Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsupvaluz2 Structured version   Visualization version   GIF version

Theorem limsupvaluz2 42367
 Description: The superior limit, when the domain of a real-valued function is a set of upper integers, and the superior limit is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupvaluz2.m (𝜑𝑀 ∈ ℤ)
limsupvaluz2.z 𝑍 = (ℤ𝑀)
limsupvaluz2.f (𝜑𝐹:𝑍⟶ℝ)
limsupvaluz2.r (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Assertion
Ref Expression
limsupvaluz2 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < ))
Distinct variable groups:   𝑘,𝐹   𝑘,𝑍
Allowed substitution hints:   𝜑(𝑘)   𝑀(𝑘)

Proof of Theorem limsupvaluz2
Dummy variables 𝑖 𝑗 𝑥 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupvaluz2.m . . 3 (𝜑𝑀 ∈ ℤ)
2 limsupvaluz2.z . . 3 𝑍 = (ℤ𝑀)
3 limsupvaluz2.f . . . 4 (𝜑𝐹:𝑍⟶ℝ)
43frexr 42006 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
51, 2, 4limsupvaluz 42337 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ))
63adantr 484 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐹:𝑍⟶ℝ)
7 id 22 . . . . . . . . . . 11 (𝑛𝑍𝑛𝑍)
82, 7uzssd2 42041 . . . . . . . . . 10 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
98adantl 485 . . . . . . . . 9 ((𝜑𝑛𝑍) → (ℤ𝑛) ⊆ 𝑍)
106, 9feqresmpt 6713 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹 ↾ (ℤ𝑛)) = (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
1110rneqd 5776 . . . . . . 7 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
1211supeq1d 8898 . . . . . 6 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ))
13 nfcv 2958 . . . . . . . . . 10 𝑚𝐹
14 limsupvaluz2.r . . . . . . . . . . 11 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
1514renepnfd 10685 . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) ≠ +∞)
1613, 2, 3, 15limsupubuz 42342 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
1716adantr 484 . . . . . . . 8 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
18 ssralv 3984 . . . . . . . . . . 11 ((ℤ𝑛) ⊆ 𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
198, 18syl 17 . . . . . . . . . 10 (𝑛𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2019adantl 485 . . . . . . . . 9 ((𝜑𝑛𝑍) → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2120reximdv 3235 . . . . . . . 8 ((𝜑𝑛𝑍) → (∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2217, 21mpd 15 . . . . . . 7 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥)
23 nfv 1915 . . . . . . . 8 𝑚(𝜑𝑛𝑍)
242eluzelz2 42027 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ ℤ)
25 uzid 12250 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
26 ne0i 4253 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
2724, 25, 263syl 18 . . . . . . . . 9 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
2827adantl 485 . . . . . . . 8 ((𝜑𝑛𝑍) → (ℤ𝑛) ≠ ∅)
296adantr 484 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐹:𝑍⟶ℝ)
309sselda 3918 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
3129, 30ffvelrnd 6833 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚) ∈ ℝ)
3223, 28, 31supxrre3rnmpt 42053 . . . . . . 7 ((𝜑𝑛𝑍) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
3322, 32mpbird 260 . . . . . 6 ((𝜑𝑛𝑍) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ)
3412, 33eqeltrd 2893 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ ℝ)
3534fmpttd 6860 . . . 4 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )):𝑍⟶ℝ)
3635frnd 6498 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⊆ ℝ)
37 nfv 1915 . . . 4 𝑛𝜑
3834elexd 3464 . . . 4 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ V)
39 eqid 2801 . . . 4 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
401, 2uzn0d 42049 . . . 4 (𝜑𝑍 ≠ ∅)
4137, 38, 39, 40rnmptn0 41837 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ≠ ∅)
42 nfcv 2958 . . . . . . . . . 10 𝑗𝐹
4342, 1, 2, 4limsupre3uz 42365 . . . . . . . . 9 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥)))
4414, 43mpbid 235 . . . . . . . 8 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥))
4544simpld 498 . . . . . . 7 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗))
46 simp-4r 783 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ)
4746rexrd 10684 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ*)
4843ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝐹:𝑍⟶ℝ*)
492uztrn2 12254 . . . . . . . . . . . . . 14 ((𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
50493adant1 1127 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
5148, 50ffvelrnd 6833 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ℝ*)
5251ad5ant134 1364 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
53 rnresss 5858 . . . . . . . . . . . . . . . 16 ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹
5453a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹)
553frnd 6498 . . . . . . . . . . . . . . . 16 (𝜑 → ran 𝐹 ⊆ ℝ)
5655adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → ran 𝐹 ⊆ ℝ)
5754, 56sstrd 3928 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ)
58 ressxr 10678 . . . . . . . . . . . . . . 15 ℝ ⊆ ℝ*
5958a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → ℝ ⊆ ℝ*)
6057, 59sstrd 3928 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
6160supxrcld 41730 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
6261ad5ant13 756 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
