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Theorem xlimliminflimsup 45388
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 483 . . . . 5 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7 xlimdm 45383 . . . . . . 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 45357 . . . 4 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
113eluzelz2 44923 . . . . . . 7 (𝑗𝑍𝑗 ∈ ℤ)
1211ad2antlr 725 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13 eqid 2725 . . . . . 6 (ℤ𝑗) = (ℤ𝑗)
14 simpr 483 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
1514frexr 44905 . . . . . . 7 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ*)
169adantr 479 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹~~>*(~~>*‘𝐹))
173, 4fuzxrpmcn 45354 . . . . . . . . . . 11 (𝜑𝐹 ∈ (ℝ*pm ℂ))
1817ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝐹 ∈ (ℝ*pm ℂ))
1911adantl 480 . . . . . . . . . 10 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → 𝑗 ∈ ℤ)
2018, 19xlimres 45347 . . . . . . . . 9 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹~~>*(~~>*‘𝐹) ↔ (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹)))
2116, 20mpbid 231 . . . . . . . 8 ((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) → (𝐹 ↾ (ℤ𝑗))~~>*(~~>*‘𝐹))
2221adantr 479 . . . . . . 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 45380 . . . . . 6 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (𝐹 ↾ (ℤ𝑗)) ∈ dom ⇝ )
2512, 13, 14, 24climliminflimsupd 45327 . . . . 5 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘(𝐹 ↾ (ℤ𝑗))))
2611adantl 480 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝑗 ∈ ℤ)
2717elexd 3483 . . . . . . . . 9 (𝜑𝐹 ∈ V)
2827adantr 479 . . . . . . . 8 ((𝜑𝑗𝑍) → 𝐹 ∈ V)
294fdmd 6733 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝑍)
3026ssd 44586 . . . . . . . . . 10 (𝜑𝑍 ⊆ ℤ)
3129, 30eqsstrd 4015 . . . . . . . . 9 (𝜑 → dom 𝐹 ⊆ ℤ)
3231adantr 479 . . . . . . . 8 ((𝜑𝑗𝑍) → dom 𝐹 ⊆ ℤ)
3326, 13, 28, 32liminfresuz2 45313 . . . . . . 7 ((𝜑𝑗𝑍) → (lim inf‘(𝐹 ↾ (ℤ𝑗))) = (lim inf‘𝐹))
3433eqcomd 2731 . . . . . 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 45235 . . . . . . 7 ((𝜑𝑗𝑍) → (lim sup‘(𝐹 ↾ (ℤ𝑗))) = (lim sup‘𝐹))
3736eqcomd 2731 . . . . . 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 2775 . . . 4 (((((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) ∧ 𝑗𝑍) ∧ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
4010, 39rexlimddv2 45349 . . 3 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41 simpll 765 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝜑)
428adantr 479 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*(~~>*‘𝐹))
43 simpr 483 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = +∞)
4442, 43breqtrd 5175 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4544adantll 712 . . . . . 6 (((𝜑𝐹 ∈ dom ~~>*) ∧ (~~>*‘𝐹) = +∞) → 𝐹~~>*+∞)
4617liminfcld 45296 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ∈ ℝ*)
4746adantr 479 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ∈ ℝ*)
4817limsupcld 45216 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
4948adantr 479 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ∈ ℝ*)
501, 3, 4liminflelimsupuz 45311 . . . . . . . 8 (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5150adantr 479 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
5249pnfged 44994 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ +∞)
531adantr 479 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝑀 ∈ ℤ)
544adantr 479 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹:𝑍⟶ℝ*)
55 simpr 483 . . . . . . . . 9 ((𝜑𝐹~~>*+∞) → 𝐹~~>*+∞)
5653, 3, 54, 55xlimpnfliminf 45386 . . . . . . . 8 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
5752, 56breqtrrd 5177 . . . . . . 7 ((𝜑𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
5847, 49, 51, 57xrletrid 13169 . . . . . 6 ((𝜑𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
5941, 45, 58syl2anc 582 . . . . 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 45348 . . . . . . . . . 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 2937 . . . . . . . . 9 (¬ (~~>*‘𝐹) = +∞ → (~~>*‘𝐹) ≠ +∞)
6867adantl 480 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
6965, 66, 68xrnpnfmnf 44995 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
7062, 69breqtrd 5175 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7170adantlll 716 . . . . 5 ((((𝜑𝐹 ∈ dom ~~>*) ∧ ¬ (~~>*‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝐹~~>*-∞)
7246adantr 479 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ∈ ℝ*)
7348adantr 479 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ∈ ℝ*)
7450adantr 479 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
751adantr 479 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝑀 ∈ ℤ)
764adantr 479 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹:𝑍⟶ℝ*)
77 simpr 483 . . . . . . . 8 ((𝜑𝐹~~>*-∞) → 𝐹~~>*-∞)
7875, 3, 76, 77xlimmnflimsup 45382 . . . . . . 7 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
