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Theorem uniiccvol 21196
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 21171.) (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 3486 . . 3  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
3 uniioombl.3 . . . 4  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
43ovollb2 21107 . . 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 21195 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
8 ioossicc 11495 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  C_  (
( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) )
9 df-ov 6206 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
10 df-ov 6206 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
118, 9, 103sstr3i 3505 . . . . . . . . . . 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 3682 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
14 ffvelrn 5953 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3465 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6726 . . . . . . . . . . . 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 5806 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  =  ( (,) `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
1917fveq2d 5806 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( [,] `  ( F `  x ) )  =  ( [,] `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
2012, 18, 193sstr4d 3510 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  C_  ( [,] `  ( F `
 x ) ) )
21 fvco3 5880 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
22 fvco3 5880 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
2320, 21, 223sstr4d 3510 . . . . . . . 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 2830 . . . . . 6  |-  ( ph  ->  A. x  e.  NN  ( ( (,)  o.  F ) `  x
)  C_  ( ( [,]  o.  F ) `  x ) )
26 ss2iun 4297 . . . . . 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 11507 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
29 ffn 5670 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3028, 29ax-mp 5 . . . . . . 7  |-  (,)  Fn  ( RR*  X.  RR* )
31 rexpssxrxp 9542 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
3213, 31sstri 3476 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
33 fss 5678 . . . . . . . 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 5688 . . . . . . 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 6076 . . . . . 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 11508 . . . . . . . 8  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
40 ffn 5670 . . . . . . . 8  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4139, 40ax-mp 5 . . . . . . 7  |-  [,]  Fn  ( RR*  X.  RR* )
42 fnfco 5688 . . . . . . 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 6076 . . . . . 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 3509 . . . 4  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
47 ovolficcss 21088 . . . . 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 21103 . . . 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 4425 . 2  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( vol* `  U. ran  ( [,]  o.  F ) ) )
52 ovolcl 21096 . . . 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 2454 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
5554, 3ovolsf 21091 . . . . . . 7  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) +oo ) )
561, 55syl 16 . . . . . 6  |-  ( ph  ->  S : NN --> ( 0 [,) +oo ) )
57 frn 5676 . . . . . 6  |-  ( S : NN --> ( 0 [,) +oo )  ->  ran  S  C_  ( 0 [,) +oo ) )
5856, 57syl 16 . . . . 5  |-  ( ph  ->  ran  S  C_  (
0 [,) +oo )
)
59 icossxr 11494 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
6058, 59syl6ss 3479 . . . 4  |-  ( ph  ->  ran  S  C_  RR* )
61 supxrcl 11391 . . . 4  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
6260, 61syl 16 . . 3  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
63 xrletri3 11243 . . 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 913 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 1370    e. wcel 1758   A.wral 2799    i^i cin 3438    C_ wss 3439   ~Pcpw 3971   <.cop 3994   U.cuni 4202   U_ciun 4282  Disj wdisj 4373   class class class wbr 4403    X. cxp 4949   ran crn 4952    o. ccom 4955    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   supcsup 7804   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399   +oocpnf 9529   RR*cxr 9531    < clt 9532    <_ cle 9533    - cmin 9709   NNcn 10436   (,)cioo 11414   [,)cico 11416   [,]cicc 11417    seqcseq 11926   abscabs 12844   vol*covol 21081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-disj 4374  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fi 7775  df-sup 7805  df-oi 7838  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-ioo 11418  df-ico 11420  df-icc 11421  df-fz 11558  df-fzo 11669  df-fl 11762  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-rlim 13088  df-sum 13285  df-rest 14483  df-topgen 14504  df-psmet 17937  df-xmet 17938  df-met 17939  df-bl 17940  df-mopn 17941  df-top 18638  df-bases 18640  df-topon 18641  df-cmp 19125  df-ovol 21083  df-vol 21084
This theorem is referenced by:  mblfinlem2  28597
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