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Theorem supcnvlimsup 42201
Description: If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
supcnvlimsup.m (𝜑𝑀 ∈ ℤ)
supcnvlimsup.z 𝑍 = (ℤ𝑀)
supcnvlimsup.f (𝜑𝐹:𝑍⟶ℝ)
supcnvlimsup.r (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Assertion
Ref Expression
supcnvlimsup (𝜑 → (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹))
Distinct variable groups:   𝑘,𝐹   𝑘,𝑍
Allowed substitution hints:   𝜑(𝑘)   𝑀(𝑘)

Proof of Theorem supcnvlimsup
Dummy variables 𝑖 𝑗 𝑥 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supcnvlimsup.z . . 3 𝑍 = (ℤ𝑀)
2 supcnvlimsup.m . . 3 (𝜑𝑀 ∈ ℤ)
3 supcnvlimsup.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶ℝ)
43adantr 484 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝐹:𝑍⟶ℝ)
5 id 22 . . . . . . . . . 10 (𝑛𝑍𝑛𝑍)
61, 5uzssd2 41873 . . . . . . . . 9 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
76adantl 485 . . . . . . . 8 ((𝜑𝑛𝑍) → (ℤ𝑛) ⊆ 𝑍)
84, 7feqresmpt 6707 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹 ↾ (ℤ𝑛)) = (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
98rneqd 5781 . . . . . 6 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
109supeq1d 8886 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ))
11 nfcv 2974 . . . . . . . . 9 𝑚𝐹
12 supcnvlimsup.r . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
1312renepnfd 10669 . . . . . . . . 9 (𝜑 → (lim sup‘𝐹) ≠ +∞)
1411, 1, 3, 13limsupubuz 42174 . . . . . . . 8 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
1514adantr 484 . . . . . . 7 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
16 ssralv 4009 . . . . . . . . . 10 ((ℤ𝑛) ⊆ 𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
176, 16syl 17 . . . . . . . . 9 (𝑛𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
1817adantl 485 . . . . . . . 8 ((𝜑𝑛𝑍) → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
1918reximdv 3259 . . . . . . 7 ((𝜑𝑛𝑍) → (∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
2015, 19mpd 15 . . . . . 6 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥)
21 nfv 1916 . . . . . . 7 𝑚(𝜑𝑛𝑍)
221eluzelz2 41858 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ ℤ)
23 uzid 12236 . . . . . . . . 9 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
24 ne0i 4273 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
2522, 23, 243syl 18 . . . . . . . 8 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
2625adantl 485 . . . . . . 7 ((𝜑𝑛𝑍) → (ℤ𝑛) ≠ ∅)
274adantr 484 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐹:𝑍⟶ℝ)
287sselda 3943 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
2927, 28ffvelrnd 6825 . . . . . . 7 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚) ∈ ℝ)
3021, 26, 29supxrre3rnmpt 41885 . . . . . 6 ((𝜑𝑛𝑍) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
3120, 30mpbird 260 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ)
3210, 31eqeltrd 2912 . . . 4 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ ℝ)
3332fmpttd 6852 . . 3 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )):𝑍⟶ℝ)
34 eqid 2821 . . . . . . . . . 10 (ℤ𝑖) = (ℤ𝑖)
351eluzelz2 41858 . . . . . . . . . 10 (𝑖𝑍𝑖 ∈ ℤ)
3635peano2zd 12068 . . . . . . . . . 10 (𝑖𝑍 → (𝑖 + 1) ∈ ℤ)
3735zred 12065 . . . . . . . . . . 11 (𝑖𝑍𝑖 ∈ ℝ)
38 lep1 11458 . . . . . . . . . . 11 (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1))
3937, 38syl 17 . . . . . . . . . 10 (𝑖𝑍𝑖 ≤ (𝑖 + 1))
4034, 35, 36, 39eluzd 41864 . . . . . . . . 9 (𝑖𝑍 → (𝑖 + 1) ∈ (ℤ𝑖))
41 uzss 12243 . . . . . . . . 9 ((𝑖 + 1) ∈ (ℤ𝑖) → (ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖))
4240, 41syl 17 . . . . . . . 8 (𝑖𝑍 → (ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖))
43 ssres2 5854 . . . . . . . 8 ((ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖) → (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)))
4442, 43syl 17 . . . . . . 7 (𝑖𝑍 → (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)))
45 rnss 5782 . . . . . . 7 ((𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
4644, 45syl 17 . . . . . 6 (𝑖𝑍 → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
4746adantl 485 . . . . 5 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
48 rnresss 41621 . . . . . . . 8 ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹
4948a1i 11 . . . . . . 7 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹)
503frnd 6494 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℝ)
5150adantr 484 . . . . . . 7 ((𝜑𝑖𝑍) → ran 𝐹 ⊆ ℝ)
5249, 51sstrd 3953 . . . . . 6 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ)
53 ressxr 10662 . . . . . . 7 ℝ ⊆ ℝ*
5453a1i 11 . . . . . 6 ((𝜑𝑖𝑍) → ℝ ⊆ ℝ*)
5552, 54sstrd 3953 . . . . 5 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
56 supxrss 12703 . . . . 5 ((ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)) ∧ ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*) → sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
5747, 55, 56syl2anc 587 . . . 4 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
58 eqidd 2822 . . . . . . 7 (𝑖𝑍 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
59 fveq2 6643 . . . . . . . . . . 11 (𝑛 = (𝑖 + 1) → (ℤ𝑛) = (ℤ‘(𝑖 + 1)))
6059reseq2d 5826 . . . . . . . . . 10 (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ‘(𝑖 + 1))))
6160rneqd 5781 . . . . . . . . 9 (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ‘(𝑖 + 1))))
6261supeq1d 8886 . . . . . . . 8 (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
6362adantl 485 . . . . . . 7 ((𝑖𝑍𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
641peano2uzs 12280 . . . . . . 7 (𝑖𝑍 → (𝑖 + 1) ∈ 𝑍)
65 xrltso 12512 . . . . . . . . 9 < Or ℝ*
6665supex 8903 . . . . . . . 8 sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ∈ V
6766a1i 11 . . . . . . 7 (𝑖𝑍 → sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ∈ V)
6858, 63, 64, 67fvmptd 6748 . . . . . 6 (𝑖𝑍 → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
6968adantl 485 . . . . 5 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
70 fveq2 6643 . . . . . . . . . . 11 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
7170reseq2d 5826 . . . . . . . . . 10 (𝑛 = 𝑖 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑖)))
7271rneqd 5781 . . . . . . . . 9 (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑖)))
