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Theorem xlimliminflimsup 42019
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 722 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
3 xlimliminflimsup.z . . . . 5 𝑍 = (ℤ𝑀)
4 xlimliminflimsup.f . . . . . 6 (𝜑𝐹:𝑍⟶ℝ*)
54ad2antrr 722 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ*)
6 simpr 485 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7 xlimdm 42014 . . . . . . 7 (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))
87biimpi 217 . . . . . 6 (𝐹 ∈ dom ~~>* → 𝐹~~>*(~~>*‘𝐹))
98ad2antlr 723 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹~~>*(~~>*‘𝐹))
102, 3, 5, 6, 9xlimxrre 41988 . . . 4 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
113eluzelz2 41552 . . . . . . 7 (𝑗𝑍𝑗 ∈ ℤ)
1211ad2antlr 723 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13 eqid 2818 . . . . . 6 (ℤ𝑗) = (ℤ𝑗)
14 simpr 485 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
1514frexr 41531 . . . . . . 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 41985 . . . . . . . . . . 11 (𝜑𝐹 ∈ (ℝ*pm ℂ))
1817ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹 ∈ (ℝ*pm ℂ))
1911adantl 482 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝑗 ∈ ℤ)
2018, 19xlimres 41978 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹~~>*(~~>*‘𝐹) ↔ (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹)))
2116, 20mpbid 233 . . . . . . . 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 772 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
2412, 13, 15, 22, 23xlimclimdm 42011 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)) ∈ dom ⇝ )
2512, 13, 14, 24climliminflimsupd 41958 . . . . 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 3512 . . . . . . . . 9 (𝜑𝐹 ∈ V)
2827adantr 481 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝐹 ∈ V)
294fdmd 6516 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝑍)
3026ssd 41221 . . . . . . . . . 10 (𝜑𝑍 ⊆ ℤ)
3129, 30eqsstrd 4002 . . . . . . . . 9 (𝜑 → dom 𝐹 ⊆ ℤ)
3231adantr 481 . . . . . . . 8 ((𝜑𝑗𝑍) → dom 𝐹 ⊆ ℤ)
3326, 13, 28, 32liminfresuz2 41944 . . . . . . 7 ((𝜑𝑗𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim inf‘𝐹))
3433eqcomd 2824 . . . . . 6 ((𝜑𝑗𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3534ad5ant14 754 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3626, 13, 28, 32limsupresuz2 41866 . . . . . . 7 ((𝜑𝑗𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘𝐹))
3736eqcomd 2824 . . . . . 6 ((𝜑𝑗𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3837ad5ant14 754 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3925, 35, 383eqtr4d 2863 . . . 4 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
4010, 39rexlimddv2 41980 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41 simpll 763 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝜑)
428adantr 481 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
43 simpr 485 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = +∞)
4442, 43breqtrd 5083 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4544adantll 710 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4617liminfcld 41927 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ∈ ℝ*)
4746adantr 481 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ∈ ℝ*)
4817limsupcld 41847 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
4948adantr 481 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ∈ ℝ*)
501, 3, 4liminflelimsupuz 41942 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5150adantr 481 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5249pnfged 41626 . . . . . . . 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 42017 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
5752, 56breqtrrd 5085 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
5847, 49, 51, 57xrletrid 12536 . . . . . 6 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
5941, 45, 58syl2anc 584 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
6059adantlr 711 . . . 4 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61 simplll 771 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
628ad2antrr 722 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
63 xlimcl 41979 . . . . . . . . . 10 (𝐹~~>*(~~>*‘𝐹) → (~~>*‘𝐹) ∈ ℝ*)
648, 63syl 17 . . . . . . . . 9 (𝐹 ∈ dom ~~>* → (~~>*‘𝐹) ∈ ℝ*)
6564ad2antrr 722 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ∈ ℝ*)
66 simplr 765 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67 neqne 3021 . . . . . . . . 9 (¬ (~~>*‘𝐹) = +∞ → (~~>*‘𝐹) ≠ +∞)
6867adantl 482 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
6965, 66, 68xrnpnfmnf 41627 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
7062, 69breqtrd 5083 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7170adantlll 714 . . . . 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 42013 . . . . . . 7 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
7972mnfled 41536 . . . . . . 7 ((𝜑𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
8078, 79eqbrtrd 5079 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
8172, 73, 74, 80xrletrid 12536 . . . . 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 809 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
8440, 83pm2.61dan 809 . 2 ((𝜑𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
