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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  cr 10529  *cxr 10667   < clt 10668  cle 10669  cz 11973  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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