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Theorem supcnvlimsup 40493
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 472 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝐹:𝑍⟶ℝ)
5 id 22 . . . . . . . . . 10 (𝑛𝑍𝑛𝑍)
61, 5uzssd2 40160 . . . . . . . . 9 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
76adantl 473 . . . . . . . 8 ((𝜑𝑛𝑍) → (ℤ𝑛) ⊆ 𝑍)
84, 7feqresmpt 6413 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹 ↾ (ℤ𝑛)) = (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
98rneqd 5508 . . . . . 6 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)))
109supeq1d 8519 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ))
11 nfcv 2902 . . . . . . . . 9 𝑚𝐹
12 supcnvlimsup.r . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
1312renepnfd 10302 . . . . . . . . 9 (𝜑 → (lim sup‘𝐹) ≠ +∞)
1411, 1, 3, 13limsupubuz 40466 . . . . . . . 8 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
1514adantr 472 . . . . . . 7 ((𝜑𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥)
16 ssralv 3807 . . . . . . . . . 10 ((ℤ𝑛) ⊆ 𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
176, 16syl 17 . . . . . . . . 9 (𝑛𝑍 → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
1817adantl 473 . . . . . . . 8 ((𝜑𝑛𝑍) → (∀𝑚𝑍 (𝐹𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
1918reximdv 3154 . . . . . . 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 1992 . . . . . . 7 𝑚(𝜑𝑛𝑍)
221eluzelz2 40143 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ ℤ)
23 uzid 11914 . . . . . . . . 9 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
24 ne0i 4064 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
2522, 23, 243syl 18 . . . . . . . 8 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
2625adantl 473 . . . . . . 7 ((𝜑𝑛𝑍) → (ℤ𝑛) ≠ ∅)
274adantr 472 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐹:𝑍⟶ℝ)
287sselda 3744 . . . . . . . 8 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
2927, 28ffvelrnd 6524 . . . . . . 7 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚) ∈ ℝ)
3021, 26, 29supxrre3rnmpt 40172 . . . . . 6 ((𝜑𝑛𝑍) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)(𝐹𝑚) ≤ 𝑥))
3120, 30mpbird 247 . . . . 5 ((𝜑𝑛𝑍) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ (𝐹𝑚)), ℝ*, < ) ∈ ℝ)
3210, 31eqeltrd 2839 . . . 4 ((𝜑𝑛𝑍) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) ∈ ℝ)
33 eqid 2760 . . . 4 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
3432, 33fmptd 6549 . . 3 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )):𝑍⟶ℝ)
35 eqid 2760 . . . . . . . . . 10 (ℤ𝑖) = (ℤ𝑖)
361eluzelz2 40143 . . . . . . . . . 10 (𝑖𝑍𝑖 ∈ ℤ)
3736peano2zd 11697 . . . . . . . . . 10 (𝑖𝑍 → (𝑖 + 1) ∈ ℤ)
3836zred 11694 . . . . . . . . . . 11 (𝑖𝑍𝑖 ∈ ℝ)
39 lep1 11074 . . . . . . . . . . 11 (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1))
4038, 39syl 17 . . . . . . . . . 10 (𝑖𝑍𝑖 ≤ (𝑖 + 1))
4135, 36, 37, 40eluzd 40151 . . . . . . . . 9 (𝑖𝑍 → (𝑖 + 1) ∈ (ℤ𝑖))
42 uzss 11920 . . . . . . . . 9 ((𝑖 + 1) ∈ (ℤ𝑖) → (ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖))
4341, 42syl 17 . . . . . . . 8 (𝑖𝑍 → (ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖))
44 ssres2 5583 . . . . . . . 8 ((ℤ‘(𝑖 + 1)) ⊆ (ℤ𝑖) → (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)))
4543, 44syl 17 . . . . . . 7 (𝑖𝑍 → (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)))
46 rnss 5509 . . . . . . 7 ((𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
4745, 46syl 17 . . . . . 6 (𝑖𝑍 → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
4847adantl 473 . . . . 5 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ𝑖)))
49 rnresss 39882 . . . . . . . 8 ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹
5049a1i 11 . . . . . . 7 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ran 𝐹)
513frnd 39943 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℝ)
5251adantr 472 . . . . . . 7 ((𝜑𝑖𝑍) → ran 𝐹 ⊆ ℝ)
5350, 52sstrd 3754 . . . . . 6 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ)
54 ressxr 10295 . . . . . . 7 ℝ ⊆ ℝ*
5554a1i 11 . . . . . 6 ((𝜑𝑖𝑍) → ℝ ⊆ ℝ*)
5653, 55sstrd 3754 . . . . 5 ((𝜑𝑖𝑍) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