8527adantr 481 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86 mnfxr 10686 . . . . . 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 42009 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
9288, 91mpbird 258 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
9385, 87, 92breldmd 5774 . . . 4 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
9493adantlr 711 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
951ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
964ad2antrr 722 . . . . . . 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 10680 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99 simplr 765 . . . . . . . . 9 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
10099, 97eqeltrd 2910 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101100renemnfd 10681 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
10295, 3, 96, 98, 101liminflimsupxrre 41974 . . . . . 6 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚𝑍 (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
1033eluzelz2 41552 . . . . . . . . 9 (𝑚𝑍𝑚 ∈ ℤ)
104103ad2antlr 723 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105 eqid 2818 . . . . . . . 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 771 . . . . . . . . . . 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 2910 . . . . . . . . . . . 12 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111110ad4ant23 749 . . . . . . . . . . 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 1127 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝑚 ∈ ℤ)
114273ad2ant1 1125 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝐹 ∈ V)
115313ad2ant1 1125 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → dom 𝐹 ⊆ ℤ)
116113, 105, 114, 115liminfresuz2 41944 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
117 simp2 1129 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘𝐹) ∈ ℝ)
118116, 117eqeltrd 2910 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
119107, 111, 112, 118syl3anc 1363 . . . . . . . . . 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 1129 . . . . . . . . . . 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 41944 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
1261253adant2 1123 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
127122, 105, 123, 124limsupresuz2 41866 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
1281273adant2 1123 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
129121, 126, 1283eqtr4d 2863 . . . . . . . . . 10 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘(𝐹 ↾ (ℤ𝑚))))
130129ad5ant124 1357 . . . . . . . . 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 41967 . . . . . . . . 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 709 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ⇝ )
133104, 105, 106, 132dmclimxlim 42008 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*)
13417ad4antr 728 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ (ℝ*pm ℂ))
135134, 104xlimresdm 42016 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*))
136133, 135mpbird 258 . . . . . 6 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ dom ~~>*)
137102, 136rexlimddv2 41980 . . . . 5 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138137adantlr 711 . . . 4 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139 simpll 763 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140 simpllr 772 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
14148ad2antrr 722 . . . . . . . 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 765 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144141, 142, 143xrnmnfpnf 41224 . . . . . . 7 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145144adantllr 715 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146140, 145eqtrd 2853 . . . . 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 10683 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 42018 . . . . . . . 8 (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151150biimpar 478 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152147, 149, 151breldmd 5774 . . . . . 6 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153152adantlr 711 . . . . 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 809 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
15694, 155pm2.61dane 3101 . 2 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
15784, 156impbida 797 1 (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  wss 3933   class class class wbr 5057  dom cdm 5548  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  cc 10523  cr 10524  +∞cpnf 10660  -∞cmnf 10661  *cxr 10662  cle 10664  cz 11969  cuz 12231  lim supclsp 14815  cli 14829  lim infclsi 41908  ~~>*clsxlim 41975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-ceil 13151  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-plusg 16566  df-mulr 16567  df-starv 16568  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-rest 16684  df-topn 16685  df-topgen 16705  df-ordt 16762  df-ps 17798  df-tsr 17799  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-lm 21765  df-haus 21851  df-xms 22857  df-ms 22858  df-liminf 41909  df-xlim 41976
This theorem is referenced by:  xlimlimsupleliminf  42020
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