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Theorem fge0iccico 42642
Description: A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
fge0iccico.f (𝜑𝐹:𝑋⟶(0[,]+∞))
fge0iccico.re (𝜑 → ¬ +∞ ∈ ran 𝐹)
Assertion
Ref Expression
fge0iccico (𝜑𝐹:𝑋⟶(0[,)+∞))

Proof of Theorem fge0iccico
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fge0iccico.f . . . 4 (𝜑𝐹:𝑋⟶(0[,]+∞))
21ffnd 6508 . . 3 (𝜑𝐹 Fn 𝑋)
3 0xr 10680 . . . . . 6 0 ∈ ℝ*
43a1i 11 . . . . 5 ((𝜑𝑥𝑋) → 0 ∈ ℝ*)
5 pnfxr 10687 . . . . . 6 +∞ ∈ ℝ*
65a1i 11 . . . . 5 ((𝜑𝑥𝑋) → +∞ ∈ ℝ*)
7 iccssxr 12811 . . . . . 6 (0[,]+∞) ⊆ ℝ*
81ffvelrnda 6844 . . . . . 6 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ (0[,]+∞))
97, 8sseldi 3963 . . . . 5 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ℝ*)
10 iccgelb 12785 . . . . . 6 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑥) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑥))
114, 6, 8, 10syl3anc 1365 . . . . 5 ((𝜑𝑥𝑋) → 0 ≤ (𝐹𝑥))
129adantr 483 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (𝐹𝑥) ∈ ℝ*)
13 simpr 487 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → ¬ (𝐹𝑥) < +∞)
145a1i 11 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ ∈ ℝ*)
1514, 12xrlenltd 10699 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (+∞ ≤ (𝐹𝑥) ↔ ¬ (𝐹𝑥) < +∞))
1613, 15mpbird 259 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ ≤ (𝐹𝑥))
1712, 16xrgepnfd 41588 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (𝐹𝑥) = +∞)
1817eqcomd 2825 . . . . . . 7 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ = (𝐹𝑥))
191ffund 6511 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
2019adantr 483 . . . . . . . . 9 ((𝜑𝑥𝑋) → Fun 𝐹)
21 simpr 487 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
22 fdm 6515 . . . . . . . . . . . . 13 (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
2322eqcomd 2825 . . . . . . . . . . . 12 (𝐹:𝑋⟶(0[,]+∞) → 𝑋 = dom 𝐹)
241, 23syl 17 . . . . . . . . . . 11 (𝜑𝑋 = dom 𝐹)
2524adantr 483 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑋 = dom 𝐹)
2621, 25eleqtrd 2913 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝑥 ∈ dom 𝐹)
27 fvelrn 6837 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
2820, 26, 27syl2anc 586 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2928adantr 483 . . . . . . 7 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (𝐹𝑥) ∈ ran 𝐹)
3018, 29eqeltrd 2911 . . . . . 6 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ ∈ ran 𝐹)
31 fge0iccico.re . . . . . . 7 (𝜑 → ¬ +∞ ∈ ran 𝐹)
3231ad2antrr 724 . . . . . 6 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → ¬ +∞ ∈ ran 𝐹)
3330, 32condan 816 . . . . 5 ((𝜑𝑥𝑋) → (𝐹𝑥) < +∞)
344, 6, 9, 11, 33elicod 12779 . . . 4 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ (0[,)+∞))
3534ralrimiva 3180 . . 3 (𝜑 → ∀𝑥𝑋 (𝐹𝑥) ∈ (0[,)+∞))
362, 35jca 514 . 2 (𝜑 → (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (0[,)+∞)))
37 ffnfv 6875 . 2 (𝐹:𝑋⟶(0[,)+∞) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (0[,)+∞)))
3836, 37sylibr 236 1 (𝜑𝐹:𝑋⟶(0[,)+∞))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1530  wcel 2107  wral 3136   class class class wbr 5057  dom cdm 5548  ran crn 5549  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7148  0cc0 10529  +∞cpnf 10664  *cxr 10666   < clt 10667  cle 10668  [,)cico 12732  [,]cicc 12733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-addrcl 10590  ax-rnegex 10600  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-ico 12736  df-icc 12737
This theorem is referenced by:  fge0iccre  42646  sge00  42648  sge0sn  42651  sge0tsms  42652  sge0cl  42653  sge0supre  42661  sge0sup  42663  sge0less  42664  sge0rnbnd  42665  sge0ltfirp  42672  sge0resplit  42678  sge0le  42679  sge0split  42681  sge0iunmptlemre  42687
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