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Theorem limsupvaluz2 45753
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 45396 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
51, 2, 4limsupvaluz 45723 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ))
63adantr 480 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐹:𝑍⟶ℝ)
72uzssd3 45437 . . . . . . . . . 10 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
87adantl 481 . . . . . . . . 9 ((𝜑𝑛𝑍) → (ℤ𝑛) ⊆ 𝑍)
96, 8feqresmpt 6978 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹 ↾ (ℤ𝑛)) = (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
109rneqd 5949 . . . . . . 7 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
1110supeq1d 9486 . . . . . 6 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ))
12 nfcv 2905 . . . . . . . . . 10 𝑚𝐹
13 limsupvaluz2.r . . . . . . . . . . 11 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
1413renepnfd 11312 . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) ≠ +∞)
1512, 2, 3, 14limsupubuz 45728 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
1615adantr 480 . . . . . . . 8 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
17 ssralv 4052 . . . . . . . . . . 11 ((ℤ𝑛) ⊆ 𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
187, 17syl 17 . . . . . . . . . 10 (𝑛𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
1918adantl 481 . . . . . . . . 9 ((𝜑𝑛𝑍) → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2019reximdv 3170 . . . . . . . 8 ((𝜑𝑛𝑍) → (∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2116, 20mpd 15 . . . . . . 7 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥)
22 nfv 1914 . . . . . . . 8 𝑚(𝜑𝑛𝑍)
232eluzelz2 45414 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ ℤ)
24 uzid 12893 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
25 ne0i 4341 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
2623, 24, 253syl 18 . . . . . . . . 9 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
2726adantl 481 . . . . . . . 8 ((𝜑𝑛𝑍) → (ℤ𝑛) ≠ ∅)
286adantr 480 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐹:𝑍⟶ℝ)
298sselda 3983 . . . . . . . . 9 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
3028, 29ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚) ∈ ℝ)
3122, 27, 30supxrre3rnmpt 45440 . . . . . . 7 ((𝜑𝑛𝑍) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
3221, 31mpbird 257 . . . . . 6 ((𝜑𝑛𝑍) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ)
3311, 32eqeltrd 2841 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ ℝ)
3433fmpttd 7135 . . . 4 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )):𝑍⟶ℝ)
3534frnd 6744 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⊆ ℝ)
36 nfv 1914 . . . 4 𝑛𝜑
37 eqid 2737 . . . 4 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
381, 2uzn0d 45436 . . . 4 (𝜑𝑍 ≠ ∅)
3936, 33, 37, 38rnmptn0 6264 . . 3 (𝜑 → ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ≠ ∅)
40 nfcv 2905 . . . . . . . . 9 𝑗𝐹
4140, 1, 2, 4limsupre3uz 45751 . . . . . . . 8 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥)))
4213, 41mpbid 232 . . . . . . 7 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥))
4342simpld 494 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗))
44 simp-4r 784 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ)
4544rexrd 11311 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ*)
4643ad2ant1 1134 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝐹:𝑍⟶ℝ*)
472uztrn2 12897 . . . . . . . . . . . . 13 ((𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
48473adant1 1131 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
4946, 48ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ℝ*)
5049ad5ant134 1369 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
51 rnresss 6035 . . . . . . . . . . . . . 14 ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹
523frnd 6744 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ ℝ)
5352adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → ran 𝐹 ⊆ ℝ)
5451, 53sstrid 3995 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ)
5554ssrexr 45443 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
5655supxrcld 45112 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
5756ad5ant13 757 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
58 simpr 484 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
59553adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
60 fvres 6925 . . . . . . . . . . . . . . 15 (𝑗 ∈ (ℤ𝑖) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) = (𝐹𝑗))
6160eqcomd 2743 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑖) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
62613ad2ant3 1136 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
633ffnd 6737 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝑍)
642uzssd3 45437 . . . . . . . . . . . . . . 15 (𝑖𝑍 → (ℤ𝑖) ⊆ 𝑍)
65 fnssres 6691 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝑍 ∧ (ℤ𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
6663, 64, 65syl2an 596 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
67 fnfvelrn 7100 . . . . . . . . . . . . . 14 (((𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖) ∧ 𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
6866, 67stoic3 1776 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
6962, 68eqeltrd 2841 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
70 eqid 2737 . . . . . . . . . . . 12 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )
7159, 69, 70supxrubd 45118 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7271ad5ant134 1369 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7345, 50, 57, 58, 72xrletrd 13204 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7473rexlimdva2 3157 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) → (∃𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
7574ralimdva 3167 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
7675reximdva 3168 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
7743, 76mpd 15 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
78 fveq2 6906 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
7978reseq2d 5997 . . . . . . . . . . 11 (𝑛 = 𝑖 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑖)))
8079rneqd 5949 . . . . . . . . . 10 (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑖)))
8180supeq1d 9486 . . . . . . . . 9 (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
82 eqcom 2744 . . . . . . . . 9 (𝑛 = 𝑖𝑖 = 𝑛)
83 eqcom 2744 . . . . . . . . 9 (sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
8481, 82, 833imtr3i 291 . . . . . . . 8 (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
8584breq2d 5155 . . . . . . 7 (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
8685cbvralvw 3237 . . . . . 6 (∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
8786rexbii 3094 . . . . 5 (∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
8877, 87sylib 218 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
8936, 33rnmptbd2 45256 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))𝑥𝑦))
9088, 89mpbid 232 . . 3 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))𝑥𝑦)
91 infxrre 13378 . . 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, < ))
9235, 39, 90, 91syl3anc 1373 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
93 fveq2 6906 . . . . . . . . 9 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
9493reseq2d 5997 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
9594rneqd 5949 . . . . . . 7 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
9695supeq1d 9486 . . . . . 6 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
9796cbvmptv 5255 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
9897rneqi 5948 . . . 4 ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
9998infeq1i 9518 . . 3 inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < )
10099a1i 11 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < ))
1015, 92, 1003eqtrd 2781 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 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333   class class class wbr 5143  cmpt 5225  ran crn 5686  cres 5687   Fn wfn 6556  wf 6557  cfv 6561  supcsup 9480  infcinf 9481  cr 11154  *cxr 11294   < clt 11295  cle 11296  cz 12613  cuz 12878  lim supclsp 15506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-ico 13393  df-fz 13548  df-fl 13832  df-ceil 13833  df-limsup 15507
This theorem is referenced by:  supcnvlimsup  45755
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