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Theorem xlimliminflimsup 44093
Description: A sequence of extended reals converges if and only if its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
Hypotheses
Ref Expression
xlimliminflimsup.m (𝜑𝑀 ∈ ℤ)
xlimliminflimsup.z 𝑍 = (ℤ𝑀)
xlimliminflimsup.f (𝜑𝐹:𝑍⟶ℝ*)
Assertion
Ref Expression
xlimliminflimsup (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Proof of Theorem xlimliminflimsup
Dummy variables 𝑗 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xlimliminflimsup.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
21ad2antrr 724 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
3 xlimliminflimsup.z . . . . 5 𝑍 = (ℤ𝑀)
4 xlimliminflimsup.f . . . . . 6 (𝜑𝐹:𝑍⟶ℝ*)
54ad2antrr 724 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ*)
6 simpr 485 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7 xlimdm 44088 . . . . . . 7 (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))
87biimpi 215 . . . . . 6 (𝐹 ∈ dom ~~>* → 𝐹~~>*(~~>*‘𝐹))
98ad2antlr 725 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹~~>*(~~>*‘𝐹))
102, 3, 5, 6, 9xlimxrre 44062 . . . 4 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
113eluzelz2 43628 . . . . . . 7 (𝑗𝑍𝑗 ∈ ℤ)
1211ad2antlr 725 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13 eqid 2736 . . . . . 6 (ℤ𝑗) = (ℤ𝑗)
14 simpr 485 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
1514frexr 43609 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ*)
169adantr 481 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹~~>*(~~>*‘𝐹))
173, 4fuzxrpmcn 44059 . . . . . . . . . . 11 (𝜑𝐹 ∈ (ℝ*pm ℂ))
1817ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹 ∈ (ℝ*pm ℂ))
1911adantl 482 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝑗 ∈ ℤ)
2018, 19xlimres 44052 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹~~>*(~~>*‘𝐹) ↔ (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹)))
2116, 20mpbid 231 . . . . . . . 8 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹))
2221adantr 481 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹))
23 simpllr 774 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
2412, 13, 15, 22, 23xlimclimdm 44085 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)) ∈ dom ⇝ )
2512, 13, 14, 24climliminflimsupd 44032 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
2611adantl 482 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝑗 ∈ ℤ)
2717elexd 3465 . . . . . . . . 9 (𝜑𝐹 ∈ V)
2827adantr 481 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝐹 ∈ V)
294fdmd 6679 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝑍)
3026ssd 43280 . . . . . . . . . 10 (𝜑𝑍 ⊆ ℤ)
3129, 30eqsstrd 3982 . . . . . . . . 9 (𝜑 → dom 𝐹 ⊆ ℤ)
3231adantr 481 . . . . . . . 8 ((𝜑𝑗𝑍) → dom 𝐹 ⊆ ℤ)
3326, 13, 28, 32liminfresuz2 44018 . . . . . . 7 ((𝜑𝑗𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim inf‘𝐹))
3433eqcomd 2742 . . . . . 6 ((𝜑𝑗𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3534ad5ant14 756 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3626, 13, 28, 32limsupresuz2 43940 . . . . . . 7 ((𝜑𝑗𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘𝐹))
3736eqcomd 2742 . . . . . 6 ((𝜑𝑗𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3837ad5ant14 756 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3925, 35, 383eqtr4d 2786 . . . 4 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
4010, 39rexlimddv2 44054 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41 simpll 765 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝜑)
428adantr 481 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
43 simpr 485 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = +∞)
4442, 43breqtrd 5131 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4544adantll 712 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4617liminfcld 44001 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ∈ ℝ*)
4746adantr 481 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ∈ ℝ*)
4817limsupcld 43921 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
4948adantr 481 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ∈ ℝ*)
501, 3, 4liminflelimsupuz 44016 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5150adantr 481 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5249pnfged 43699 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ +∞)
531adantr 481 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝑀 ∈ ℤ)
544adantr 481 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹:𝑍⟶ℝ*)
55 simpr 485 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹~~>*+∞)
5653, 3, 54, 55xlimpnfliminf 44091 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
5752, 56breqtrrd 5133 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
5847, 49, 51, 57xrletrid 13074 . . . . . 6 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
5941, 45, 58syl2anc 584 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
6059adantlr 713 . . . 4 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61 simplll 773 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
628ad2antrr 724 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
63 xlimcl 44053 . . . . . . . . . 10 (𝐹~~>*(~~>*‘𝐹) → (~~>*‘𝐹) ∈ ℝ*)
648, 63syl 17 . . . . . . . . 9 (𝐹 ∈ dom ~~>* → (~~>*‘𝐹) ∈ ℝ*)
6564ad2antrr 724 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ∈ ℝ*)
66 simplr 767 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67 neqne 2951 . . . . . . . . 9 (¬ (~~>*‘𝐹) = +∞ → (~~>*‘𝐹) ≠ +∞)
6867adantl 482 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
6965, 66, 68xrnpnfmnf 43700 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
7062, 69breqtrd 5131 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7170adantlll 716 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7246adantr 481 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ∈ ℝ*)
7348adantr 481 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ∈ ℝ*)
7450adantr 481 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
751adantr 481 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝑀 ∈ ℤ)
764adantr 481 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹:𝑍⟶ℝ*)
77 simpr 485 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹~~>*-∞)
7875, 3, 76, 77xlimmnflimsup 44087 . . . . . . 7 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
7972mnfled 43613 . . . . . . 7 ((𝜑𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
8078, 79eqbrtrd 5127 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
8172, 73, 74, 80xrletrid 13074 . . . . 5 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
8261, 71, 81syl2anc 584 . . . 4 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
8360, 82pm2.61dan 811 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
