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Theorem limsupvaluz2 40473
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 40102 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
51, 2, 4limsupvaluz 40443 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ))
63adantr 472 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐹:𝑍⟶ℝ)
7 id 22 . . . . . . . . . . 11 (𝑛𝑍𝑛𝑍)
82, 7uzssd2 40142 . . . . . . . . . 10 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
98adantl 473 . . . . . . . . 9 ((𝜑𝑛𝑍) → (ℤ𝑛) ⊆ 𝑍)
106, 9feqresmpt 6412 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹 ↾ (ℤ𝑛)) = (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
1110rneqd 5508 . . . . . . 7 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
1211supeq1d 8517 . . . . . 6 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ))
13 nfcv 2902 . . . . . . . . . 10 𝑚𝐹
14 limsupvaluz2.r . . . . . . . . . . 11 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
1514renepnfd 10282 . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) ≠ +∞)
1613, 2, 3, 15limsupubuz 40448 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
1716adantr 472 . . . . . . . 8 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
18 ssralv 3807 . . . . . . . . . . 11 ((ℤ𝑛) ⊆ 𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
198, 18syl 17 . . . . . . . . . 10 (𝑛𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2019adantl 473 . . . . . . . . 9 ((𝜑𝑛𝑍) → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2120reximdv 3154 . . . . . . . 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 1992 . . . . . . . 8 𝑚(𝜑𝑛𝑍)
242eluzelz2 40125 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ ℤ)
25 uzid 11894 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
26 ne0i 4064 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
2724, 25, 263syl 18 . . . . . . . . 9 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
2827adantl 473 . . . . . . . 8 ((𝜑𝑛𝑍) → (ℤ𝑛) ≠ ∅)
296adantr 472 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐹:𝑍⟶ℝ)
309sselda 3744 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
3129, 30ffvelrnd 6523 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚) ∈ ℝ)
3223, 28, 31supxrre3rnmpt 40154 . . . . . . 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 2839 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ ℝ)
35 eqid 2760 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
3634, 35fmptd 6548 . . . 4 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )):𝑍⟶ℝ)
3736frnd 39925 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⊆ ℝ)
38 nfv 1992 . . . 4 𝑛𝜑
3934elexd 3354 . . . 4 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ V)
401, 2uzn0d 40150 . . . 4 (𝜑𝑍 ≠ ∅)
4138, 39, 35, 40rnmptn0 39912 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ≠ ∅)
42 nfcv 2902 . . . . . . . . . 10 𝑗𝐹
4342, 1, 2, 4limsupre3uz 40471 . . . . . . . . 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 477 . . . . . . 7 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗))
46 simp-4r 827 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ)
4746rexrd 10281 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ*)
4843ad2ant1 1128 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝐹:𝑍⟶ℝ*)
492uztrn2 11897 . . . . . . . . . . . . . . 15 ((𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
50493adant1 1125 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
5148, 50ffvelrnd 6523 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ℝ*)
5251ad5ant134 1467 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
53 rnresss 39864 . . . . . . . . . . . . . . . . 17 ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹
5453a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹)
553frnd 39925 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ⊆ ℝ)
5655adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → ran 𝐹 ⊆ ℝ)
5754, 56sstrd 3754 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ)
58 ressxr 10275 . . . . . . . . . . . . . . . 16 ℝ ⊆ ℝ*
5958a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → ℝ ⊆ ℝ*)
6057, 59sstrd 3754 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
6160supxrcld 39789 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
6261ad5ant13 1216 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
63 simpr 479 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
64603adant3 1127 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
65 fvres 6368 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑖) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) = (𝐹𝑗))
6665eqcomd 2766 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑖) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
67663ad2ant3 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
683ffnd 6207 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 Fn 𝑍)
6968adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → 𝐹 Fn 𝑍)
70 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑖𝑍𝑖𝑍)
712, 70uzssd2 40142 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑍 → (ℤ𝑖) ⊆ 𝑍)
7271adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → (ℤ𝑖) ⊆ 𝑍)
73 fnssres 6165 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝑍 ∧ (ℤ𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
7469, 72, 73syl2anc 696 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
75743adant3 1127 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
76 simp3 1133 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗 ∈ (ℤ𝑖))
77 fnfvelrn 6519 . . . . . . . . . . . . . . . 16 (((𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖) ∧ 𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
7875, 76, 77syl2anc 696 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
7967, 78eqeltrd 2839 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
80 eqid 2760 . . . . . . . . . . . . . 14 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )
8164, 79, 80supxrubd 39796 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8281ad5ant134 1467 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8347, 52, 62, 63, 82xrletrd 12186 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8483ex 449 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) → (𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8584rexlimdva 3169 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) → (∃𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8685ralimdva 3100 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8786reximdva 3155 . . . . . . 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 6352 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
9190reseq2d 5551 . . . . . . . . . . 11 (𝑛 = 𝑖 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑖)))
9291rneqd 5508 . . . . . . . . . 10 (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑖)))
9392supeq1d 8517 . . . . . . . . 9 (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
94 eqcom 2767 . . . . . . . . . . 11 (𝑛 = 𝑖𝑖 = 𝑛)
9594imbi1i 338 . . . . . . . . . 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 2767 . . . . . . . . . . 11 (sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
9796imbi2i 325 . . . . . . . . . 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 4816 . . . . . . 7 (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
101100cbvralv 3310 . . . . . 6 (∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
102101rexbii 3179 . . . . 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 39963 . . . 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 12359 . . 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 1477 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
108 fveq2 6352 . . . . . . . . 9 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
109108reseq2d 5551 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
110109rneqd 5508 . . . . . . 7 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
111110supeq1d 8517 . . . . . 6 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
112111cbvmptv 4902 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
113112rneqi 5507 . . . 4 ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
114113infeq1i 8549 . . 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 2798 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 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  wss 3715  c0 4058   class class class wbr 4804  cmpt 4881  ran crn 5267  cres 5268   Fn wfn 6044  wf 6045  cfv 6049  supcsup 8511  infcinf 8512  cr 10127  *cxr 10265   < clt 10266  cle 10267  cz 11569  cuz 11879  lim supclsp 14400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-ico 12374  df-fz 12520  df-fl 12787  df-ceil 12788  df-limsup 14401
This theorem is referenced by:  supcnvlimsup  40475
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