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Theorem xrge0f 22304
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 5713 . . 3  |-  ( F : RR --> RR  ->  F  Fn  RR )
21adantr 463 . 2  |-  ( ( F : RR --> RR  /\  0p  oR 
<_  F )  ->  F  Fn  RR )
3 ax-resscn 9538 . . . . . 6  |-  RR  C_  CC
43a1i 11 . . . . 5  |-  ( F : RR --> RR  ->  RR  C_  CC )
54, 10pledm 22246 . . . 4  |-  ( F : RR --> RR  ->  ( 0p  oR  <_  F  <->  ( RR  X.  { 0 } )  oR  <_  F
) )
6 0re 9585 . . . . . 6  |-  0  e.  RR
7 fnconstg 5755 . . . . . 6  |-  ( 0  e.  RR  ->  ( RR  X.  { 0 } )  Fn  RR )
86, 7mp1i 12 . . . . 5  |-  ( F : RR --> RR  ->  ( RR  X.  { 0 } )  Fn  RR )
9 reex 9572 . . . . . 6  |-  RR  e.  _V
109a1i 11 . . . . 5  |-  ( F : RR --> RR  ->  RR  e.  _V )
11 inidm 3693 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
12 c0ex 9579 . . . . . . 7  |-  0  e.  _V
1312fvconst2 6103 . . . . . 6  |-  ( x  e.  RR  ->  (
( RR  X.  {
0 } ) `  x )  =  0 )
1413adantl 464 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( ( RR  X.  { 0 } ) `
 x )  =  0 )
15 eqidd 2455 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
168, 1, 10, 10, 11, 14, 15ofrfval 6521 . . . 4  |-  ( F : RR --> RR  ->  ( ( RR  X.  {
0 } )  oR  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
17 ffvelrn 6005 . . . . . . . 8  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
1817rexrd 9632 . . . . . . 7  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR* )
1918biantrurd 506 . . . . . 6  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( 0  <_  ( F `  x )  <->  ( ( F `  x
)  e.  RR*  /\  0  <_  ( F `  x
) ) ) )
20 elxrge0 11632 . . . . . 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 2890 . . . 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 482 . 2  |-  ( ( F : RR --> RR  /\  0p  oR 
<_  F )  ->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,] +oo )
)
25 ffnfv 6033 . 2  |-  ( F : RR --> ( 0 [,] +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,] +oo )
) )
262, 24, 25sylanbrc 662 1  |-  ( ( F : RR --> RR  /\  0p  oR 
<_  F )  ->  F : RR --> ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    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   RR*cxr 9616    <_ cle 9618   [,]cicc 11535   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
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-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-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-icc 11539  df-0p 22243
This theorem is referenced by:  itg2itg1  22309
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