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Theorem uniiccvol 21035
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 21010.) (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 3370 . . 3  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
3 uniioombl.3 . . . 4  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
43ovollb2 20947 . . 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 21034 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
8 ioossicc 11373 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  C_  (
( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) )
9 df-ov 6089 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
10 df-ov 6089 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
118, 9, 103sstr3i 3389 . . . . . . . . . . 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 3566 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
14 ffvelrn 5836 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3349 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6608 . . . . . . . . . . . 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 5690 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  =  ( (,) `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
1917fveq2d 5690 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( [,] `  ( F `  x ) )  =  ( [,] `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
2012, 18, 193sstr4d 3394 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  C_  ( [,] `  ( F `
 x ) ) )
21 fvco3 5763 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
22 fvco3 5763 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
2320, 21, 223sstr4d 3394 . . . . . . . 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 2794 . . . . . 6  |-  ( ph  ->  A. x  e.  NN  ( ( (,)  o.  F ) `  x
)  C_  ( ( [,]  o.  F ) `  x ) )
26 ss2iun 4181 . . . . . 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 11379 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
29 ffn 5554 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3028, 29ax-mp 5 . . . . . . 7  |-  (,)  Fn  ( RR*  X.  RR* )
31 rexpssxrxp 9420 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
3213, 31sstri 3360 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
33 fss 5562 . . . . . . . 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 5572 . . . . . . 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 5959 . . . . . 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 11380 . . . . . . . 8  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
40 ffn 5554 . . . . . . . 8  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4139, 40ax-mp 5 . . . . . . 7  |-  [,]  Fn  ( RR*  X.  RR* )
42 fnfco 5572 . . . . . . 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 5959 . . . . . 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 3393 . . . 4  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
47 ovolficcss 20928 . . . . 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 20943 . . . 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 4309 . 2  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( vol* `  U. ran  ( [,]  o.  F ) ) )
52 ovolcl 20936 . . . 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 2438 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
5554, 3ovolsf 20931 . . . . . . 7  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) +oo ) )
561, 55syl 16 . . . . . 6  |-  ( ph  ->  S : NN --> ( 0 [,) +oo ) )
57 frn 5560 . . . . . 6  |-  ( S : NN --> ( 0 [,) +oo )  ->  ran  S  C_  ( 0 [,) +oo ) )
5856, 57syl 16 . . . . 5  |-  ( ph  ->  ran  S  C_  (
0 [,) +oo )
)
59 icossxr 11372 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
6058, 59syl6ss 3363 . . . 4  |-  ( ph  ->  ran  S  C_  RR* )
61 supxrcl 11269 . . . 4  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
6260, 61syl 16 . . 3  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
63 xrletri3 11121 . . 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 1369    e. wcel 1756   A.wral 2710    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   <.cop 3878   U.cuni 4086   U_ciun 4166  Disj wdisj 4257   class class class wbr 4287    X. cxp 4833   ran crn 4836    o. ccom 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   supcsup 7682   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277   +oocpnf 9407   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587   NNcn 10314   (,)cioo 11292   [,)cico 11294   [,]cicc 11295    seqcseq 11798   abscabs 12715   vol*covol 20921
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-rest 14353  df-topgen 14374  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-top 18478  df-bases 18480  df-topon 18481  df-cmp 18965  df-ovol 20923  df-vol 20924
This theorem is referenced by:  mblfinlem2  28382
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