57 supxrss 12375 . . . . 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*, < ))
5848, 56, 57syl2anc 696 . . . 4 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
59 eqidd 2761 . . . . . . 7 (𝑖𝑍 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
60 fveq2 6353 . . . . . . . . . . 11 (𝑛 = (𝑖 + 1) → (ℤ𝑛) = (ℤ‘(𝑖 + 1)))
6160reseq2d 5551 . . . . . . . . . 10 (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ‘(𝑖 + 1))))
6261rneqd 5508 . . . . . . . . 9 (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ‘(𝑖 + 1))))
6362supeq1d 8519 . . . . . . . 8 (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
6463adantl 473 . . . . . . 7 ((𝑖𝑍𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
651peano2uzs 11955 . . . . . . 7 (𝑖𝑍 → (𝑖 + 1) ∈ 𝑍)
66 xrltso 12187 . . . . . . . . 9 < Or ℝ*
6766supex 8536 . . . . . . . 8 sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ∈ V
6867a1i 11 . . . . . . 7 (𝑖𝑍 → sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ) ∈ V)
6959, 64, 65, 68fvmptd 6451 . . . . . 6 (𝑖𝑍 → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
7069adantl 473 . . . . 5 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ‘(𝑖 + 1))), ℝ*, < ))
71 fveq2 6353 . . . . . . . . . . 11 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
7271reseq2d 5551 . . . . . . . . . 10 (𝑛 = 𝑖 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑖)))
7372rneqd 5508 . . . . . . . . 9 (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑖)))
7473supeq1d 8519 . . . . . . . 8 (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
7574adantl 473 . . . . . . 7 ((𝑖𝑍𝑛 = 𝑖) → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
76 id 22 . . . . . . 7 (𝑖𝑍𝑖𝑍)
7766supex 8536 . . . . . . . 8 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ V
7877a1i 11 . . . . . . 7 (𝑖𝑍 → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ V)
7959, 75, 76, 78fvmptd 6451 . . . . . 6 (𝑖𝑍 → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8079adantl 473 . . . . 5 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
8170, 80breq12d 4817 . . . 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*, < )))
8258, 81mpbird 247 . . 3 ((𝜑𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘(𝑖 + 1)) ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖))
83 nfcv 2902 . . . . . . . 8 𝑗𝐹
843frexr 40120 . . . . . . . 8 (𝜑𝐹:𝑍⟶ℝ*)
8583, 2, 1, 84limsupre3uz 40489 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥)))
8612, 85mpbid 222 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖𝑍𝑗 ∈ (ℤ𝑖)(𝐹𝑗) ≤ 𝑥))
8786simpld 477 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗))
88 simp-4r 827 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ)
8988rexrd 10301 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ∈ ℝ*)
90843ad2ant1 1128 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝐹:𝑍⟶ℝ*)
911uztrn2 11917 . . . . . . . . . . . . 13 ((𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
92913adant1 1125 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗𝑍)
9390, 92ffvelrnd 6524 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ℝ*)
9493ad5ant134 1467 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
9556supxrcld 39807 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
9695ad5ant13 1216 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) ∈ ℝ*)
97 simpr 479 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
98563adant3 1127 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ran (𝐹 ↾ (ℤ𝑖)) ⊆ ℝ*)
99 fvres 6369 . . . . . . . . . . . . . . 15 (𝑗 ∈ (ℤ𝑖) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) = (𝐹𝑗))
10099eqcomd 2766 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑖) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
1011003ad2ant3 1130 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) = ((𝐹 ↾ (ℤ𝑖))‘𝑗))
1023ffnd 6207 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 Fn 𝑍)
103102adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → 𝐹 Fn 𝑍)
1041, 76uzssd2 40160 . . . . . . . . . . . . . . . . 17 (𝑖𝑍 → (ℤ𝑖) ⊆ 𝑍)
105104adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → (ℤ𝑖) ⊆ 𝑍)
