MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolficc Unicode version

Theorem ovolficc 18828
Description: Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolficc  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U. ran  ( [,]  o.  F )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
Distinct variable groups:    z, n, A    n, F, z

Proof of Theorem ovolficc
StepHypRef Expression
1 iccf 10742 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
2 inss2 3390 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ressxr 8876 . . . . . . . . 9  |-  RR  C_  RR*
4 xpss12 4792 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
53, 3, 4mp2an 653 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
62, 5sstri 3188 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
7 fss 5397 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
86, 7mpan2 652 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( RR*  X.  RR* ) )
9 fco 5398 . . . . . 6  |-  ( ( [,] : ( RR*  X. 
RR* ) --> ~P RR*  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( [,]  o.  F
) : NN --> ~P RR* )
101, 8, 9sylancr 644 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( [,]  o.  F ) : NN --> ~P RR* )
11 ffn 5389 . . . . 5  |-  ( ( [,]  o.  F ) : NN --> ~P RR*  ->  ( [,]  o.  F
)  Fn  NN )
12 fniunfv 5773 . . . . 5  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  =  U. ran  ( [,]  o.  F
) )
1310, 11, 123syl 18 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  =  U. ran  ( [,]  o.  F
) )
1413sseq2d 3206 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( A  C_  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  <->  A  C_  U. ran  ( [,]  o.  F ) ) )
1514adantl 452 . 2  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  <->  A  C_  U. ran  ( [,]  o.  F ) ) )
16 dfss3 3170 . . 3  |-  ( A 
C_  U_ n  e.  NN  ( ( [,]  o.  F ) `  n
)  <->  A. z  e.  A  z  e.  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n ) )
17 ssel2 3175 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
18 eliun 3909 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( [,]  o.  F
) `  n )
)
19 fvco3 5596 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( [,] `  ( F `  n )
) )
20 ffvelrn 5663 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
212, 20sseldi 3178 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
22 1st2nd2 6159 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( RR  X.  RR )  ->  ( F `
 n )  = 
<. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2321, 22syl 15 . . . . . . . . . . . . . . 15  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2423fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( [,] `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
25 df-ov 5861 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) [,] ( 2nd `  ( F `  n )
) )  =  ( [,] `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2624, 25syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) [,] ( 2nd `  ( F `  n
) ) ) )
2719, 26eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) [,] ( 2nd `  ( F `  n ) ) ) )
2827eleq2d 2350 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
z  e.  ( ( [,]  o.  F ) `
 n )  <->  z  e.  ( ( 1st `  ( F `  n )
) [,] ( 2nd `  ( F `  n
) ) ) ) )
29 ovolfcl 18826 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR  /\  ( 1st `  ( F `  n ) )  <_ 
( 2nd `  ( F `  n )
) ) )
30 elicc2 10715 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR )  -> 
( z  e.  ( ( 1st `  ( F `  n )
) [,] ( 2nd `  ( F `  n
) ) )  <->  ( z  e.  RR  /\  ( 1st `  ( F `  n
) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n
) ) ) ) )
31 3anass 938 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR  /\  ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
3230, 31syl6bb 252 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR )  -> 
( z  e.  ( ( 1st `  ( F `  n )
) [,] ( 2nd `  ( F `  n
) ) )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) ) )
33323adant3 975 . . . . . . . . . . . 12  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR  /\  ( 1st `  ( F `  n ) )  <_ 
( 2nd `  ( F `  n )
) )  ->  (
z  e.  ( ( 1st `  ( F `
 n ) ) [,] ( 2nd `  ( F `  n )
) )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) ) )
3429, 33syl 15 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
z  e.  ( ( 1st `  ( F `
 n ) ) [,] ( 2nd `  ( F `  n )
) )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) ) )
3528, 34bitrd 244 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
z  e.  ( ( [,]  o.  F ) `
 n )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) ) )
3635adantll 694 . . . . . . . . 9  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  ( z  e.  ( ( [,]  o.  F
) `  n )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) ) )
37 simpll 730 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  z  e.  RR )
3837biantrurd 494 . . . . . . . . 9  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  ( ( ( 1st `  ( F `  n
) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n
) ) )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) ) )
3936, 38bitr4d 247 . . . . . . . 8  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  ( z  e.  ( ( [,]  o.  F
) `  n )  <->  ( ( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
4039rexbidva 2560 . . . . . . 7  |-  ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( E. n  e.  NN  z  e.  ( ( [,]  o.  F
) `  n )  <->  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
4118, 40syl5bb 248 . . . . . 6  |-  ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( z  e.  U_ n  e.  NN  (
( [,]  o.  F
) `  n )  <->  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
4217, 41sylan 457 . . . . 5  |-  ( ( ( A  C_  RR  /\  z  e.  A )  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  ->  ( z  e. 
U_ n  e.  NN  ( ( [,]  o.  F ) `  n
)  <->  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
4342an32s 779 . . . 4  |-  ( ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  z  e.  A )  ->  ( z  e.  U_ n  e.  NN  (
( [,]  o.  F
) `  n )  <->  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
4443ralbidva 2559 . . 3  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A. z  e.  A  z  e.  U_ n  e.  NN  (
( [,]  o.  F
) `  n )  <->  A. z  e.  A  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
4516, 44syl5bb 248 . 2  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n
) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n
) ) ) ) )
4615, 45bitr3d 246 1  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U. ran  ( [,]  o.  F )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023    X. cxp 4687   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RRcr 8736   RR*cxr 8866    <_ cle 8868   NNcn 9746   [,]cicc 10659
This theorem is referenced by:  ovollb2lem  18847  ovolctb  18849  ovolicc1  18875  ioombl1lem4  18918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
  Copyright terms: Public domain W3C validator