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Theorem 0plef 19556
Description: Two ways to say that the function  F on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)
Assertion
Ref Expression
0plef  |-  ( F : RR --> ( 0 [,)  +oo )  <->  ( F : RR --> RR  /\  0 p  o R  <_  F
) )

Proof of Theorem 0plef
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0re 9083 . . . 4  |-  0  e.  RR
2 pnfxr 10705 . . . 4  |-  +oo  e.  RR*
3 icossre 10983 . . . 4  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
41, 2, 3mp2an 654 . . 3  |-  ( 0 [,)  +oo )  C_  RR
5 fss 5591 . . 3  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  RR )  ->  F : RR
--> RR )
64, 5mpan2 653 . 2  |-  ( F : RR --> ( 0 [,)  +oo )  ->  F : RR --> RR )
7 ffvelrn 5860 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
8 elrege0 10999 . . . . . 6  |-  ( ( F `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
98baib 872 . . . . 5  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( 0 [,)  +oo )  <->  0  <_  ( F `  x ) ) )
107, 9syl 16 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( ( F `  x )  e.  ( 0 [,)  +oo )  <->  0  <_  ( F `  x ) ) )
1110ralbidva 2713 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) 
+oo )  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
12 ffn 5583 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
13 ffnfv 5886 . . . . 5  |-  ( F : RR --> ( 0 [,)  +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,)  +oo )
) )
1413baib 872 . . . 4  |-  ( F  Fn  RR  ->  ( F : RR --> ( 0 [,)  +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,)  +oo )
) )
1512, 14syl 16 . . 3  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,)  +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,)  +oo )
) )
16 0cn 9076 . . . . . . 7  |-  0  e.  CC
17 fnconstg 5623 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1816, 17ax-mp 8 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
19 df-0p 19554 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
2019fneq1i 5531 . . . . . 6  |-  ( 0 p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
2118, 20mpbir 201 . . . . 5  |-  0 p  Fn  CC
2221a1i 11 . . . 4  |-  ( F : RR --> RR  ->  0 p  Fn  CC )
23 cnex 9063 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
25 reex 9073 . . . . 5  |-  RR  e.  _V
2625a1i 11 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
27 ax-resscn 9039 . . . . 5  |-  RR  C_  CC
28 sseqin2 3552 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2927, 28mpbi 200 . . . 4  |-  ( CC 
i^i  RR )  =  RR
30 0pval 19555 . . . . 5  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
3130adantl 453 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
32 eqidd 2436 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3322, 12, 24, 26, 29, 31, 32ofrfval 6305 . . 3  |-  ( F : RR --> RR  ->  ( 0 p  o R  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
3411, 15, 333bitr4d 277 . 2  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,)  +oo )  <->  0 p  o R  <_  F ) )
356, 34biadan2 624 1  |-  ( F : RR --> ( 0 [,)  +oo )  <->  ( F : RR --> RR  /\  0 p  o R  <_  F
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {csn 3806   class class class wbr 4204    X. cxp 4868    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Rcofr 6296   CCcc 8980   RRcr 8981   0cc0 8982    +oocpnf 9109   RR*cxr 9111    <_ cle 9113   [,)cico 10910   0 pc0p 19553
This theorem is referenced by:  itg2i1fseq  19639  itg2addlem  19642  ftc1anclem8  26277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-ofr 6298  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-ico 10914  df-0p 19554
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