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Theorem 0plef 19043
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 8854 . . . 4  |-  0  e.  RR
2 pnfxr 10471 . . . 4  |-  +oo  e.  RR*
3 icossre 10746 . . . 4  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
41, 2, 3mp2an 653 . . 3  |-  ( 0 [,)  +oo )  C_  RR
5 fss 5413 . . 3  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  RR )  ->  F : RR
--> RR )
64, 5mpan2 652 . 2  |-  ( F : RR --> ( 0 [,)  +oo )  ->  F : RR --> RR )
7 ffvelrn 5679 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
8 elrege0 10762 . . . . . 6  |-  ( ( F `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
98baib 871 . . . . 5  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( 0 [,)  +oo )  <->  0  <_  ( F `  x ) ) )
107, 9syl 15 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( ( F `  x )  e.  ( 0 [,)  +oo )  <->  0  <_  ( F `  x ) ) )
1110ralbidva 2572 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) 
+oo )  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
12 ffn 5405 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
13 ffnfv 5701 . . . . 5  |-  ( F : RR --> ( 0 [,)  +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,)  +oo )
) )
1413baib 871 . . . 4  |-  ( F  Fn  RR  ->  ( F : RR --> ( 0 [,)  +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,)  +oo )
) )
1512, 14syl 15 . . 3  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,)  +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,)  +oo )
) )
16 0cn 8847 . . . . . . 7  |-  0  e.  CC
17 fnconstg 5445 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1816, 17ax-mp 8 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
19 df-0p 19041 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
2019fneq1i 5354 . . . . . 6  |-  ( 0 p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
2118, 20mpbir 200 . . . . 5  |-  0 p  Fn  CC
2221a1i 10 . . . 4  |-  ( F : RR --> RR  ->  0 p  Fn  CC )
23 cnex 8834 . . . . 5  |-  CC  e.  _V
2423a1i 10 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
25 reex 8844 . . . . 5  |-  RR  e.  _V
2625a1i 10 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
27 ax-resscn 8810 . . . . 5  |-  RR  C_  CC
28 sseqin2 3401 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2927, 28mpbi 199 . . . 4  |-  ( CC 
i^i  RR )  =  RR
30 0pval 19042 . . . . 5  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
3130adantl 452 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
32 eqidd 2297 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3322, 12, 24, 26, 29, 31, 32ofrfval 6102 . . 3  |-  ( F : RR --> RR  ->  ( 0 p  o R  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
3411, 15, 333bitr4d 276 . 2  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,)  +oo )  <->  0 p  o R  <_  F ) )
356, 34biadan2 623 1  |-  ( F : RR --> ( 0 [,)  +oo )  <->  ( F : RR --> RR  /\  0 p  o R  <_  F
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   class class class wbr 4039    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Rcofr 6093   CCcc 8751   RRcr 8752   0cc0 8753    +oocpnf 8880   RR*cxr 8882    <_ cle 8884   [,)cico 10674   0 pc0p 19040
This theorem is referenced by:  itg2i1fseq  19126  itg2addlem  19129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ofr 6095  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ico 10678  df-0p 19041
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