7372supeq1d 8886 . . . . . . . 8 (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7473adantl 485 . . . . . . 7 ((𝑖𝑍𝑛 = 𝑖) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
75 id 22 . . . . . . 7 (𝑖𝑍𝑖𝑍)
7665supex 8903 . . . . . . . 8 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ V
7776a1i 11 . . . . . . 7 (𝑖𝑍 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ V)
7858, 74, 75, 77fvmptd 6748 . . . . . 6 (𝑖𝑍 → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7978adantl 485 . . . . 5 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8069, 79breq12d 5052 . . . 4 ((𝜑𝑖𝑍) → (((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
8157, 80mpbird 260 . . 3 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖))
82 nfcv 2974 . . . . . . . 8 𝑗𝐹
833frexr 41837 . . . . . . . 8 (𝜑𝐹:𝑍⟶ℝ*)
8482, 2, 1, 83limsupre3uz 42197 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥)))
8512, 84mpbid 235 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥))
8685simpld 498 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗))
87 simp-4r 783 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ)
8887rexrd 10668 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ*)
89833ad2ant1 1130 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝐹:𝑍⟶ℝ*)
901uztrn2 12240 . . . . . . . . . . . 12 ((𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
91903adant1 1127 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
9289, 91ffvelrnd 6825 . . . . . . . . . 10 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ℝ*)
9392ad5ant134 1364 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
9455supxrcld 41556 . . . . . . . . . 10 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
9594ad5ant13 756 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
96 simpr 488 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
97553adant3 1129 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
98 fvres 6662 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑖) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) = (𝐹𝑗))
9998eqcomd 2827 . . . . . . . . . . . . 13 (𝑗 ∈ (ℤ𝑖) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
100993ad2ant3 1132 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
1013ffnd 6488 . . . . . . . . . . . . . . . 16 (𝜑𝐹 Fn 𝑍)
102101adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝐹 Fn 𝑍)
1031, 75uzssd2 41873 . . . . . . . . . . . . . . . 16 (𝑖𝑍 → (ℤ𝑖) ⊆ 𝑍)
104103adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → (ℤ𝑖) ⊆ 𝑍)
105 fnssres 6443 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝑍 ∧ (ℤ𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
106102, 104, 105syl2anc 587 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
1071063adant3 1129 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
108 simp3 1135 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗 ∈ (ℤ𝑖))
109 fnfvelrn 6821 . . . . . . . . . . . . 13 (((𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖) ∧ 𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
110107, 108, 109syl2anc 587 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
111100, 110eqeltrd 2912 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
112 eqid 2821 . . . . . . . . . . 11 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )
11397, 111, 112supxrubd 41562 . . . . . . . . . 10 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
114113ad5ant134 1364 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
11588, 93, 95, 96, 114xrletrd 12533 . . . . . . . 8 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
116115rexlimdva2 3273 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) → (∃𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
117116ralimdva 3165 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
118117reximdva 3260 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
11986, 118mpd 15 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
120 simpl 486 . . . . . . 7 ((𝑦 = 𝑥𝑖𝑍) → 𝑦 = 𝑥)
12178adantl 485 . . . . . . 7 ((𝑦 = 𝑥𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
122120, 121breq12d 5052 . . . . . 6 ((𝑦 = 𝑥𝑖𝑍) → (𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
123122ralbidva 3184 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
124123cbvrexvw 3427 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
125119, 124sylibr 237 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖))
1261, 2, 33, 81, 125climinf 42067 . 2 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⇝ inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
127 fveq2 6643 . . . . . . . 8 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
128127reseq2d 5826 . . . . . . 7 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
129128rneqd 5781 . . . . . 6 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
130129supeq1d 8886 . . . . 5 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
131130cbvmptv 5142 . . . 4 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
132131a1i 11 . . 3 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )))
1332, 1, 3, 12limsupvaluz2 42199 . . . 4 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
134133eqcomd 2827 . . 3 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = (lim sup‘𝐹))
135132, 134breq12d 5052 . 2 (𝜑 → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⇝ inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) ↔ (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹)))
136126, 135mpbid 235 1 (𝜑 → (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wral 3126  wrex 3127  Vcvv 3471  wss 3910  c0 4266   class class class wbr 5039  cmpt 5119  ran crn 5529  cres 5530   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7130  supcsup 8880  infcinf 8881  cr 10513  1c1 10515   + caddc 10517  *cxr 10651   < clt 10652  cle 10653  cz 11959  cuz 12221  lim supclsp 14806  cli 14820
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  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 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-sup 8882  df-inf 8883  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-n0 11876  df-z 11960  df-uz 12222  df-rp 12368  df-ico 12722  df-fz 12876  df-fl 13145  df-ceil 13146  df-seq 13353  df-exp 13414  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-limsup 14807  df-clim 14824
This theorem is referenced by:  supcnvlimsupmpt  42202
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