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Theorem 0plef 21284
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 0re 9498 . . . 4  |-  0  e.  RR
2 pnfxr 11204 . . . 4  |- +oo  e.  RR*
3 icossre 11488 . . . 4  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
41, 2, 3mp2an 672 . . 3  |-  ( 0 [,) +oo )  C_  RR
5 fss 5676 . . 3  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : RR --> RR )
64, 5mpan2 671 . 2  |-  ( F : RR --> ( 0 [,) +oo )  ->  F : RR --> RR )
7 ffvelrn 5951 . . . . 5  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  RR )
8 elrege0 11510 . . . . . 6  |-  ( ( F `  x )  e.  ( 0 [,) +oo )  <->  ( ( F `
 x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
98baib 896 . . . . 5  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( 0 [,) +oo )  <->  0  <_  ( F `  x ) ) )
107, 9syl 16 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( ( F `  x )  e.  ( 0 [,) +oo )  <->  0  <_  ( F `  x ) ) )
1110ralbidva 2844 . . 3  |-  ( F : RR --> RR  ->  ( A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )  <->  A. x  e.  RR  0  <_  ( F `  x ) ) )
12 ffn 5668 . . . 4  |-  ( F : RR --> RR  ->  F  Fn  RR )
13 ffnfv 5979 . . . . 5  |-  ( F : RR --> ( 0 [,) +oo )  <->  ( F  Fn  RR  /\  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
1413baib 896 . . . 4  |-  ( F  Fn  RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
1512, 14syl 16 . . 3  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  A. x  e.  RR  ( F `  x )  e.  ( 0 [,) +oo )
) )
16 0cn 9490 . . . . . . 7  |-  0  e.  CC
17 fnconstg 5707 . . . . . . 7  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
1816, 17ax-mp 5 . . . . . 6  |-  ( CC 
X.  { 0 } )  Fn  CC
19 df-0p 21282 . . . . . . 7  |-  0p  =  ( CC  X.  { 0 } )
2019fneq1i 5614 . . . . . 6  |-  ( 0p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
2118, 20mpbir 209 . . . . 5  |-  0p  Fn  CC
2221a1i 11 . . . 4  |-  ( F : RR --> RR  ->  0p  Fn  CC )
23 cnex 9475 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( F : RR --> RR  ->  CC  e.  _V )
25 reex 9485 . . . . 5  |-  RR  e.  _V
2625a1i 11 . . . 4  |-  ( F : RR --> RR  ->  RR  e.  _V )
27 ax-resscn 9451 . . . . 5  |-  RR  C_  CC
28 sseqin2 3678 . . . . 5  |-  ( RR  C_  CC  <->  ( CC  i^i  RR )  =  RR )
2927, 28mpbi 208 . . . 4  |-  ( CC 
i^i  RR )  =  RR
30 0pval 21283 . . . . 5  |-  ( x  e.  CC  ->  (
0p `  x
)  =  0 )
3130adantl 466 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  CC )  ->  ( 0p `  x )  =  0 )
32 eqidd 2455 . . . 4  |-  ( ( F : RR --> RR  /\  x  e.  RR )  ->  ( F `  x
)  =  ( F `
 x ) )
3322, 12, 24, 26, 29, 31, 32ofrfval 6439 . . 3  |-  ( F : RR --> RR  ->  ( 0p  oR  <_  F  <->  A. x  e.  RR  0  <_  ( F `  x )
) )
3411, 15, 333bitr4d 285 . 2  |-  ( F : RR --> RR  ->  ( F : RR --> ( 0 [,) +oo )  <->  0p  oR  <_  F ) )
356, 34biadan2 642 1  |-  ( F : RR --> ( 0 [,) +oo )  <->  ( F : RR --> RR  /\  0p  oR  <_  F
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    i^i cin 3436    C_ wss 3437   {csn 3986   class class class wbr 4401    X. cxp 4947    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201    oRcofr 6430   CCcc 9392   RRcr 9393   0cc0 9394   +oocpnf 9527   RR*cxr 9529    <_ cle 9531   [,)cico 11414   0pc0p 21281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-i2m1 9462  ax-1ne0 9463  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-ofr 6432  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-ico 11418  df-0p 21282
This theorem is referenced by:  itg2i1fseq  21367  itg2addlem  21370  ftc1anclem8  28623
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