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Theorem ovolfsf 22366
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 13344 . . . . . 6  |-  abs : CC
--> RR
2 subf 9828 . . . . . 6  |-  -  :
( CC  X.  CC )
--> CC
3 fco 5699 . . . . . 6  |-  ( ( abs : CC --> RR  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
41, 2, 3mp2an 676 . . . . 5  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> RR
5 inss2 3626 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ax-resscn 9547 . . . . . . . 8  |-  RR  C_  CC
7 xpss12 4902 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR  X.  RR )  C_  ( CC  X.  CC ) )
86, 6, 7mp2an 676 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( CC  X.  CC )
95, 8sstri 3416 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC )
10 fss 5697 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC ) )  ->  F : NN --> ( CC  X.  CC ) )
119, 10mpan2 675 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( CC  X.  CC ) )
12 fco 5699 . . . . 5  |-  ( ( ( abs  o.  -  ) : ( CC  X.  CC ) --> RR  /\  F : NN --> ( CC  X.  CC ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
134, 11, 12sylancr 667 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
14 ovolfs.1 . . . . 5  |-  G  =  ( ( abs  o.  -  )  o.  F
)
1514feq1i 5681 . . . 4  |-  ( G : NN --> RR  <->  ( ( abs  o.  -  )  o.  F ) : NN --> RR )
1613, 15sylibr 215 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> RR )
17 ffn 5689 . . 3  |-  ( G : NN --> RR  ->  G  Fn  NN )
1816, 17syl 17 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  Fn  NN )
1916ffvelrnda 5981 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
20 ovolfcl 22361 . . . . . 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 )
) ) )
21 subge0 10078 . . . . . . . 8  |-  ( ( ( 2nd `  ( F `  x )
)  e.  RR  /\  ( 1st `  ( F `
 x ) )  e.  RR )  -> 
( 0  <_  (
( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) )  <->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
2221ancoms 454 . . . . . . 7  |-  ( ( ( 1st `  ( F `  x )
)  e.  RR  /\  ( 2nd `  ( F `
 x ) )  e.  RR )  -> 
( 0  <_  (
( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) )  <->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
2322biimp3ar 1365 . . . . . 6  |-  ( ( ( 1st `  ( F `  x )
)  e.  RR  /\  ( 2nd `  ( F `
 x ) )  e.  RR  /\  ( 1st `  ( F `  x ) )  <_ 
( 2nd `  ( F `  x )
) )  ->  0  <_  ( ( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) ) )
2420, 23syl 17 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( ( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) ) )
2514ovolfsval 22365 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  =  ( ( 2nd `  ( F `  x
) )  -  ( 1st `  ( F `  x ) ) ) )
2624, 25breqtrrd 4393 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( G `  x
) )
27 elrege0 11689 . . . 4  |-  ( ( G `  x )  e.  ( 0 [,) +oo )  <->  ( ( G `
 x )  e.  RR  /\  0  <_ 
( G `  x
) ) )
2819, 26, 27sylanbrc 668 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( 0 [,) +oo ) )
2928ralrimiva 2779 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  NN  ( G `  x )  e.  ( 0 [,) +oo )
)
30 ffnfv 6008 . 2  |-  ( G : NN --> ( 0 [,) +oo )  <->  ( G  Fn  NN  /\  A. x  e.  NN  ( G `  x )  e.  ( 0 [,) +oo )
) )
3118, 29, 30sylanbrc 668 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2714    i^i cin 3378    C_ wss 3379   class class class wbr 4366    X. cxp 4794    o. ccom 4800    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   1stc1st 6749   2ndc2nd 6750   CCcc 9488   RRcr 9489   0cc0 9490   +oocpnf 9623    <_ cle 9627    - cmin 9811   NNcn 10560   [,)cico 11588   abscabs 13241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-ico 11592  df-seq 12164  df-exp 12223  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243
This theorem is referenced by:  ovolsf  22367  ovollb2lem  22383  ovolunlem1a  22391  ovoliunlem1  22397  ovolshftlem1  22404  ovolicc2lem4OLD  22415  ovolicc2lem4  22416  ioombl1lem4  22456  ovolfs2  22465  uniioombllem2  22482  uniioombllem2OLD  22483  uniioombllem6  22488
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