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Theorem uniiccvol 22414
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 22384.) (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 3489 . . 3  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
3 uniioombl.3 . . . 4  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
43ovollb2 22320 . . 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 666 . 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 22413 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
8 ioossicc 11720 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  C_  (
( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) )
9 df-ov 6308 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
10 df-ov 6308 . . . . . . . . . . . 12  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
118, 9, 103sstr3i 3508 . . . . . . . . . . 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 3689 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
14 ffvelrn 6035 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3468 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6844 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  ( RR  X.  RR )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1715, 16syl 17 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  =  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1817fveq2d 5885 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  =  ( (,) `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
1917fveq2d 5885 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( [,] `  ( F `  x ) )  =  ( [,] `  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
2012, 18, 193sstr4d 3513 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( (,) `  ( F `  x ) )  C_  ( [,] `  ( F `
 x ) ) )
21 fvco3 5958 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
22 fvco3 5958 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
2320, 21, 223sstr4d 3513 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  C_  ( ( [,]  o.  F ) `  x
) )
241, 23sylan 473 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  C_  ( ( [,]  o.  F ) `  x
) )
2524ralrimiva 2846 . . . . . 6  |-  ( ph  ->  A. x  e.  NN  ( ( (,)  o.  F ) `  x
)  C_  ( ( [,]  o.  F ) `  x ) )
26 ss2iun 4318 . . . . . 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 17 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  C_  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x ) )
28 ioof 11732 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
29 ffn 5746 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3028, 29ax-mp 5 . . . . . . 7  |-  (,)  Fn  ( RR*  X.  RR* )
31 rexpssxrxp 9684 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
3213, 31sstri 3479 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
33 fss 5754 . . . . . . . 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 666 . . . . . . 7  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
35 fnfco 5765 . . . . . . 7  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
3630, 34, 35sylancr 667 . . . . . 6  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
37 fniunfv 6167 . . . . . 6  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  =  U. ran  ( (,)  o.  F
) )
3836, 37syl 17 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  =  U. ran  ( (,)  o.  F ) )
39 iccf 11733 . . . . . . . 8  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
40 ffn 5746 . . . . . . . 8  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4139, 40ax-mp 5 . . . . . . 7  |-  [,]  Fn  ( RR*  X.  RR* )
42 fnfco 5765 . . . . . . 7  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4341, 34, 42sylancr 667 . . . . . 6  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
44 fniunfv 6167 . . . . . 6  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x )  =  U. ran  ( [,]  o.  F
) )
4543, 44syl 17 . . . . 5  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U. ran  ( [,]  o.  F ) )
4627, 38, 453sstr3d 3512 . . . 4  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
47 ovolficcss 22301 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
481, 47syl 17 . . . 4  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  RR )
49 ovolss 22316 . . . 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 665 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  F ) )  <_ 
( vol* `  U. ran  ( [,]  o.  F ) ) )
517, 50eqbrtrrd 4448 . 2  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( vol* `  U. ran  ( [,]  o.  F ) ) )
52 ovolcl 22309 . . . 4  |-  ( U. ran  ( [,]  o.  F
)  C_  RR  ->  ( vol* `  U. ran  ( [,]  o.  F
) )  e.  RR* )
5348, 52syl 17 . . 3  |-  ( ph  ->  ( vol* `  U. ran  ( [,]  o.  F ) )  e. 
RR* )
54 eqid 2429 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
5554, 3ovolsf 22304 . . . . . . 7  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) +oo ) )
561, 55syl 17 . . . . . 6  |-  ( ph  ->  S : NN --> ( 0 [,) +oo ) )
57 frn 5752 . . . . . 6  |-  ( S : NN --> ( 0 [,) +oo )  ->  ran  S  C_  ( 0 [,) +oo ) )
5856, 57syl 17 . . . . 5  |-  ( ph  ->  ran  S  C_  (
0 [,) +oo )
)
59 icossxr 11719 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
6058, 59syl6ss 3482 . . . 4  |-  ( ph  ->  ran  S  C_  RR* )
61 supxrcl 11600 . . . 4  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
6260, 61syl 17 . . 3  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
63 xrletri3 11451 . . 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 665 . 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 930 1  |-  ( ph  ->  ( vol* `  U. ran  ( [,]  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    i^i cin 3441    C_ wss 3442   ~Pcpw 3985   <.cop 4008   U.cuni 4222   U_ciun 4302  Disj wdisj 4397   class class class wbr 4426    X. cxp 4852   ran crn 4855    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   supcsup 7960   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859   NNcn 10609   (,)cioo 11635   [,)cico 11637   [,]cicc 11638    seqcseq 12210   abscabs 13276   vol*covol 22294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-rest 15280  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-top 19852  df-bases 19853  df-topon 19854  df-cmp 20333  df-ovol 22296  df-vol 22297
This theorem is referenced by:  mblfinlem2  31682
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