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Theorem ovolfsf 22009
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 13182 . . . . . 6  |-  abs : CC
--> RR
2 subf 9841 . . . . . 6  |-  -  :
( CC  X.  CC )
--> CC
3 fco 5747 . . . . . 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 3715 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ax-resscn 9566 . . . . . . . 8  |-  RR  C_  CC
7 xpss12 5117 . . . . . . . 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 3508 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC )
10 fss 5745 . . . . . 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 5747 . . . . 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 5729 . . . 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 5737 . . 3  |-  ( G : NN --> RR  ->  G  Fn  NN )
1816, 17syl 16 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  Fn  NN )
1916ffvelrnda 6032 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
20 ovolfcl 22004 . . . . . 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 10086 . . . . . . . 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 1329 . . . . . 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 22008 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  =  ( ( 2nd `  ( F `  x
) )  -  ( 1st `  ( F `  x ) ) ) )
2624, 25breqtrrd 4482 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( G `  x
) )
27 elrege0 11652 . . . 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 2871 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  NN  ( G `  x )  e.  ( 0 [,) +oo )
)
30 ffnfv 6058 . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807    i^i cin 3470    C_ wss 3471   class class class wbr 4456    X. cxp 5006    o. ccom 5012    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   CCcc 9507   RRcr 9508   0cc0 9509   +oocpnf 9642    <_ cle 9646    - cmin 9824   NNcn 10556   [,)cico 11556   abscabs 13079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081
This theorem is referenced by:  ovolsf  22010  ovollb2lem  22025  ovolunlem1a  22033  ovoliunlem1  22039  ovolshftlem1  22046  ovolicc2lem4  22057  ioombl1lem4  22097  ovolfs2  22106  uniioombllem2  22118  uniioombllem6  22123
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