Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fge0iccico Structured version   Visualization version   GIF version

Theorem fge0iccico 41330
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 6257 . . 3 (𝜑𝐹 Fn 𝑋)
3 0xr 10375 . . . . . 6 0 ∈ ℝ*
43a1i 11 . . . . 5 ((𝜑𝑥𝑋) → 0 ∈ ℝ*)
5 pnfxr 10382 . . . . . 6 +∞ ∈ ℝ*
65a1i 11 . . . . 5 ((𝜑𝑥𝑋) → +∞ ∈ ℝ*)
7 iccssxr 12505 . . . . . 6 (0[,]+∞) ⊆ ℝ*
81ffvelrnda 6585 . . . . . 6 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ (0[,]+∞))
97, 8sseldi 3796 . . . . 5 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ℝ*)
10 iccgelb 12479 . . . . . 6 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑥) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑥))
114, 6, 8, 10syl3anc 1491 . . . . 5 ((𝜑𝑥𝑋) → 0 ≤ (𝐹𝑥))
129adantr 473 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (𝐹𝑥) ∈ ℝ*)
13 simpr 478 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → ¬ (𝐹𝑥) < +∞)
145a1i 11 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ ∈ ℝ*)
1514, 12xrlenltd 10394 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (+∞ ≤ (𝐹𝑥) ↔ ¬ (𝐹𝑥) < +∞))
1613, 15mpbird 249 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ ≤ (𝐹𝑥))
1712, 16xrgepnfd 40291 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (𝐹𝑥) = +∞)
1817eqcomd 2805 . . . . . . 7 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ = (𝐹𝑥))
191ffund 6260 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
2019adantr 473 . . . . . . . . 9 ((𝜑𝑥𝑋) → Fun 𝐹)
21 simpr 478 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
22 fdm 6264 . . . . . . . . . . . . 13 (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
2322eqcomd 2805 . . . . . . . . . . . 12 (𝐹:𝑋⟶(0[,]+∞) → 𝑋 = dom 𝐹)
241, 23syl 17 . . . . . . . . . . 11 (𝜑𝑋 = dom 𝐹)
2524adantr 473 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑋 = dom 𝐹)
2621, 25eleqtrd 2880 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝑥 ∈ dom 𝐹)
27 fvelrn 6578 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
2820, 26, 27syl2anc 580 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2928adantr 473 . . . . . . 7 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → (𝐹𝑥) ∈ ran 𝐹)
3018, 29eqeltrd 2878 . . . . . 6 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → +∞ ∈ ran 𝐹)
31 fge0iccico.re . . . . . . 7 (𝜑 → ¬ +∞ ∈ ran 𝐹)
3231ad2antrr 718 . . . . . 6 (((𝜑𝑥𝑋) ∧ ¬ (𝐹𝑥) < +∞) → ¬ +∞ ∈ ran 𝐹)
3330, 32condan 853 . . . . 5 ((𝜑𝑥𝑋) → (𝐹𝑥) < +∞)
344, 6, 9, 11, 33elicod 12473 . . . 4 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ (0[,)+∞))
3534ralrimiva 3147 . . 3 (𝜑 → ∀𝑥𝑋 (𝐹𝑥) ∈ (0[,)+∞))
362, 35jca 508 . 2 (𝜑 → (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (0[,)+∞)))
37 ffnfv 6614 . 2 (𝐹:𝑋⟶(0[,)+∞) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (0[,)+∞)))
3836, 37sylibr 226 1 (𝜑𝐹:𝑋⟶(0[,)+∞))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089   class class class wbr 4843  dom cdm 5312  ran crn 5313  Fun wfun 6095   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  0cc0 10224  +∞cpnf 10360  *cxr 10362   < clt 10363  cle 10364  [,)cico 12426  [,]cicc 12427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-addrcl 10285  ax-rnegex 10295  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-po 5233  df-so 5234  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-ico 12430  df-icc 12431
This theorem is referenced by:  fge0iccre  41334  sge00  41336  sge0sn  41339  sge0tsms  41340  sge0cl  41341  sge0supre  41349  sge0sup  41351  sge0less  41352  sge0rnbnd  41353  sge0ltfirp  41360  sge0resplit  41366  sge0le  41367  sge0split  41369  sge0iunmptlemre  41375
  Copyright terms: Public domain W3C validator