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Theorem fge0npnf 38323
Description: If  F maps to nonnegative reals, then +oo is not in its range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
fge0npnf.1  |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )
Assertion
Ref Expression
fge0npnf  |-  ( ph  ->  -. +oo  e.  ran  F )

Proof of Theorem fge0npnf
StepHypRef Expression
1 fge0npnf.1 . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )
2 frn 5747 . . . . 5  |-  ( F : X --> ( 0 [,) +oo )  ->  ran  F  C_  ( 0 [,) +oo ) )
31, 2syl 17 . . . 4  |-  ( ph  ->  ran  F  C_  (
0 [,) +oo )
)
43adantr 472 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ran  F  C_  ( 0 [,) +oo ) )
5 simpr 468 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
64, 5sseldd 3419 . 2  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ( 0 [,) +oo )
)
7 0xr 9705 . . . 4  |-  0  e.  RR*
8 icoub 37723 . . . 4  |-  ( 0  e.  RR*  ->  -. +oo  e.  ( 0 [,) +oo ) )
97, 8ax-mp 5 . . 3  |-  -. +oo  e.  ( 0 [,) +oo )
109a1i 11 . 2  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  -. +oo  e.  ( 0 [,) +oo ) )
116, 10pm2.65da 586 1  |-  ( ph  ->  -. +oo  e.  ran  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    e. wcel 1904    C_ wss 3390   ran crn 4840   -->wf 5585  (class class class)co 6308   0cc0 9557   +oocpnf 9690   RR*cxr 9692   [,)cico 11662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-ico 11666
This theorem is referenced by:  sge0reval  38328  sge0fsum  38343
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