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Theorem ovolfsf 21071
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 12927 . . . . . 6  |-  abs : CC
--> RR
2 subf 9713 . . . . . 6  |-  -  :
( CC  X.  CC )
--> CC
3 fco 5666 . . . . . 6  |-  ( ( abs : CC --> RR  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
41, 2, 3mp2an 672 . . . . 5  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> RR
5 inss2 3669 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ax-resscn 9440 . . . . . . . 8  |-  RR  C_  CC
7 xpss12 5043 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR  X.  RR )  C_  ( CC  X.  CC ) )
86, 6, 7mp2an 672 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( CC  X.  CC )
95, 8sstri 3463 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC )
10 fss 5665 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC ) )  ->  F : NN --> ( CC  X.  CC ) )
119, 10mpan2 671 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( CC  X.  CC ) )
12 fco 5666 . . . . 5  |-  ( ( ( abs  o.  -  ) : ( CC  X.  CC ) --> RR  /\  F : NN --> ( CC  X.  CC ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
134, 11, 12sylancr 663 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
14 ovolfs.1 . . . . 5  |-  G  =  ( ( abs  o.  -  )  o.  F
)
1514feq1i 5649 . . . 4  |-  ( G : NN --> RR  <->  ( ( abs  o.  -  )  o.  F ) : NN --> RR )
1613, 15sylibr 212 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> RR )
17 ffn 5657 . . 3  |-  ( G : NN --> RR  ->  G  Fn  NN )
1816, 17syl 16 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  Fn  NN )
1916ffvelrnda 5942 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
20 ovolfcl 21066 . . . . . 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 9953 . . . . . . . 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 453 . . . . . . 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 1320 . . . . . 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 16 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( ( 2nd `  ( F `  x )
)  -  ( 1st `  ( F `  x
) ) ) )
2514ovolfsval 21070 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  =  ( ( 2nd `  ( F `  x
) )  -  ( 1st `  ( F `  x ) ) ) )
2624, 25breqtrrd 4416 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( G `  x
) )
27 elrege0 11493 . . . 4  |-  ( ( G `  x )  e.  ( 0 [,) +oo )  <->  ( ( G `
 x )  e.  RR  /\  0  <_ 
( G `  x
) ) )
2819, 26, 27sylanbrc 664 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( 0 [,) +oo ) )
2928ralrimiva 2822 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  NN  ( G `  x )  e.  ( 0 [,) +oo )
)
30 ffnfv 5968 . 2  |-  ( G : NN --> ( 0 [,) +oo )  <->  ( G  Fn  NN  /\  A. x  e.  NN  ( G `  x )  e.  ( 0 [,) +oo )
) )
3118, 29, 30sylanbrc 664 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    i^i cin 3425    C_ wss 3426   class class class wbr 4390    X. cxp 4936    o. ccom 4942    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   1stc1st 6675   2ndc2nd 6676   CCcc 9381   RRcr 9382   0cc0 9383   +oocpnf 9516    <_ cle 9520    - cmin 9696   NNcn 10423   [,)cico 11403   abscabs 12825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-ico 11407  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827
This theorem is referenced by:  ovolsf  21072  ovollb2lem  21087  ovolunlem1a  21095  ovoliunlem1  21101  ovolshftlem1  21108  ovolicc2lem4  21119  ioombl1lem4  21158  ovolfs2  21167  uniioombllem2  21179  uniioombllem6  21184
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