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Theorem xlimliminflimsup 42502
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 725 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
3 xlimliminflimsup.z . . . . 5 𝑍 = (ℤ𝑀)
4 xlimliminflimsup.f . . . . . 6 (𝜑𝐹:𝑍⟶ℝ*)
54ad2antrr 725 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ*)
6 simpr 488 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7 xlimdm 42497 . . . . . . 7 (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))
87biimpi 219 . . . . . 6 (𝐹 ∈ dom ~~>* → 𝐹~~>*(~~>*‘𝐹))
98ad2antlr 726 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹~~>*(~~>*‘𝐹))
102, 3, 5, 6, 9xlimxrre 42471 . . . 4 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
113eluzelz2 42038 . . . . . . 7 (𝑗𝑍𝑗 ∈ ℤ)
1211ad2antlr 726 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13 eqid 2798 . . . . . 6 (ℤ𝑗) = (ℤ𝑗)
14 simpr 488 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
1514frexr 42017 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ*)
169adantr 484 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹~~>*(~~>*‘𝐹))
173, 4fuzxrpmcn 42468 . . . . . . . . . . 11 (𝜑𝐹 ∈ (ℝ*pm ℂ))
1817ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹 ∈ (ℝ*pm ℂ))
1911adantl 485 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝑗 ∈ ℤ)
2018, 19xlimres 42461 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹~~>*(~~>*‘𝐹) ↔ (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹)))
2116, 20mpbid 235 . . . . . . . 8 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹))
2221adantr 484 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹))
23 simpllr 775 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
2412, 13, 15, 22, 23xlimclimdm 42494 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)) ∈ dom ⇝ )
2512, 13, 14, 24climliminflimsupd 42441 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
2611adantl 485 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝑗 ∈ ℤ)
2717elexd 3461 . . . . . . . . 9 (𝜑𝐹 ∈ V)
2827adantr 484 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝐹 ∈ V)
294fdmd 6497 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝑍)
3026ssd 41714 . . . . . . . . . 10 (𝜑𝑍 ⊆ ℤ)
3129, 30eqsstrd 3953 . . . . . . . . 9 (𝜑 → dom 𝐹 ⊆ ℤ)
3231adantr 484 . . . . . . . 8 ((𝜑𝑗𝑍) → dom 𝐹 ⊆ ℤ)
3326, 13, 28, 32liminfresuz2 42427 . . . . . . 7 ((𝜑𝑗𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim inf‘𝐹))
3433eqcomd 2804 . . . . . 6 ((𝜑𝑗𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3534ad5ant14 757 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3626, 13, 28, 32limsupresuz2 42349 . . . . . . 7 ((𝜑𝑗𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘𝐹))
3736eqcomd 2804 . . . . . 6 ((𝜑𝑗𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3837ad5ant14 757 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3925, 35, 383eqtr4d 2843 . . . 4 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
4010, 39rexlimddv2 42463 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41 simpll 766 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝜑)
428adantr 484 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
43 simpr 488 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = +∞)
4442, 43breqtrd 5056 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4544adantll 713 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4617liminfcld 42410 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ∈ ℝ*)
4746adantr 484 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ∈ ℝ*)
4817limsupcld 42330 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
4948adantr 484 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ∈ ℝ*)
501, 3, 4liminflelimsupuz 42425 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5150adantr 484 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5249pnfged 42111 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ +∞)
531adantr 484 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝑀 ∈ ℤ)
544adantr 484 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹:𝑍⟶ℝ*)
55 simpr 488 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹~~>*+∞)
5653, 3, 54, 55xlimpnfliminf 42500 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
5752, 56breqtrrd 5058 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
5847, 49, 51, 57xrletrid 12536 . . . . . 6 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
5941, 45, 58syl2anc 587 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
6059adantlr 714 . . . 4 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61 simplll 774 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
628ad2antrr 725 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
63 xlimcl 42462 . . . . . . . . . 10 (𝐹~~>*(~~>*‘𝐹) → (~~>*‘𝐹) ∈ ℝ*)
648, 63syl 17 . . . . . . . . 9 (𝐹 ∈ dom ~~>* → (~~>*‘𝐹) ∈ ℝ*)
6564ad2antrr 725 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ∈ ℝ*)
66 simplr 768 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67 neqne 2995 . . . . . . . . 9 (¬ (~~>*‘𝐹) = +∞ → (~~>*‘𝐹) ≠ +∞)
6867adantl 485 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
6965, 66, 68xrnpnfmnf 42112 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
7062, 69breqtrd 5056 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7170adantlll 717 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7246adantr 484 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ∈ ℝ*)
7348adantr 484 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ∈ ℝ*)
7450adantr 484 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
751adantr 484 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝑀 ∈ ℤ)
764adantr 484 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹:𝑍⟶ℝ*)
77 simpr 488 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹~~>*-∞)
7875, 3, 76, 77xlimmnflimsup 42496 . . . . . . 7 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
7972mnfled 42022 . . . . . . 7 ((𝜑𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
8078, 79eqbrtrd 5052 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
8172, 73, 74, 80xrletrid 12536 . . . . 5 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
8261, 71, 81syl2anc 587 . . . 4 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
8360, 82pm2.61dan 812 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
8440, 83pm2.61dan 812 . 2 ((𝜑𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
8527adantr 484 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86 mnfxr 10687 . . . . . 6 -∞ ∈ ℝ*
8786a1i 11 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → -∞ ∈ ℝ*)
