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Theorem xlimliminflimsup 41580
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 713 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
3 xlimliminflimsup.z . . . . 5 𝑍 = (ℤ𝑀)
4 xlimliminflimsup.f . . . . . 6 (𝜑𝐹:𝑍⟶ℝ*)
54ad2antrr 713 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ*)
6 simpr 477 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7 xlimdm 41575 . . . . . . 7 (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))
87biimpi 208 . . . . . 6 (𝐹 ∈ dom ~~>* → 𝐹~~>*(~~>*‘𝐹))
98ad2antlr 714 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → 𝐹~~>*(~~>*‘𝐹))
102, 3, 5, 6, 9xlimxrre 41549 . . . 4 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
113eluzelz2 41112 . . . . . . 7 (𝑗𝑍𝑗 ∈ ℤ)
1211ad2antlr 714 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13 eqid 2778 . . . . . 6 (ℤ𝑗) = (ℤ𝑗)
14 simpr 477 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
1514frexr 41091 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ*)
169adantr 473 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹~~>*(~~>*‘𝐹))
173, 4fuzxrpmcn 41546 . . . . . . . . . . 11 (𝜑𝐹 ∈ (ℝ*pm ℂ))
1817ad3antrrr 717 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹 ∈ (ℝ*pm ℂ))
1911adantl 474 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝑗 ∈ ℤ)
2018, 19xlimres 41539 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹~~>*(~~>*‘𝐹) ↔ (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹)))
2116, 20mpbid 224 . . . . . . . 8 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹))
2221adantr 473 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹))
23 simpllr 763 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
2412, 13, 15, 22, 23xlimclimdm 41572 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)) ∈ dom ⇝ )
2512, 13, 14, 24climliminflimsupd 41519 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
2611adantl 474 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝑗 ∈ ℤ)
2717elexd 3435 . . . . . . . . 9 (𝜑𝐹 ∈ V)
2827adantr 473 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝐹 ∈ V)
294fdmd 6353 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝑍)
3026ssd 40769 . . . . . . . . . 10 (𝜑𝑍 ⊆ ℤ)
3129, 30eqsstrd 3895 . . . . . . . . 9 (𝜑 → dom 𝐹 ⊆ ℤ)
3231adantr 473 . . . . . . . 8 ((𝜑𝑗𝑍) → dom 𝐹 ⊆ ℤ)
3326, 13, 28, 32liminfresuz2 41505 . . . . . . 7 ((𝜑𝑗𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim inf‘𝐹))
3433eqcomd 2784 . . . . . 6 ((𝜑𝑗𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3534ad5ant14 745 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ𝑗))))
3626, 13, 28, 32limsupresuz2 41427 . . . . . . 7 ((𝜑𝑗𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘𝐹))
3736eqcomd 2784 . . . . . 6 ((𝜑𝑗𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3837ad5ant14 745 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
3925, 35, 383eqtr4d 2824 . . . 4 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
4010, 39rexlimddv2 41541 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41 simpll 754 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝜑)
428adantr 473 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
43 simpr 477 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = +∞)
4442, 43breqtrd 4955 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4544adantll 701 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4617liminfcld 41488 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ∈ ℝ*)
4746adantr 473 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ∈ ℝ*)
4817limsupcld 41408 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
4948adantr 473 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ∈ ℝ*)
501, 3, 4liminflelimsupuz 41503 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5150adantr 473 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5249pnfged 41187 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ +∞)
531adantr 473 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝑀 ∈ ℤ)
544adantr 473 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹:𝑍⟶ℝ*)
55 simpr 477 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹~~>*+∞)
5653, 3, 54, 55xlimpnfliminf 41578 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
5752, 56breqtrrd 4957 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
5847, 49, 51, 57xrletrid 12365 . . . . . 6 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
5941, 45, 58syl2anc 576 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
6059adantlr 702 . . . 4 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61 simplll 762 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
628ad2antrr 713 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
63 xlimcl 41540 . . . . . . . . . 10 (𝐹~~>*(~~>*‘𝐹) → (~~>*‘𝐹) ∈ ℝ*)
648, 63syl 17 . . . . . . . . 9 (𝐹 ∈ dom ~~>* → (~~>*‘𝐹) ∈ ℝ*)
6564ad2antrr 713 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ∈ ℝ*)
66 simplr 756 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67 neqne 2975 . . . . . . . . 9 (¬ (~~>*‘𝐹) = +∞ → (~~>*‘𝐹) ≠ +∞)
6867adantl 474 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
6965, 66, 68xrnpnfmnf 41188 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
7062, 69breqtrd 4955 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7170adantlll 705 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7246adantr 473 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ∈ ℝ*)
7348adantr 473 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ∈ ℝ*)
7450adantr 473 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
751adantr 473 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝑀 ∈ ℤ)
764adantr 473 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹:𝑍⟶ℝ*)
77 simpr 477 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹~~>*-∞)
7875, 3, 76, 77xlimmnflimsup 41574 . . . . . . 7 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
7972mnfled 41096 . . . . . . 7 ((𝜑𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
8078, 79eqbrtrd 4951 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
8172, 73, 74, 80xrletrid 12365 . . . . 5 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
8261, 71, 81syl2anc 576 . . . 4 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
8360, 82pm2.61dan 800 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
8440, 83pm2.61dan 800 . 2 ((𝜑𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
8527adantr 473 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86 mnfxr 10498 . . . . . 6 -∞ ∈ ℝ*
8786a1i 11 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → -∞ ∈ ℝ*)
88 simpr 477 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (lim sup‘𝐹) = -∞)
891adantr 473 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝑀 ∈ ℤ)
904adantr 473 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹:𝑍⟶ℝ*)
9189, 3, 90xlimmnflimsup2 41570 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
9288, 91mpbird 249 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