7972mnfled 13150 . . . . . . 7 ((𝜑𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
8078, 79eqbrtrd 5171 . . . . . 6 ((𝜑𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
8172, 73, 74, 80xrletrid 13169 . . . . 5 ((𝜑𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
8261, 71, 81syl2anc 582 . . . 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 479 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86 mnfxr 11303 . . . . . 6 -∞ ∈ ℝ*
8786a1i 11 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → -∞ ∈ ℝ*)
88 simpr 483 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (lim sup‘𝐹) = -∞)
891adantr 479 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝑀 ∈ ℤ)
904adantr 479 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹:𝑍⟶ℝ*)
9189, 3, 90xlimmnflimsup2 45378 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
9288, 91mpbird 256 . . . . 5 ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
9385, 87, 92breldmd 5915 . . . 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 483 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
9897renepnfd 11297 . . . . . . 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 2825 . . . . . . . 8 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101100renemnfd 11298 . . . . . . 7 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
10295, 3, 96, 98, 101liminflimsupxrre 45343 . . . . . 6 (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚𝑍 (𝐹 ↾ (ℤ𝑚)):(ℤ𝑚)⟶ℝ)
1033eluzelz2 44923 . . . . . . . . 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 2725 . . . . . . . 8 (ℤ𝑚) = (ℤ𝑚)
106 simpr 483 . . . . . . . 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 481 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
109 simpr 483 . . . . . . . . . . . . 13 (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
110108, 109eqeltrd 2825 . . . . . . . . . . . 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 483 . . . . . . . . . . 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 45313 . . . . . . . . . . . 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 2825 . . . . . . . . . . 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 479 . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑚 ∈ ℤ)
12327adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝐹 ∈ V)
12431adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → dom 𝐹 ⊆ ℤ)
125122, 105, 123, 124liminfresuz2 45313 . . . . . . . . . . . 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 45235 . . . . . . . . . . . 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 2775 . . . . . . . . . 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 45336 . . . . . . . . 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 45377 . . . . . . 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 45385 . . . . . . 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 45349 . . . . 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 483 . . . . . . . 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 44589 . . . . . . 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 2765 . . . . 5 ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = +∞)
14727adantr 479 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ V)
148 pnfxr 11300 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 45387 . . . . . . . 8 (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151150biimpar 476 . . . . . . 7 ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152147, 149, 151breldmd 5915 . . . . . 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 582 . . . 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 3018 . 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 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  Vcvv 3461  wss 3944   class class class wbr 5149  dom cdm 5678  cres 5680  wf 6545  cfv 6549  (class class class)co 7419  pm cpm 8846  cc 11138  cr 11139  +∞cpnf 11277  -∞cmnf 11278  *cxr 11279  cle 11281  cz 12591  cuz 12855  lim supclsp 15450  cli 15464  lim infclsi 45277  ~~>*clsxlim 45344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9436  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ioo 13363  df-ioc 13364  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-ceil 13794  df-seq 14003  df-exp 14063  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-limsup 15451  df-clim 15468  df-rlim 15469  df-struct 17119  df-slot 17154  df-ndx 17166  df-base 17184  df-plusg 17249  df-mulr 17250  df-starv 17251  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-rest 17407  df-topn 17408  df-topgen 17428  df-ordt 17486  df-ps 18561  df-tsr 18562  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-cnfld 21297  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-lm 23177  df-haus 23263  df-xms 24270  df-ms 24271  df-liminf 45278  df-xlim 45345
This theorem is referenced by:  xlimlimsupleliminf  45389
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