Theorem limsupvald 45858

Description: The superior limit of a sequence 𝐹 of extended real numbers is the infimum of the set of suprema of all restrictions of 𝐹 to an upperset of reals . (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
limsupvald.1	⊢ (𝜑 → 𝐹 ∈ 𝑉)
limsupvald.2	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))

Assertion

Ref	Expression
limsupvald	⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))

Distinct variable group:   𝑘,𝐹

Allowed substitution hints:   𝜑(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem limsupvald

Step	Hyp	Ref	Expression
1		limsupvald.1	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		limsupvald.2	. . 3 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
3	2	limsupval 15387	. 2 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
4	1, 3	syl 17	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ∩ cin 3896   ↦ cmpt 5174  ran crn 5620   “ cima 5622  ‘cfv 6487  (class class class)co 7352  supcsup 9330  infcinf 9331  ℝcr 11011  +∞cpnf 11149  ℝ^*cxr 11151   < clt 11152  [,)cico 13253  lim supclsp 15383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-pre-lttri 11086  ax-pre-lttrn 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-sup 9332  df-inf 9333  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-limsup 15384

This theorem is referenced by:  liminflelimsuplem  45878