88 simpr 488 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (lim sup‘𝐹) = -∞)
891adantr 484 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝑀 ∈ ℤ)
904adantr 484 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹:𝑍⟶ℝ*)
9189, 3, 90xlimmnflimsup2 42492 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
9288, 91mpbird 260 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
9385, 87, 92breldmd 5745 . . . 4 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
9493adantlr 714 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
951ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
964ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ*)
97 simpr 488 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
9897renepnfd 10681 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99 simplr 768 . . . . . . . . 9 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
10099, 97eqeltrd 2890 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101100renemnfd 10682 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
10295, 3, 96, 98, 101liminflimsupxrre 42457 . . . . . 6 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚𝑍 (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
1033eluzelz2 42038 . . . . . . . . 9 (𝑚𝑍𝑚 ∈ ℤ)
104103ad2antlr 726 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105 eqid 2798 . . . . . . . 8 (ℤ𝑚) = (ℤ𝑚)
106 simpr 488 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
107 simplll 774 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → 𝜑)
108 simpl 486 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
109 simpr 488 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
110108, 109eqeltrd 2890 . . . . . . . . . . . 12 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111110ad4ant23 752 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → (lim inf‘𝐹) ∈ ℝ)
112 simpr 488 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → 𝑚𝑍)
1131033ad2ant3 1132 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝑚 ∈ ℤ)
114273ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝐹 ∈ V)
115313ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → dom 𝐹 ⊆ ℤ)
116113, 105, 114, 115liminfresuz2 42427 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
117 simp2 1134 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘𝐹) ∈ ℝ)
118116, 117eqeltrd 2890 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
119107, 111, 112, 118syl3anc 1368 . . . . . . . . . 10 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
120119adantr 484 . . . . . . . . 9 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
121 simp2 1134 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘𝐹) = (lim sup‘𝐹))
122103adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑚 ∈ ℤ)
12327adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝐹 ∈ V)
12431adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → dom 𝐹 ⊆ ℤ)
125122, 105, 123, 124liminfresuz2 42427 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
1261253adant2 1128 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
127122, 105, 123, 124limsupresuz2 42349 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
1281273adant2 1128 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
129121, 126, 1283eqtr4d 2843 . . . . . . . . . 10 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘(𝐹 ↾ (ℤ𝑚))))
130129ad5ant124 1362 . . . . . . . . 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 42450 . . . . . . . . 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 712 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ⇝ )
133104, 105, 106, 132dmclimxlim 42491 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*)
13417ad4antr 731 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ (ℝ*pm ℂ))
135134, 104xlimresdm 42499 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*))
136133, 135mpbird 260 . . . . . 6 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ dom ~~>*)
137102, 136rexlimddv2 42463 . . . . 5 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138137adantlr 714 . . . 4 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139 simpll 766 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140 simpllr 775 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
14148ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
142 simpr 488 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → ¬ (lim sup‘𝐹) ∈ ℝ)
143 simplr 768 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144141, 142, 143xrnmnfpnf 41717 . . . . . . 7 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145144adantllr 718 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146140, 145eqtrd 2833 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = +∞)
14727adantr 484 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ V)
148 pnfxr 10684 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 42501 . . . . . . . 8 (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151150biimpar 481 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152147, 149, 151breldmd 5745 . . . . . 6 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153152adantlr 714 . . . . 5 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154139, 146, 153syl2anc 587 . . . 4 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
155138, 154pm2.61dan 812 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
15694, 155pm2.61dane 3074 . 2 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
15784, 156impbida 800 1 (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987  Vcvv 3441  wss 3881   class class class wbr 5030  dom cdm 5519  cres 5521  wf 6320  cfv 6324  (class class class)co 7135  pm cpm 8390  cc 10524  cr 10525  +∞cpnf 10661  -∞cmnf 10662  *cxr 10663  cle 10665  cz 11969  cuz 12231  lim supclsp 14819  cli 14833  lim infclsi 42391  ~~>*clsxlim 42458
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  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 12886  df-fzo 13029  df-fl 13157  df-ceil 13158  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-plusg 16570  df-mulr 16571  df-starv 16572  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-rest 16688  df-topn 16689  df-topgen 16709  df-ordt 16766  df-ps 17802  df-tsr 17803  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-cnfld 20092  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-lm 21834  df-haus 21920  df-xms 22927  df-ms 22928  df-liminf 42392  df-xlim 42459
This theorem is referenced by:  xlimlimsupleliminf  42503
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