9385, 87, 92breldmd 40851 . . . 4 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
9493adantlr 702 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
951ad2antrr 713 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
964ad2antrr 713 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ*)
97 simpr 477 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
9897renepnfd 10491 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99 simplr 756 . . . . . . . . 9 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
10099, 97eqeltrd 2866 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101100renemnfd 10492 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
10295, 3, 96, 98, 101liminflimsupxrre 41535 . . . . . 6 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚𝑍 (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
1033eluzelz2 41112 . . . . . . . . 9 (𝑚𝑍𝑚 ∈ ℤ)
104103ad2antlr 714 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105 eqid 2778 . . . . . . . 8 (ℤ𝑚) = (ℤ𝑚)
106 simpr 477 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
107 simplll 762 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → 𝜑)
108 simpl 475 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
109 simpr 477 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
110108, 109eqeltrd 2866 . . . . . . . . . . . 12 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111110ad4ant23 740 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → (lim inf‘𝐹) ∈ ℝ)
112 simpr 477 . . . . . . . . . . 11 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → 𝑚𝑍)
1131033ad2ant3 1115 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝑚 ∈ ℤ)
114273ad2ant1 1113 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → 𝐹 ∈ V)
115313ad2ant1 1113 . . . . . . . . . . . . 13 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → dom 𝐹 ⊆ ℤ)
116113, 105, 114, 115liminfresuz2 41505 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
117 simp2 1117 . . . . . . . . . . . 12 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘𝐹) ∈ ℝ)
118116, 117eqeltrd 2866 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
119107, 111, 112, 118syl3anc 1351 . . . . . . . . . 10 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
120119adantr 473 . . . . . . . . 9 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) ∈ ℝ)
121 simp2 1117 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘𝐹) = (lim sup‘𝐹))
122103adantl 474 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑚 ∈ ℤ)
12327adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝐹 ∈ V)
12431adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → dom 𝐹 ⊆ ℤ)
125122, 105, 123, 124liminfresuz2 41505 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
1261253adant2 1111 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim inf‘𝐹))
127122, 105, 123, 124limsupresuz2 41427 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
1281273adant2 1111 . . . . . . . . . . 11 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘𝐹))
129121, 126, 1283eqtr4d 2824 . . . . . . . . . 10 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑚))) = (lim sup‘(𝐹 ↾ (ℤ𝑚))))
130129ad5ant124 1345 . . . . . . . . 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 41528 . . . . . . . . 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 700 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ⇝ )
133104, 105, 106, 132dmclimxlim 41569 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*)
13417ad4antr 719 . . . . . . . 8 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ (ℝ*pm ℂ))
135134, 104xlimresdm 41577 . . . . . . 7 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ𝑚)) ∈ dom ~~>*))
136133, 135mpbird 249 . . . . . 6 (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚𝑍) ∧ (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ) → 𝐹 ∈ dom ~~>*)
137102, 136rexlimddv2 41541 . . . . 5 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138137adantlr 702 . . . 4 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139 simpll 754 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140 simpllr 763 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
14148ad2antrr 713 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
142 simpr 477 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → ¬ (lim sup‘𝐹) ∈ ℝ)
143 simplr 756 . . . . . . . 8 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144141, 142, 143xrnmnfpnf 40772 . . . . . . 7 (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145144adantllr 706 . . . . . 6 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146140, 145eqtrd 2814 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = +∞)
14727adantr 473 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ V)
148 pnfxr 10494 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 41579 . . . . . . . 8 (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151150biimpar 470 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152147, 149, 151breldmd 40851 . . . . . 6 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153152adantlr 702 . . . . 5 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154139, 146, 153syl2anc 576 . . . 4 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
155138, 154pm2.61dan 800 . . 3 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
15694, 155pm2.61dane 3055 . 2 ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
15784, 156impbida 788 1 (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  Vcvv 3415  wss 3829   class class class wbr 4929  dom cdm 5407  cres 5409  wf 6184  cfv 6188  (class class class)co 6976  pm cpm 8207  cc 10333  cr 10334  +∞cpnf 10471  -∞cmnf 10472  *cxr 10473  cle 10475  cz 11793  cuz 12058  lim supclsp 14688  cli 14702  lim infclsi 41469  ~~>*clsxlim 41536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-pm 8209  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fi 8670  df-sup 8701  df-inf 8702  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-2 11503  df-3 11504  df-4 11505  df-5 11506  df-6 11507  df-7 11508  df-8 11509  df-9 11510  df-n0 11708  df-z 11794  df-dec 11912  df-uz 12059  df-q 12163  df-rp 12205  df-xneg 12324  df-xadd 12325  df-xmul 12326  df-ioo 12558  df-ioc 12559  df-ico 12560  df-icc 12561  df-fz 12709  df-fzo 12850  df-fl 12977  df-ceil 12978  df-seq 13185  df-exp 13245  df-cj 14319  df-re 14320  df-im 14321  df-sqrt 14455  df-abs 14456  df-limsup 14689  df-clim 14706  df-rlim 14707  df-struct 16341  df-ndx 16342  df-slot 16343  df-base 16345  df-plusg 16434  df-mulr 16435  df-starv 16436  df-tset 16440  df-ple 16441  df-ds 16443  df-unif 16444  df-rest 16552  df-topn 16553  df-topgen 16573  df-ordt 16630  df-ps 17668  df-tsr 17669  df-psmet 20239  df-xmet 20240  df-met 20241  df-bl 20242  df-mopn 20243  df-cnfld 20248  df-top 21206  df-topon 21223  df-topsp 21245  df-bases 21258  df-lm 21541  df-haus 21627  df-xms 22633  df-ms 22634  df-liminf 41470  df-xlim 41537
This theorem is referenced by: (None)
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