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Theorem 0plef 19567
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 9096 . . . 4  |-  0  e.  RR
2 pnfxr 10718 . . . 4  |-  +oo  e.  RR*
3 icossre 10996 . . . 4  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
41, 2, 3mp2an 655 . . 3  |-  ( 0 [,)  +oo )  C_  RR
5 fss 5602 . . 3  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  RR )  ->  F : RR
--> RR )
64, 5mpan2 654 . 2  |-  ( F : RR --> ( 0 [,)  +oo )  ->  F : RR --> RR )
7 ffvelrn 5871 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
8 elrege0 11012 . . . . . 6  |-  ( ( F `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
98baib 873 . . . . 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 2723 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) 
+oo )  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
12 ffn 5594 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
13 ffnfv 5897 . . . . 5  |-  ( F : RR --> ( 0 [,)  +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,)  +oo )
) )
1413baib 873 . . . 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 9089 . . . . . . 7  |-  0  e.  CC
17 fnconstg 5634 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1816, 17ax-mp 5 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
19 df-0p 19565 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
2019fneq1i 5542 . . . . . 6  |-  ( 0 p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
2118, 20mpbir 202 . . . . 5  |-  0 p  Fn  CC
2221a1i 11 . . . 4  |-  ( F : RR --> RR  ->  0 p  Fn  CC )
23 cnex 9076 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
25 reex 9086 . . . . 5  |-  RR  e.  _V
2625a1i 11 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
27 ax-resscn 9052 . . . . 5  |-  RR  C_  CC
28 sseqin2 3562 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2927, 28mpbi 201 . . . 4  |-  ( CC 
i^i  RR )  =  RR
30 0pval 19566 . . . . 5  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
3130adantl 454 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
32 eqidd 2439 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3322, 12, 24, 26, 29, 31, 32ofrfval 6316 . . 3  |-  ( F : RR --> RR  ->  ( 0 p  o R  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
3411, 15, 333bitr4d 278 . 2  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,)  +oo )  <->  0 p  o R  <_  F ) )
356, 34biadan2 625 1  |-  ( F : RR --> ( 0 [,)  +oo )  <->  ( F : RR --> RR  /\  0 p  o R  <_  F
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    i^i cin 3321    C_ wss 3322   {csn 3816   class class class wbr 4215    X. cxp 4879    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Rcofr 6307   CCcc 8993   RRcr 8994   0cc0 8995    +oocpnf 9122   RR*cxr 9124    <_ cle 9126   [,)cico 10923   0 pc0p 19564
This theorem is referenced by:  itg2i1fseq  19650  itg2addlem  19653  ftc1anclem8  26301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-i2m1 9063  ax-1ne0 9064  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 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-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-ofr 6309  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-ico 10927  df-0p 19565
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