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Theorem xrge0f 21873
Description: A real function is a nonnegative extended real function if all its values are greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)
Assertion
Ref Expression
xrge0f  |-  ( ( F : RR --> RR  /\  0p  oR 
<_  F )  ->  F : RR --> ( 0 [,] +oo ) )

Proof of Theorem xrge0f
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ffn 5729 . . 3  |-  ( F : RR --> RR  ->  F  Fn  RR )
21adantr 465 . 2  |-  ( ( F : RR --> RR  /\  0p  oR 
<_  F )  ->  F  Fn  RR )
3 ax-resscn 9545 . . . . . 6  |-  RR  C_  CC
43a1i 11 . . . . 5  |-  ( F : RR --> RR  ->  RR  C_  CC )
54, 10pledm 21815 . . . 4  |-  ( F : RR --> RR  ->  ( 0p  oR  <_  F  <->  ( RR  X.  { 0 } )  oR  <_  F
) )
6 0re 9592 . . . . . 6  |-  0  e.  RR
7 fnconstg 5771 . . . . . 6  |-  ( 0  e.  RR  ->  ( RR  X.  { 0 } )  Fn  RR )
86, 7mp1i 12 . . . . 5  |-  ( F : RR --> RR  ->  ( RR  X.  { 0 } )  Fn  RR )
9 reex 9579 . . . . . 6  |-  RR  e.  _V
109a1i 11 . . . . 5  |-  ( F : RR --> RR  ->  RR  e.  _V )
11 inidm 3707 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
12 c0ex 9586 . . . . . . 7  |-  0  e.  _V
1312fvconst2 6114 . . . . . 6  |-  ( x  e.  RR  ->  (
( RR  X.  {
0 } ) `  x )  =  0 )
1413adantl 466 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( ( RR  X.  { 0 } ) `
 x )  =  0 )
15 eqidd 2468 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
168, 1, 10, 10, 11, 14, 15ofrfval 6530 . . . 4  |-  ( F : RR --> RR  ->  ( ( RR  X.  {
0 } )  oR  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
17 ffvelrn 6017 . . . . . . . 8  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
1817rexrd 9639 . . . . . . 7  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR* )
1918biantrurd 508 . . . . . 6  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( 0  <_  ( F `  x )  <->  ( ( F `  x
)  e.  RR*  /\  0  <_  ( F `  x
) ) ) )
20 elxrge0 11625 . . . . . 6  |-  ( ( F `  x )  e.  ( 0 [,] +oo )  <->  ( ( F `
 x )  e. 
RR*  /\  0  <_  ( F `  x ) ) )
2119, 20syl6bbr 263 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( 0  <_  ( F `  x )  <->  ( F `  x )  e.  ( 0 [,] +oo ) ) )
2221ralbidva 2900 . . . 4  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  0  <_  ( F `  x )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,] +oo )
) )
235, 16, 223bitrd 279 . . 3  |-  ( F : RR --> RR  ->  ( 0p  oR  <_  F  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,] +oo )
) )
2423biimpa 484 . 2  |-  ( ( F : RR --> RR  /\  0p  oR 
<_  F )  ->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,] +oo )
)
25 ffnfv 6045 . 2  |-  ( F : RR --> ( 0 [,] +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,] +oo )
) )
262, 24, 25sylanbrc 664 1  |-  ( ( F : RR --> RR  /\  0p  oR 
<_  F )  ->  F : RR --> ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   {csn 4027   class class class wbr 4447    X. cxp 4997    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    oRcofr 6521   CCcc 9486   RRcr 9487   0cc0 9488   +oocpnf 9621   RR*cxr 9623    <_ cle 9625   [,]cicc 11528   0pc0p 21811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-ofr 6523  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-icc 11532  df-0p 21812
This theorem is referenced by:  itg2itg1  21878
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