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Theorem xlimliminflimsup 44193
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 (πœ‘ β†’ 𝑀 ∈ β„€)
xlimliminflimsup.z 𝑍 = (β„€β‰₯β€˜π‘€)
xlimliminflimsup.f (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)
Assertion
Ref Expression
xlimliminflimsup (πœ‘ β†’ (𝐹 ∈ dom ~~>* ↔ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)))

Proof of Theorem xlimliminflimsup
Dummy variables 𝑗 π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xlimliminflimsup.m . . . . . 6 (πœ‘ β†’ 𝑀 ∈ β„€)
21ad2antrr 725 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ 𝑀 ∈ β„€)
3 xlimliminflimsup.z . . . . 5 𝑍 = (β„€β‰₯β€˜π‘€)
4 xlimliminflimsup.f . . . . . 6 (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)
54ad2antrr 725 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ 𝐹:π‘βŸΆβ„*)
6 simpr 486 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ (~~>*β€˜πΉ) ∈ ℝ)
7 xlimdm 44188 . . . . . . 7 (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*β€˜πΉ))
87biimpi 215 . . . . . 6 (𝐹 ∈ dom ~~>* β†’ 𝐹~~>*(~~>*β€˜πΉ))
98ad2antlr 726 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ 𝐹~~>*(~~>*β€˜πΉ))
102, 3, 5, 6, 9xlimxrre 44162 . . . 4 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ βˆƒπ‘— ∈ 𝑍 (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„)
113eluzelz2 43728 . . . . . . 7 (𝑗 ∈ 𝑍 β†’ 𝑗 ∈ β„€)
1211ad2antlr 726 . . . . . 6 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ 𝑗 ∈ β„€)
13 eqid 2733 . . . . . 6 (β„€β‰₯β€˜π‘—) = (β„€β‰₯β€˜π‘—)
14 simpr 486 . . . . . 6 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„)
1514frexr 43710 . . . . . . 7 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„*)
169adantr 482 . . . . . . . . 9 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝐹~~>*(~~>*β€˜πΉ))
173, 4fuzxrpmcn 44159 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))
1817ad3antrrr 729 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))
1911adantl 483 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝑗 ∈ β„€)
2018, 19xlimres 44152 . . . . . . . . 9 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ (𝐹~~>*(~~>*β€˜πΉ) ↔ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—))~~>*(~~>*β€˜πΉ)))
2116, 20mpbid 231 . . . . . . . 8 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—))~~>*(~~>*β€˜πΉ))
2221adantr 482 . . . . . . 7 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—))~~>*(~~>*β€˜πΉ))
23 simpllr 775 . . . . . . 7 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (~~>*β€˜πΉ) ∈ ℝ)
2412, 13, 15, 22, 23xlimclimdm 44185 . . . . . 6 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)) ∈ dom ⇝ )
2512, 13, 14, 24climliminflimsupd 44132 . . . . 5 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
2611adantl 483 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝑗 ∈ β„€)
2717elexd 3467 . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ V)
2827adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝐹 ∈ V)
294fdmd 6683 . . . . . . . . . 10 (πœ‘ β†’ dom 𝐹 = 𝑍)
3026ssd 43382 . . . . . . . . . 10 (πœ‘ β†’ 𝑍 βŠ† β„€)
3129, 30eqsstrd 3986 . . . . . . . . 9 (πœ‘ β†’ dom 𝐹 βŠ† β„€)
3231adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
3326, 13, 28, 32liminfresuz2 44118 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))) = (lim infβ€˜πΉ))
3433eqcomd 2739 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim infβ€˜πΉ) = (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3534ad5ant14 757 . . . . 5 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (lim infβ€˜πΉ) = (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3626, 13, 28, 32limsupresuz2 44040 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))) = (lim supβ€˜πΉ))
3736eqcomd 2739 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim supβ€˜πΉ) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3837ad5ant14 757 . . . . 5 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (lim supβ€˜πΉ) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3925, 35, 383eqtr4d 2783 . . . 4 (((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
4010, 39rexlimddv2 44154 . . 3 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
41 simpll 766 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ πœ‘)
428adantr 482 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*(~~>*β€˜πΉ))
43 simpr 486 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) = +∞)
4442, 43breqtrd 5135 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4544adantll 713 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4617liminfcld 44101 . . . . . . . 8 (πœ‘ β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4746adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4817limsupcld 44021 . . . . . . . 8 (πœ‘ β†’ (lim supβ€˜πΉ) ∈ ℝ*)
4948adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
501, 3, 4liminflelimsupuz 44116 . . . . . . . 8 (πœ‘ β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
5150adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
5249pnfged 43799 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ≀ +∞)
531adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝑀 ∈ β„€)
544adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝐹:π‘βŸΆβ„*)
55 simpr 486 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝐹~~>*+∞)
5653, 3, 54, 55xlimpnfliminf 44191 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) = +∞)
5752, 56breqtrrd 5137 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
