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Theorem 0plef 23439
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 12280 . . 3 |- ( 0 [,) +oo ) C_ RR
2 fss 6056 . . 3 |- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ RR ) -> F : RR --> RR )
31, 2mpan2 707 . 2 |- ( F : RR --> ( 0 [,) +oo ) -> F : RR --> RR )
4 ffvelrn 6357 . . . . 5 |- ( ( F : RR --> RR /\ x e. RR ) -> ( F ` x ) e. RR )
5 elrege0 12278 . . . . . 6 |- ( ( F ` x ) e. ( 0 [,) +oo ) <-> ( ( F ` x ) e. RR /\ 0 <_ ( F ` x ) ) )
65baib 944 . . . . 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 2985 . . 3 |- ( F : RR --> RR -> ( A. x e. RR ( F ` x ) e. ( 0 [,) +oo ) <-> A. x e. RR 0 <_ ( F ` x ) ) )
9 ffn 6045 . . . 4 |- ( F : RR --> RR -> F Fn RR )
10 ffnfv 6388 . . . . 5 |- ( F : RR --> ( 0 [,) +oo ) <-> ( F Fn RR /\ A. x e. RR ( F ` x ) e. ( 0 [,) +oo ) ) )
1110baib 944 . . . 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 10032 . . . . . . 7 |- 0 e. CC
14 fnconstg 6093 . . . . . . 7 |- ( 0 e. CC -> ( CC X. { 0 } ) Fn CC )
1513, 14ax-mp 5 . . . . . 6 |- ( CC X. { 0 } ) Fn CC
16 df-0p 23437 . . . . . . 7 |- 0p = ( CC X. { 0 } )
1716fneq1i 5985 . . . . . 6 |- ( 0p Fn CC <-> ( CC X. { 0 } ) Fn CC )
1815, 17mpbir 221 . . . . 5 |- 0p Fn CC
1918a1i 11 . . . 4 |- ( F : RR --> RR -> 0p Fn CC )
20 cnex 10017 . . . . 5 |- CC e. _V
2120a1i 11 . . . 4 |- ( F : RR --> RR -> CC e. _V )
22 reex 10027 . . . . 5 |- RR e. _V
2322a1i 11 . . . 4 |- ( F : RR --> RR -> RR e. _V )
24 ax-resscn 9993 . . . . 5 |- RR C_ CC
25 sseqin2 3817 . . . . 5 |- ( RR C_ CC <-> ( CC i^i RR ) = RR )
2624, 25mpbi 220 . . . 4 |- ( CC i^i RR ) = RR
27 0pval 23438 . . . . 5 |- ( x e. CC -> ( 0p ` x ) = 0 )
2827adantl 482 . . . 4 |- ( ( F : RR --> RR /\ x e. CC ) -> ( 0p ` x ) = 0 )
29 eqidd 2623 . . . 4 |- ( ( F : RR --> RR /\ x e. RR ) -> ( F ` x ) = ( F ` x ) )
3019, 9, 21, 23, 26, 28, 29ofrfval 6905 . . 3 |- ( F : RR --> RR -> ( 0p oR <_ F <-> A. x e. RR 0 <_ ( F ` x ) ) )
318, 12, 303bitr4d 300 . 2 |- ( F : RR --> RR -> ( F : RR --> ( 0 [,) +oo ) <-> 0p oR <_ F ) )
323, 31biadan2 674 1 |- ( F : RR --> ( 0 [,) +oo ) <-> ( F : RR --> RR /\ 0p oR <_ F ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oRcofr 6896   CCcc 9934   RRcr 9935   0cc0 9936   +oocpnf 10071   <_ cle 10075   [,)cico 12177   0pc0p 23436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ofr 6898  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181  df-0p 23437
This theorem is referenced by:  itg2i1fseq  23522  itg2addlem  23525  ftc1anclem8  33492
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