106 fnssres 6165 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑍 ∧ (ℤ𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
107103, 105, 106syl2anc 696 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
1081073adant3 1127 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖))
109 simp3 1133 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → 𝑗 ∈ (ℤ𝑖))
110 fnfvelrn 6520 . . . . . . . . . . . . . 14 (((𝐹 ↾ (ℤ𝑖)) Fn (ℤ𝑖) ∧ 𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
111108, 109, 110syl2anc 696 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → ((𝐹 ↾ (ℤ𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
112101, 111eqeltrd 2839 . . . . . . . . . . . 12 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ∈ ran (𝐹 ↾ (ℤ𝑖)))
113 eqid 2760 . . . . . . . . . . . 12 sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )
11498, 112, 113supxrubd 39814 . . . . . . . . . . 11 ((𝜑𝑖𝑍𝑗 ∈ (ℤ𝑖)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
115114ad5ant134 1467 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝐹𝑗) ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
11689, 94, 96, 97, 115xrletrd 12206 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
117116ex 449 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑗 ∈ (ℤ𝑖)) → (𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
118117rexlimdva 3169 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ 𝑖𝑍) → (∃𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
119118ralimdva 3100 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
120119reximdva 3155 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍𝑗 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
12187, 120mpd 15 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
122 simpl 474 . . . . . . 7 ((𝑦 = 𝑥𝑖𝑍) → 𝑦 = 𝑥)
12379adantl 473 . . . . . . 7 ((𝑦 = 𝑥𝑖𝑍) → ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
124122, 123breq12d 4817 . . . . . 6 ((𝑦 = 𝑥𝑖𝑍) → (𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
125124ralbidva 3123 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < )))
126125cbvrexv 3311 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖) ↔ ∃𝑥 ∈ ℝ ∀𝑖𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑖)), ℝ*, < ))
127121, 126sylibr 224 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ ((𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))‘𝑖))
1281, 2, 34, 82, 127climinf 40359 . 2 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) ⇝ inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
129 fveq2 6353 . . . . . . . 8 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
130129reseq2d 5551 . . . . . . 7 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
131130rneqd 5508 . . . . . 6 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
132131supeq1d 8519 . . . . 5 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
133132cbvmptv 4902 . . . 4 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
134133a1i 11 . . 3 (𝜑 → (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )))
1352, 1, 3, 12limsupvaluz2 40491 . . . 4 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ))
136135eqcomd 2766 . . 3 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ, < ) = (lim sup‘𝐹))
137134, 136breq12d 4817 . 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)))
138128, 137mpbid 222 1 (𝜑 → (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹))
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  (class class class)co 6814  supcsup 8513  infcinf 8514  cr 10147  1c1 10149   + caddc 10151  *cxr 10285   < clt 10286  cle 10287  cz 11589  cuz 11899  lim supclsp 14420  cli 14434
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 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
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 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-ico 12394  df-fz 12540  df-fl 12807  df-ceil 12808  df-seq 13016  df-exp 13075  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-limsup 14421  df-clim 14438
This theorem is referenced by:  supcnvlimsupmpt  40494
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