Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  supcnvlimsup Structured version   Visualization version   GIF version

Theorem supcnvlimsup 45361
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 479 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝐹:𝑍⟶ℝ)
5 id 22 . . . . . . . . . 10 (𝑛𝑍𝑛𝑍)
61, 5uzssd2 45032 . . . . . . . . 9 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
76adantl 480 . . . . . . . 8 ((𝜑𝑛𝑍) → (ℤ𝑛) ⊆ 𝑍)
84, 7feqresmpt 6972 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹 ↾ (ℤ𝑛)) = (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
98rneqd 5944 . . . . . 6 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
109supeq1d 9489 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ))
11 nfcv 2892 . . . . . . . . 9 𝑚𝐹
12 supcnvlimsup.r . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
1312renepnfd 11315 . . . . . . . . 9 (𝜑 → (lim sup‘𝐹) ≠ +∞)
1411, 1, 3, 13limsupubuz 45334 . . . . . . . 8 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
1514adantr 479 . . . . . . 7 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
16 ssralv 4048 . . . . . . . . . 10 ((ℤ𝑛) ⊆ 𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
176, 16syl 17 . . . . . . . . 9 (𝑛𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
1817adantl 480 . . . . . . . 8 ((𝜑𝑛𝑍) → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
1918reximdv 3160 . . . . . . 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 1910 . . . . . . 7 𝑚(𝜑𝑛𝑍)
221eluzelz2 45018 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ ℤ)
23 uzid 12889 . . . . . . . . 9 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
24 ne0i 4337 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
2522, 23, 243syl 18 . . . . . . . 8 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
2625adantl 480 . . . . . . 7 ((𝜑𝑛𝑍) → (ℤ𝑛) ≠ ∅)
274adantr 479 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐹:𝑍⟶ℝ)
287sselda 3979 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
2927, 28ffvelcdmd 7099 . . . . . . 7 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚) ∈ ℝ)
3021, 26, 29supxrre3rnmpt 45044 . . . . . 6 ((𝜑𝑛𝑍) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
3120, 30mpbird 256 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ)
3210, 31eqeltrd 2826 . . . 4 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ ℝ)
3332fmpttd 7129 . . 3 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )):𝑍⟶ℝ)
34 eqid 2726 . . . . . . . 8 (ℤ𝑖) = (ℤ𝑖)
351eluzelz2 45018 . . . . . . . 8 (𝑖𝑍𝑖 ∈ ℤ)
3635peano2zd 12721 . . . . . . . 8 (𝑖𝑍 → (𝑖 + 1) ∈ ℤ)
3735zred 12718 . . . . . . . . 9 (𝑖𝑍𝑖 ∈ ℝ)
38 lep1 12106 . . . . . . . . 9 (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1))
3937, 38syl 17 . . . . . . . 8 (𝑖𝑍𝑖 ≤ (𝑖 + 1))
4034, 35, 36, 39eluzd 45024 . . . . . . 7 (𝑖𝑍 → (𝑖 + 1) ∈ (ℤ𝑖))
41 uzss 12897 . . . . . . 7 ((𝑖 + 1) ∈ (ℤ𝑖) → (ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖))
42 ssres2 6014 . . . . . . 7 ((ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖) → (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)))
43 rnss 5945 . . . . . . 7 ((𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
4440, 41, 42, 434syl 19 . . . . . 6 (𝑖𝑍 → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
4544adantl 480 . . . . 5 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
46 rnresss 6026 . . . . . . . 8 ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹
4746a1i 11 . . . . . . 7 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹)
483frnd 6736 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℝ)
4948adantr 479 . . . . . . 7 ((𝜑𝑖𝑍) → ran 𝐹 ⊆ ℝ)
5047, 49sstrd 3990 . . . . . 6 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ)
51 ressxr 11308 . . . . . . 7 ℝ ⊆ ℝ*
5251a1i 11 . . . . . 6 ((𝜑𝑖𝑍) → ℝ ⊆ ℝ*)
5350, 52sstrd 3990 . . . . 5 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
54 supxrss 13365 . . . . 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*, < ))
5545, 53, 54syl2anc 582 . . . 4 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
56 eqidd 2727 . . . . . . 7 (𝑖𝑍 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
57 fveq2 6901 . . . . . . . . . . 11 (𝑛 = (𝑖 + 1) → (ℤ𝑛) = (ℤ‘(𝑖 + 1)))
5857reseq2d 5989 . . . . . . . . . 10 (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ‘(𝑖 + 1))))
5958rneqd 5944 . . . . . . . . 9 (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ‘(𝑖 + 1))))
6059supeq1d 9489 . . . . . . . 8 (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
6160adantl 480 . . . . . . 7 ((𝑖𝑍𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
621peano2uzs 12938 . . . . . . 7 (𝑖𝑍 → (𝑖 + 1) ∈ 𝑍)
63 xrltso 13174 . . . . . . . . 9 < Or ℝ*
6463supex 9506 . . . . . . . 8 sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ∈ V
6564a1i 11 . . . . . . 7 (𝑖𝑍 → sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ∈ V)
6656, 61, 62, 65fvmptd 7016 . . . . . 6 (𝑖𝑍 → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
6766adantl 480 . . . . 5 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
68 fveq2 6901 . . . . . . . . . . 11 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
6968reseq2d 5989 . . . . . . . . . 10 (𝑛 = 𝑖 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑖)))
7069rneqd 5944 . . . . . . . . 9 (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑖)))
7170supeq1d 9489 . . . . . . . 8 (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7271adantl 480 . . . . . . 7 ((𝑖𝑍𝑛 = 𝑖) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
73 id 22 . . . . . . 7 (𝑖𝑍𝑖𝑍)
7463supex 9506 . . . . . . . 8 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ V
