-rest 14695  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-cmp 19755  df-ovol 21744  df-vol 21745
This theorem is referenced by:  mblfinlem2  29970
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