63 simpr 488 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
64603adant3 1129 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
65 fvres 6668 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑖) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) = (𝐹𝑗))
6665eqcomd 2807 . . . . . . . . . . . . . . 15 (𝑗 ∈ (ℤ𝑖) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
67663ad2ant3 1132 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
683ffnd 6492 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn 𝑍)
6968adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝐹 Fn 𝑍)
70 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑍𝑖𝑍)
712, 70uzssd2 42041 . . . . . . . . . . . . . . . . . 18 (𝑖𝑍 → (ℤ𝑖) ⊆ 𝑍)
7271adantl 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → (ℤ𝑖) ⊆ 𝑍)
73 fnssres 6446 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝑍 ∧ (ℤ𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
7469, 72, 73syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
75743adant3 1129 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
76 simp3 1135 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗 ∈ (ℤ𝑖))
77 fnfvelrn 6829 . . . . . . . . . . . . . . 15 (((𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖) ∧ 𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
7875, 76, 77syl2anc 587 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
7967, 78eqeltrd 2893 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
80 eqid 2801 . . . . . . . . . . . . 13 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )
8164, 79, 80supxrubd 41736 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8281ad5ant134 1364 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8347, 52, 62, 63, 82xrletrd 12547 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8483rexlimdva2 3249 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) → (∃𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8584ralimdva 3147 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8685reximdva 3236 . . . . . . 7 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8745, 86mpd 15 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8887idi 1 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
89 fveq2 6649 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
9089reseq2d 5822 . . . . . . . . . . 11 (𝑛 = 𝑖 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑖)))
9190rneqd 5776 . . . . . . . . . 10 (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑖)))
9291supeq1d 8898 . . . . . . . . 9 (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
93 eqcom 2808 . . . . . . . . . . 11 (𝑛 = 𝑖𝑖 = 𝑛)
9493imbi1i 353 . . . . . . . . . 10 ((𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )) ↔ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
95 eqcom 2808 . . . . . . . . . . 11 (sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
9695imbi2i 339 . . . . . . . . . 10 ((𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )) ↔ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
9794, 96bitri 278 . . . . . . . . 9 ((𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )) ↔ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
9892, 97mpbi 233 . . . . . . . 8 (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
9998breq2d 5045 . . . . . . 7 (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
10099cbvralvw 3399 . . . . . 6 (∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
101100rexbii 3213 . . . . 5 (∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
10288, 101sylib 221 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
10337, 38rnmptbd2 41874 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))𝑥𝑦))
104102, 103mpbid 235 . . 3 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))𝑥𝑦)
105 infxrre 12721 . . 3 ((ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⊆ ℝ ∧ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))𝑥𝑦) → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
10636, 41, 104, 105syl3anc 1368 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
107 fveq2 6649 . . . . . . . . 9 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
108107reseq2d 5822 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
109108rneqd 5776 . . . . . . 7 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
110109supeq1d 8898 . . . . . 6 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
111110cbvmptv 5136 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
112111rneqi 5775 . . . 4 ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
113112infeq1i 8930 . . 3 inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < )
114113a1i 11 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < ))
1155, 106, 1143eqtrd 2840 1 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  Vcvv 3444   ⊆ wss 3884  ∅c0 4246   class class class wbr 5033   ↦ cmpt 5113  ran crn 5524   ↾ cres 5525   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  supcsup 8892  infcinf 8893  ℝcr 10529  ℝ*cxr 10667   < clt 10668   ≤ cle 10669  ℤcz 11973  ℤ≥cuz 12235  lim supclsp 14822 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-ico 12736  df-fz 12890  df-fl 13161  df-ceil 13162  df-limsup 14823 This theorem is referenced by:  supcnvlimsup  42369
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