7574a1i 11 . . . . . . 7 (𝑖𝑍 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ V)
7656, 72, 73, 75fvmptd 7016 . . . . . 6 (𝑖𝑍 → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7776adantl 480 . . . . 5 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7867, 77breq12d 5166 . . . 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*, < )))
7955, 78mpbird 256 . . 3 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖))
80 nfcv 2892 . . . . . . . 8 𝑗𝐹
813frexr 45000 . . . . . . . 8 (𝜑𝐹:𝑍⟶ℝ*)
8280, 2, 1, 81limsupre3uz 45357 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥)))
8312, 82mpbid 231 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥))
8483simpld 493 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗))
85 simp-4r 782 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ)
8685rexrd 11314 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ*)
87813ad2ant1 1130 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝐹:𝑍⟶ℝ*)
881uztrn2 12893 . . . . . . . . . . . 12 ((𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
89883adant1 1127 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
9087, 89ffvelcdmd 7099 . . . . . . . . . 10 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ℝ*)
9190ad5ant134 1364 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
9253supxrcld 44708 . . . . . . . . . 10 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
9392ad5ant13 755 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
94 simpr 483 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
95533adant3 1129 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
96 fvres 6920 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑖) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) = (𝐹𝑗))
9796eqcomd 2732 . . . . . . . . . . . . 13 (𝑗 ∈ (ℤ𝑖) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
98973ad2ant3 1132 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
993ffnd 6729 . . . . . . . . . . . . . . . 16 (𝜑𝐹 Fn 𝑍)
10099adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝐹 Fn 𝑍)
1011, 73uzssd2 45032 . . . . . . . . . . . . . . . 16 (𝑖𝑍 → (ℤ𝑖) ⊆ 𝑍)
102101adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → (ℤ𝑖) ⊆ 𝑍)
103 fnssres 6684 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝑍 ∧ (ℤ𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
104100, 102, 103syl2anc 582 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
1051043adant3 1129 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
106 simp3 1135 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗 ∈ (ℤ𝑖))
107 fnfvelrn 7094 . . . . . . . . . . . . 13 (((𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖) ∧ 𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
108105, 106, 107syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
10998, 108eqeltrd 2826 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
110 eqid 2726 . . . . . . . . . . 11 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )
11195, 109, 110supxrubd 44714 . . . . . . . . . 10 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
112111ad5ant134 1364 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
11386, 91, 93, 94, 112xrletrd 13195 . . . . . . . 8 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
114113rexlimdva2 3147 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) → (∃𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
115114ralimdva 3157 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
116115reximdva 3158 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
11784, 116mpd 15 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
118 simpl 481 . . . . . . 7 ((𝑦 = 𝑥𝑖𝑍) → 𝑦 = 𝑥)
11976adantl 480 . . . . . . 7 ((𝑦 = 𝑥𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
120118, 119breq12d 5166 . . . . . 6 ((𝑦 = 𝑥𝑖𝑍) → (𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
121120ralbidva 3166 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
122121cbvrexvw 3226 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
123117, 122sylibr 233 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖))
1241, 2, 33, 79, 123climinf 45227 . 2 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⇝ inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
125 fveq2 6901 . . . . . . . 8 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
126125reseq2d 5989 . . . . . . 7 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
127126rneqd 5944 . . . . . 6 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
128127supeq1d 9489 . . . . 5 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
129128cbvmptv 5266 . . . 4 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
130129a1i 11 . . 3 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )))
1312, 1, 3, 12limsupvaluz2 45359 . . . 4 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
132131eqcomd 2732 . . 3 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = (lim sup‘𝐹))
133130, 132breq12d 5166 . 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)))
134124, 133mpbid 231 1 (𝜑 → (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3462  wss 3947  c0 4325   class class class wbr 5153  cmpt 5236  ran crn 5683  cres 5684   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  supcsup 9483  infcinf 9484  cr 11157  1c1 11159   + caddc 11161  *cxr 11297   < clt 11298  cle 11299  cz 12610  cuz 12874  lim supclsp 15472  cli 15486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-ico 13384  df-fz 13539  df-fl 13812  df-ceil 13813  df-seq 14022  df-exp 14082  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-limsup 15473  df-clim 15490
This theorem is referenced by:  supcnvlimsupmpt  45362
  Copyright terms: Public domain W3C validator