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Theorem 0plef 19131
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 8928 . . . 4  |-  0  e.  RR
2 pnfxr 10547 . . . 4  |-  +oo  e.  RR*
3 icossre 10822 . . . 4  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
41, 2, 3mp2an 653 . . 3  |-  ( 0 [,)  +oo )  C_  RR
5 fss 5480 . . 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 5746 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
8 elrege0 10838 . . . . . 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 2635 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) 
+oo )  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
12 ffn 5472 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
13 ffnfv 5768 . . . . 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 8921 . . . . . . 7  |-  0  e.  CC
17 fnconstg 5512 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1816, 17ax-mp 8 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
19 df-0p 19129 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
2019fneq1i 5420 . . . . . 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 8908 . . . . 5  |-  CC  e.  _V
2423a1i 10 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
25 reex 8918 . . . . 5  |-  RR  e.  _V
2625a1i 10 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
27 ax-resscn 8884 . . . . 5  |-  RR  C_  CC
28 sseqin2 3464 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2927, 28mpbi 199 . . . 4  |-  ( CC 
i^i  RR )  =  RR
30 0pval 19130 . . . . 5  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
3130adantl 452 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
32 eqidd 2359 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3322, 12, 24, 26, 29, 31, 32ofrfval 6173 . . 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 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    i^i cin 3227    C_ wss 3228   {csn 3716   class class class wbr 4104    X. cxp 4769    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Rcofr 6164   CCcc 8825   RRcr 8826   0cc0 8827    +oocpnf 8954   RR*cxr 8956    <_ cle 8958   [,)cico 10750   0 pc0p 19128
This theorem is referenced by:  itg2i1fseq  19214  itg2addlem  19217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-i2m1 8895  ax-1ne0 8896  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-ofr 6166  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-ico 10754  df-0p 19129
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