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Theorem 0plef 22245
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  /\  0p  oR  <_  F
) )

Proof of Theorem 0plef
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rge0ssre 11631 . . 3  |-  ( 0 [,) +oo )  C_  RR
2 fss 5721 . . 3  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : RR --> RR )
31, 2mpan2 669 . 2  |-  ( F : RR --> ( 0 [,) +oo )  ->  F : RR --> RR )
4 ffvelrn 6005 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
5 elrege0 11630 . . . . . 6  |-  ( ( F `  x )  e.  ( 0 [,) +oo )  <->  ( ( F `
 x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
65baib 901 . . . . 5  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( 0 [,) +oo )  <->  0  <_  ( F `  x ) ) )
74, 6syl 16 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( ( F `  x )  e.  ( 0 [,) +oo )  <->  0  <_  ( F `  x ) ) )
87ralbidva 2890 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )  <->  A. x  e.  RR  0  <_  ( F `  x ) ) )
9 ffn 5713 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
10 ffnfv 6033 . . . . 5  |-  ( F : RR --> ( 0 [,) +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
1110baib 901 . . . 4  |-  ( F  Fn  RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
129, 11syl 16 . . 3  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
13 0cn 9577 . . . . . . 7  |-  0  e.  CC
14 fnconstg 5755 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1513, 14ax-mp 5 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
16 df-0p 22243 . . . . . . 7  |-  0p  =  ( CC  X.  { 0 } )
1716fneq1i 5657 . . . . . 6  |-  ( 0p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
1815, 17mpbir 209 . . . . 5  |-  0p  Fn  CC
1918a1i 11 . . . 4  |-  ( F : RR --> RR  ->  0p  Fn  CC )
20 cnex 9562 . . . . 5  |-  CC  e.  _V
2120a1i 11 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
22 reex 9572 . . . . 5  |-  RR  e.  _V
2322a1i 11 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
24 ax-resscn 9538 . . . . 5  |-  RR  C_  CC
25 sseqin2 3703 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2624, 25mpbi 208 . . . 4  |-  ( CC 
i^i  RR )  =  RR
27 0pval 22244 . . . . 5  |-  ( x  e.  CC  ->  (
0p `  x
)  =  0 )
2827adantl 464 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0p `  x )  =  0 )
29 eqidd 2455 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3019, 9, 21, 23, 26, 28, 29ofrfval 6521 . . 3  |-  ( F : RR --> RR  ->  ( 0p  oR  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
318, 12, 303bitr4d 285 . 2  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  0p  oR  <_  F ) )
323, 31biadan2 640 1  |-  ( F : RR --> ( 0 [,) +oo )  <->  ( F : RR --> RR  /\  0p  oR  <_  F
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    i^i cin 3460    C_ wss 3461   {csn 4016   class class class wbr 4439    X. cxp 4986    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oRcofr 6512   CCcc 9479   RRcr 9480   0cc0 9481   +oocpnf 9614    <_ cle 9618   [,)cico 11534   0pc0p 22242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-ofr 6514  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ico 11538  df-0p 22243
This theorem is referenced by:  itg2i1fseq  22328  itg2addlem  22331  ftc1anclem8  30337
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