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Theorem 0plef 22567
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 11686 . . 3  |-  ( 0 [,) +oo )  C_  RR
2 fss 5692 . . 3  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : RR --> RR )
31, 2mpan2 675 . 2  |-  ( F : RR --> ( 0 [,) +oo )  ->  F : RR --> RR )
4 ffvelrn 5974 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
5 elrege0 11684 . . . . . 6  |-  ( ( F `  x )  e.  ( 0 [,) +oo )  <->  ( ( F `
 x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
65baib 911 . . . . 5  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( 0 [,) +oo )  <->  0  <_  ( F `  x ) ) )
74, 6syl 17 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( ( F `  x )  e.  ( 0 [,) +oo )  <->  0  <_  ( F `  x ) ) )
87ralbidva 2796 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )  <->  A. x  e.  RR  0  <_  ( F `  x ) ) )
9 ffn 5684 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
10 ffnfv 6003 . . . . 5  |-  ( F : RR --> ( 0 [,) +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
1110baib 911 . . . 4  |-  ( F  Fn  RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
129, 11syl 17 . . 3  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
13 0cn 9581 . . . . . . 7  |-  0  e.  CC
14 fnconstg 5726 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1513, 14ax-mp 5 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
16 df-0p 22565 . . . . . . 7  |-  0p  =  ( CC  X.  { 0 } )
1716fneq1i 5626 . . . . . 6  |-  ( 0p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
1815, 17mpbir 212 . . . . 5  |-  0p  Fn  CC
1918a1i 11 . . . 4  |-  ( F : RR --> RR  ->  0p  Fn  CC )
20 cnex 9566 . . . . 5  |-  CC  e.  _V
2120a1i 11 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
22 reex 9576 . . . . 5  |-  RR  e.  _V
2322a1i 11 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
24 ax-resscn 9542 . . . . 5  |-  RR  C_  CC
25 sseqin2 3619 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2624, 25mpbi 211 . . . 4  |-  ( CC 
i^i  RR )  =  RR
27 0pval 22566 . . . . 5  |-  ( x  e.  CC  ->  (
0p `  x
)  =  0 )
2827adantl 467 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0p `  x )  =  0 )
29 eqidd 2424 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3019, 9, 21, 23, 26, 28, 29ofrfval 6492 . . 3  |-  ( F : RR --> RR  ->  ( 0p  oR  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
318, 12, 303bitr4d 288 . 2  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  0p  oR  <_  F ) )
323, 31biadan2 646 1  |-  ( F : RR --> ( 0 [,) +oo )  <->  ( F : RR --> RR  /\  0p  oR  <_  F
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2709   _Vcvv 3017    i^i cin 3373    C_ wss 3374   {csn 3936   class class class wbr 4361    X. cxp 4789    Fn wfn 5534   -->wf 5535   ` cfv 5539  (class class class)co 6244    oRcofr 6483   CCcc 9483   RRcr 9484   0cc0 9485   +oocpnf 9618    <_ cle 9622   [,)cico 11583   0pc0p 22564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-i2m1 9553  ax-1ne0 9554  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-po 4712  df-so 4713  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-ofr 6485  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-ico 11587  df-0p 22565
This theorem is referenced by:  itg2i1fseq  22650  itg2addlem  22653  ftc1anclem8  31931
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