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Theorem ovolfsf 19370
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 12143 . . . . . 6  |-  abs : CC
--> RR
2 subf 9309 . . . . . 6  |-  -  :
( CC  X.  CC )
--> CC
3 fco 5602 . . . . . 6  |-  ( ( abs : CC --> RR  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
41, 2, 3mp2an 655 . . . . 5  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> RR
5 inss2 3564 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ax-resscn 9049 . . . . . . . 8  |-  RR  C_  CC
7 xpss12 4983 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR  X.  RR )  C_  ( CC  X.  CC ) )
86, 6, 7mp2an 655 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( CC  X.  CC )
95, 8sstri 3359 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC )
10 fss 5601 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( CC  X.  CC ) )  ->  F : NN --> ( CC  X.  CC ) )
119, 10mpan2 654 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( CC  X.  CC ) )
12 fco 5602 . . . . 5  |-  ( ( ( abs  o.  -  ) : ( CC  X.  CC ) --> RR  /\  F : NN --> ( CC  X.  CC ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
134, 11, 12sylancr 646 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> RR )
14 ovolfs.1 . . . . 5  |-  G  =  ( ( abs  o.  -  )  o.  F
)
1514feq1i 5587 . . . 4  |-  ( G : NN --> RR  <->  ( ( abs  o.  -  )  o.  F ) : NN --> RR )
1613, 15sylibr 205 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> RR )
17 ffn 5593 . . 3  |-  ( G : NN --> RR  ->  G  Fn  NN )
1816, 17syl 16 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  Fn  NN )
1916ffvelrnda 5872 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
20 ovolfcl 19365 . . . . . 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 9543 . . . . . . . 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 441 . . . . . . 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 1285 . . . . . 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 19369 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  =  ( ( 2nd `  ( F `  x
) )  -  ( 1st `  ( F `  x ) ) ) )
2624, 25breqtrrd 4240 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  0  <_  ( G `  x
) )
27 elrege0 11009 . . . 4  |-  ( ( G `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( G `  x )  e.  RR  /\  0  <_ 
( G `  x
) ) )
2819, 26, 27sylanbrc 647 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( 0 [,)  +oo ) )
2928ralrimiva 2791 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  NN  ( G `  x )  e.  ( 0 [,)  +oo )
)
30 ffnfv 5896 . 2  |-  ( G : NN --> ( 0 [,)  +oo )  <->  ( G  Fn  NN  /\  A. x  e.  NN  ( G `  x )  e.  ( 0 [,)  +oo )
) )
3118, 29, 30sylanbrc 647 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) 
+oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    C_ wss 3322   class class class wbr 4214    X. cxp 4878    o. ccom 4884    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   CCcc 8990   RRcr 8991   0cc0 8992    +oocpnf 9119    <_ cle 9123    - cmin 9293   NNcn 10002   [,)cico 10920   abscabs 12041
This theorem is referenced by:  ovolsf  19371  ovollb2lem  19386  ovolunlem1a  19394  ovoliunlem1  19400  ovolshftlem1  19407  ovolicc2lem4  19418  ioombl1lem4  19457  ovolfs2  19465  uniioombllem2  19477  uniioombllem6  19482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043
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