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Theorem 0plef 19517
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 9047 . . . 4  |-  0  e.  RR
2 pnfxr 10669 . . . 4  |-  +oo  e.  RR*
3 icossre 10947 . . . 4  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
41, 2, 3mp2an 654 . . 3  |-  ( 0 [,)  +oo )  C_  RR
5 fss 5558 . . 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 5827 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
8 elrege0 10963 . . . . . 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 2682 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) 
+oo )  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
12 ffn 5550 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
13 ffnfv 5853 . . . . 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 9040 . . . . . . 7  |-  0  e.  CC
17 fnconstg 5590 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1816, 17ax-mp 8 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
19 df-0p 19515 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
2019fneq1i 5498 . . . . . 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 9027 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
25 reex 9037 . . . . 5  |-  RR  e.  _V
2625a1i 11 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
27 ax-resscn 9003 . . . . 5  |-  RR  C_  CC
28 sseqin2 3520 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2927, 28mpbi 200 . . . 4  |-  ( CC 
i^i  RR )  =  RR
30 0pval 19516 . . . . 5  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
3130adantl 453 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
32 eqidd 2405 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3322, 12, 24, 26, 29, 31, 32ofrfval 6272 . . 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 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    i^i cin 3279    C_ wss 3280   {csn 3774   class class class wbr 4172    X. cxp 4835    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Rcofr 6263   CCcc 8944   RRcr 8945   0cc0 8946    +oocpnf 9073   RR*cxr 9075    <_ cle 9077   [,)cico 10874   0 pc0p 19514
This theorem is referenced by:  itg2i1fseq  19600  itg2addlem  19603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-i2m1 9014  ax-1ne0 9015  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-ofr 6265  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-ico 10878  df-0p 19515
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