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Theorem limsupvaluz2 39406
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 39103 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
51, 2, 4limsupvaluz 39376 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ))
63adantr 481 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐹:𝑍⟶ℝ)
7 id 22 . . . . . . . . . . 11 (𝑛𝑍𝑛𝑍)
82, 7uzssd2 39143 . . . . . . . . . 10 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
98adantl 482 . . . . . . . . 9 ((𝜑𝑛𝑍) → (ℤ𝑛) ⊆ 𝑍)
106, 9feqresmpt 6217 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹 ↾ (ℤ𝑛)) = (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
1110rneqd 5323 . . . . . . 7 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
1211supeq1d 8312 . . . . . 6 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ))
13 nfcv 2761 . . . . . . . . . 10 𝑚𝐹
14 limsupvaluz2.r . . . . . . . . . . 11 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
1514renepnfd 10050 . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) ≠ +∞)
1613, 2, 3, 15limsupubuz 39381 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
1716adantr 481 . . . . . . . 8 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
18 ssralv 3651 . . . . . . . . . . 11 ((ℤ𝑛) ⊆ 𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
198, 18syl 17 . . . . . . . . . 10 (𝑛𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2019adantl 482 . . . . . . . . 9 ((𝜑𝑛𝑍) → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2120reximdv 3012 . . . . . . . 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 1840 . . . . . . . 8 𝑚(𝜑𝑛𝑍)
242eluzelz2 39126 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ ℤ)
25 uzid 11662 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
26 ne0i 3903 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
2724, 25, 263syl 18 . . . . . . . . 9 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
2827adantl 482 . . . . . . . 8 ((𝜑𝑛𝑍) → (ℤ𝑛) ≠ ∅)
296adantr 481 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐹:𝑍⟶ℝ)
309sselda 3588 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
3129, 30ffvelrnd 6326 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚) ∈ ℝ)
3223, 28, 31supxrre3rnmpt 39155 . . . . . . 7 ((𝜑𝑛𝑍) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
3322, 32mpbird 247 . . . . . 6 ((𝜑𝑛𝑍) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ)
3412, 33eqeltrd 2698 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ ℝ)
35 eqid 2621 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
3634, 35fmptd 6351 . . . 4 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )):𝑍⟶ℝ)
3736frnd 38935 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⊆ ℝ)
38 nfv 1840 . . . 4 𝑛𝜑
3934elexd 3204 . . . 4 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ V)
401, 2uzn0d 39151 . . . 4 (𝜑𝑍 ≠ ∅)
4138, 39, 35, 40rnmptn0 38922 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ≠ ∅)
42 nfcv 2761 . . . . . . . . . 10 𝑗𝐹
4342, 1, 2, 4limsupre3uz 39404 . . . . . . . . 9 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥)))
4414, 43mpbid 222 . . . . . . . 8 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥))
4544simpld 475 . . . . . . 7 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗))
46 simp-4r 806 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ)
4746rexrd 10049 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ*)
4843ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝐹:𝑍⟶ℝ*)
492uztrn2 11665 . . . . . . . . . . . . . . 15 ((𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
50493adant1 1077 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
5148, 50ffvelrnd 6326 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ℝ*)
5251ad5ant134 1310 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
53 rnresss 38874 . . . . . . . . . . . . . . . . 17 ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹
5453a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹)
553frnd 38935 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ⊆ ℝ)
5655adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → ran 𝐹 ⊆ ℝ)
5754, 56sstrd 3598 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ)
58 ressxr 10043 . . . . . . . . . . . . . . . 16 ℝ ⊆ ℝ*
5958a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → ℝ ⊆ ℝ*)
6057, 59sstrd 3598 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
6160supxrcld 38814 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
6261ad5ant13 1298 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
63 simpr 477 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
64603adant3 1079 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
65 fvres 6174 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑖) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) = (𝐹𝑗))
6665eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑖) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
67663ad2ant3 1082 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
683ffnd 6013 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 Fn 𝑍)
6968adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → 𝐹 Fn 𝑍)
70 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑖𝑍𝑖𝑍)
712, 70uzssd2 39143 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑍 → (ℤ𝑖) ⊆ 𝑍)
7271adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → (ℤ𝑖) ⊆ 𝑍)
73 fnssres 5972 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝑍 ∧ (ℤ𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
7469, 72, 73syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
75743adant3 1079 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
76 simp3 1061 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗 ∈ (ℤ𝑖))
77 fnfvelrn 6322 . . . . . . . . . . . . . . . 16 (((𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖) ∧ 𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
7875, 76, 77syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
7967, 78eqeltrd 2698 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
80 eqid 2621 . . . . . . . . . . . . . 14 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )
8164, 79, 80supxrubd 38821 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8281ad5ant134 1310 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8347, 52, 62, 63, 82xrletrd 11953 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8483ex 450 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) → (𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8584rexlimdva 3026 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) → (∃𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8685ralimdva 2958 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8786reximdva 3013 . . . . . . 7 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8845, 87mpd 15 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8988idi 2 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
90 fveq2 6158 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
9190reseq2d 5366 . . . . . . . . . . 11 (𝑛 = 𝑖 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑖)))
9291rneqd 5323 . . . . . . . . . 10 (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑖)))
9392supeq1d 8312 . . . . . . . . 9 (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
94 eqcom 2628 . . . . . . . . . . 11 (𝑛 = 𝑖𝑖 = 𝑛)
9594imbi1i 339 . . . . . . . . . 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*, < )))
96 eqcom 2628 . . . . . . . . . . 11 (sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
9796imbi2i 326 . . . . . . . . . 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*, < )))
9895, 97bitri 264 . . . . . . . . 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*, < )))
9993, 98mpbi 220 . . . . . . . 8 (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
10099breq2d 4635 . . . . . . 7 (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
101100cbvralv 3163 . . . . . 6 (∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
102101rexbii 3036 . . . . 5 (∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
10389, 102sylib 208 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
10438, 39rnmptbd2 38975 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))𝑥𝑦))
105103, 104mpbid 222 . . 3 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))𝑥𝑦)
106 infxrre 12125 . . 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, < ))
10737, 41, 105, 106syl3anc 1323 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
108 fveq2 6158 . . . . . . . . 9 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
109108reseq2d 5366 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
110109rneqd 5323 . . . . . . 7 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
111110supeq1d 8312 . . . . . 6 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
112111cbvmptv 4720 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
113112rneqi 5322 . . . 4 ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
114113infeq1i 8344 . . 3 inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < )
115114a1i 11 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < ))
1165, 107, 1153eqtrd 2659 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  Vcvv 3190  wss 3560  c0 3897   class class class wbr 4623  cmpt 4683  ran crn 5085  cres 5086   Fn wfn 5852  wf 5853  cfv 5857  supcsup 8306  infcinf 8307  cr 9895  *cxr 10033   < clt 10034  cle 10035  cz 11337  cuz 11647  lim supclsp 14151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-ico 12139  df-fz 12285  df-fl 12549  df-ceil 12550  df-limsup 14152
This theorem is referenced by:  supcnvlimsup  39408
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