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Theorem ovolfsf 18831
Description: Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypothesis
Ref Expression
ovolfs.1  |-  G  =  ( ( abs  o.  -  )  o.  F
)
Assertion
Ref Expression
ovolfsf  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) 
+oo ) )

Proof of Theorem ovolfsf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 absf 11821 . . . . . 6  |-  abs : CC
--> RR
2 subf 9053 . . . . . 6  |-  -  :
( CC  X.  CC )
--> CC
3 fco 5398 . . . . . 6  |-  ( ( abs : CC --> RR  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
41, 2, 3mp2an 653 . . . . 5  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> RR
5 inss2 3390 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ax-resscn 8794 . . . . . . . 8  |-  RR  C_  CC
7 xpss12 4792 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR  X.  RR )  C_  ( CC  X.  CC ) )
86, 6, 7mp2an 653 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( CC  X.  CC )
95, 8sstri 3188 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC )
10 fss 5397 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC ) )  ->  F : NN --> ( CC  X.  CC ) )
119, 10mpan2 652 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( CC  X.  CC ) )
12 fco 5398 . . . . 5  |-  ( ( ( abs  o.  -  ) : ( CC  X.  CC ) --> RR  /\  F : NN --> ( CC  X.  CC ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
134, 11, 12sylancr 644 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
14 ovolfs.1 . . . . 5  |-  G  =  ( ( abs  o.  -  )  o.  F
)
1514feq1i 5383 . . . 4  |-  ( G : NN --> RR  <->  ( ( abs  o.  -  )  o.  F ) : NN --> RR )
1613, 15sylibr 203 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> RR )
17 ffn 5389 . . 3  |-  ( G : NN --> RR  ->  G  Fn  NN )
1816, 17syl 15 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  Fn  NN )
19 ffvelrn 5663 . . . . 5  |-  ( ( G : NN --> RR  /\  x  e.  NN )  ->  ( G `  x
)  e.  RR )
2016, 19sylan 457 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
21 ovolfcl 18826 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 1st `  ( F `  x )
)  e.  RR  /\  ( 2nd `  ( F `
 x ) )  e.  RR  /\  ( 1st `  ( F `  x ) )  <_ 
( 2nd `  ( F `  x )
) ) )
22 subge0 9287 . . . . . . . 8  |-  ( ( ( 2nd `  ( F `  x )
)  e.  RR  /\  ( 1st `  ( F `
 x ) )  e.  RR )  -> 
( 0  <_  (
( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) )  <->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
2322ancoms 439 . . . . . . 7  |-  ( ( ( 1st `  ( F `  x )
)  e.  RR  /\  ( 2nd `  ( F `
 x ) )  e.  RR )  -> 
( 0  <_  (
( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) )  <->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
2423biimp3ar 1282 . . . . . 6  |-  ( ( ( 1st `  ( F `  x )
)  e.  RR  /\  ( 2nd `  ( F `
 x ) )  e.  RR  /\  ( 1st `  ( F `  x ) )  <_ 
( 2nd `  ( F `  x )
) )  ->  0  <_  ( ( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) ) )
2521, 24syl 15 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( ( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) ) )
2614ovolfsval 18830 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  =  ( ( 2nd `  ( F `  x
) )  -  ( 1st `  ( F `  x ) ) ) )
2725, 26breqtrrd 4049 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( G `  x
) )
28 elrege0 10746 . . . 4  |-  ( ( G `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( G `  x )  e.  RR  /\  0  <_ 
( G `  x
) ) )
2920, 27, 28sylanbrc 645 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( 0 [,)  +oo ) )
3029ralrimiva 2626 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  NN  ( G `  x )  e.  ( 0 [,)  +oo )
)
31 ffnfv 5685 . 2  |-  ( G : NN --> ( 0 [,)  +oo )  <->  ( G  Fn  NN  /\  A. x  e.  NN  ( G `  x )  e.  ( 0 [,)  +oo )
) )
3218, 30, 31sylanbrc 645 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) 
+oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023    X. cxp 4687    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   CCcc 8735   RRcr 8736   0cc0 8737    +oocpnf 8864    <_ cle 8868    - cmin 9037   NNcn 9746   [,)cico 10658   abscabs 11719
This theorem is referenced by:  ovolsf  18832  ovollb2lem  18847  ovolunlem1a  18855  ovoliunlem1  18861  ovolshftlem1  18868  ovolicc2lem4  18879  ioombl1lem4  18918  ovolfs2  18926  uniioombllem2  18938  uniioombllem6  18943
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-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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