8440, 83pm2.61dan 811 . 2 ((𝜑𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
8527adantr 481 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86 mnfxr 11212 . . . . . 6 -∞ ∈ ℝ*
8786a1i 11 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → -∞ ∈ ℝ*)
88 simpr 485 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (lim sup‘𝐹) = -∞)
891adantr 481 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝑀 ∈ ℤ)
904adantr 481 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹:𝑍⟶ℝ*)
9189, 3, 90xlimmnflimsup2 44083 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
9288, 91mpbird 256 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
9385, 87, 92breldmd 5868 . . . 4 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
9493adantlr 713 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
951ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
964ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ*)
97 simpr 485 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
9897renepnfd 11206 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99 simplr 767 . . . . . . . . 9 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
10099, 97eqeltrd 2838 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101100renemnfd 11207 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
10295, 3, 96, 98, 101liminflimsupxrre 44048 . . . . . 6 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚𝑍 (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
1033eluzelz2 43628 . . . . . . . . 9 (𝑚𝑍𝑚 ∈ ℤ)
104103ad2antlr 725 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105 eqid 2736 . . . . . . . 8 (ℤ𝑚) = (ℤ𝑚)
106 simpr 485 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
107 simplll 773 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → 𝜑)
108 simpl 483 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
109 simpr 485 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
110108, 109eqeltrd 2838 . . . . . . . . . . . 12 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111110ad4ant23 751 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → (lim inf‘𝐹) ∈ ℝ)
112 simpr 485 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → 𝑚𝑍)
1131033ad2ant3 1135 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝑚 ∈ ℤ)
114273ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝐹 ∈ V)
115313ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → dom 𝐹 ⊆ ℤ)
116113, 105, 114, 115liminfresuz2 44018 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
117 simp2 1137 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘𝐹) ∈ ℝ)
118116, 117eqeltrd 2838 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
119107, 111, 112, 118syl3anc 1371 . . . . . . . . . 10 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
120119adantr 481 . . . . . . . . 9 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
121 simp2 1137 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘𝐹) = (lim sup‘𝐹))
122103adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑚 ∈ ℤ)
12327adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝐹 ∈ V)
12431adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → dom 𝐹 ⊆ ℤ)
125122, 105, 123, 124liminfresuz2 44018 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
1261253adant2 1131 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
127122, 105, 123, 124limsupresuz2 43940 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
1281273adant2 1131 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
129121, 126, 1283eqtr4d 2786 . . . . . . . . . 10 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘(𝐹 ↾ (ℤ𝑚))))
130129ad5ant124 1365 . . . . . . . . 9 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘(𝐹 ↾ (ℤ𝑚))))
131104, 105, 106climliminflimsup3 44041 . . . . . . . . 9 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → ((𝐹 ↾ (ℤ𝑚)) ∈ dom ⇝ ↔ ((lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ ∧ (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘(𝐹 ↾ (ℤ𝑚))))))
132120, 130, 131mpbir2and 711 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ⇝ )
133104, 105, 106, 132dmclimxlim 44082 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*)
13417ad4antr 730 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ (ℝ*pm ℂ))
135134, 104xlimresdm 44090 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*))
136133, 135mpbird 256 . . . . . 6 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ dom ~~>*)
137102, 136rexlimddv2 44054 . . . . 5 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138137adantlr 713 . . . 4 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139 simpll 765 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140 simpllr 774 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
14148ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
142 simpr 485 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → ¬ (lim sup‘𝐹) ∈ ℝ)
143 simplr 767 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144141, 142, 143xrnmnfpnf 43283 . . . . . . 7 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145144adantllr 717 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146140, 145eqtrd 2776 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = +∞)
14727adantr 481 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ V)
148 pnfxr 11209 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 44092 . . . . . . . 8 (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151150biimpar 478 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152147, 149, 151breldmd 5868 . . . . . 6 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153152adantlr 713 . . . . 5 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154139, 146, 153syl2anc 584 . . . 4 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
155138, 154pm2.61dan 811 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
15694, 155pm2.61dane 3032 . 2 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
15784, 156impbida 799 1 (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  wss 3910   class class class wbr 5105  dom cdm 5633  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  pm cpm 8766  cc 11049  cr 11050  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188  cle 11190  cz 12499  cuz 12763  lim supclsp 15352  cli 15366  lim infclsi 43982  ~~>*clsxlim 44049
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-ceil 13698  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-rest 17304  df-topn 17305  df-topgen 17325  df-ordt 17383  df-ps 18455  df-tsr 18456  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-lm 22580  df-haus 22666  df-xms 23673  df-ms 23674  df-liminf 43983  df-xlim 44050
This theorem is referenced by:  xlimlimsupleliminf  44094
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