5847, 49, 51, 57xrletrid 13083 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
5941, 45, 58syl2anc 585 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
6059adantlr 714 . . . 4 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ (~~>*β€˜πΉ) = +∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
61 simplll 774 . . . . 5 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ πœ‘)
628ad2antrr 725 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*(~~>*β€˜πΉ))
63 xlimcl 44153 . . . . . . . . . 10 (𝐹~~>*(~~>*β€˜πΉ) β†’ (~~>*β€˜πΉ) ∈ ℝ*)
648, 63syl 17 . . . . . . . . 9 (𝐹 ∈ dom ~~>* β†’ (~~>*β€˜πΉ) ∈ ℝ*)
6564ad2antrr 725 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) ∈ ℝ*)
66 simplr 768 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ Β¬ (~~>*β€˜πΉ) ∈ ℝ)
67 neqne 2948 . . . . . . . . 9 (Β¬ (~~>*β€˜πΉ) = +∞ β†’ (~~>*β€˜πΉ) β‰  +∞)
6867adantl 483 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) β‰  +∞)
6965, 66, 68xrnpnfmnf 43800 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) = -∞)
7062, 69breqtrd 5135 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*-∞)
7170adantlll 717 . . . . 5 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*-∞)
7246adantr 482 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) ∈ ℝ*)
7348adantr 482 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
7450adantr 482 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
751adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝑀 ∈ β„€)
764adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝐹:π‘βŸΆβ„*)
77 simpr 486 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝐹~~>*-∞)
7875, 3, 76, 77xlimmnflimsup 44187 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) = -∞)
7972mnfled 13064 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ -∞ ≀ (lim infβ€˜πΉ))
8078, 79eqbrtrd 5131 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
8172, 73, 74, 80xrletrid 13083 . . . . 5 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8261, 71, 81syl2anc 585 . . . 4 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8360, 82pm2.61dan 812 . . 3 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8440, 83pm2.61dan 812 . 2 ((πœ‘ ∧ 𝐹 ∈ dom ~~>*) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8527adantr 482 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ V)
86 mnfxr 11220 . . . . . 6 -∞ ∈ ℝ*
8786a1i 11 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ -∞ ∈ ℝ*)
88 simpr 486 . . . . . 6 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ (lim supβ€˜πΉ) = -∞)
891adantr 482 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝑀 ∈ β„€)
904adantr 482 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹:π‘βŸΆβ„*)
9189, 3, 90xlimmnflimsup2 44183 . . . . . 6 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ (𝐹~~>*-∞ ↔ (lim supβ€˜πΉ) = -∞))
9288, 91mpbird 257 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹~~>*-∞)
9385, 87, 92breldmd 5872 . . . 4 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ dom ~~>*)
9493adantlr 714 . . 3 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ dom ~~>*)
951ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝑀 ∈ β„€)
964ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹:π‘βŸΆβ„*)
97 simpr 486 . . . . . . . 8 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ)
9897renepnfd 11214 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) β‰  +∞)
99 simplr 768 . . . . . . . . 9 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
10099, 97eqeltrd 2834 . . . . . . . 8 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) ∈ ℝ)
101100renemnfd 11215 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) β‰  -∞)
10295, 3, 96, 98, 101liminflimsupxrre 44148 . . . . . 6 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ βˆƒπ‘š ∈ 𝑍 (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„)
1033eluzelz2 43728 . . . . . . . . 9 (π‘š ∈ 𝑍 β†’ π‘š ∈ β„€)
104103ad2antlr 726 . . . . . . . 8 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ π‘š ∈ β„€)
105 eqid 2733 . . . . . . . 8 (β„€β‰₯β€˜π‘š) = (β„€β‰₯β€˜π‘š)
106 simpr 486 . . . . . . . 8 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„)
107 simplll 774 . . . . . . . . . . 11 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ πœ‘)
108 simpl 484 . . . . . . . . . . . . 13 (((lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
109 simpr 486 . . . . . . . . . . . . 13 (((lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ)
110108, 109eqeltrd 2834 . . . . . . . . . . . 12 (((lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) ∈ ℝ)
111110ad4ant23 752 . . . . . . . . . . 11 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜πΉ) ∈ ℝ)
112 simpr 486 . . . . . . . . . . 11 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ 𝑍)
1131033ad2ant3 1136 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ β„€)
114273ad2ant1 1134 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ 𝐹 ∈ V)
115313ad2ant1 1134 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
116113, 105, 114, 115liminfresuz2 44118 . . . . . . . . . . . 12 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim infβ€˜πΉ))
117 simp2 1138 . . . . . . . . . . . 12 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜πΉ) ∈ ℝ)
118116, 117eqeltrd 2834 . . . . . . . . . . 11 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) ∈ ℝ)
119107, 111, 112, 118syl3anc 1372 . . . . . . . . . 10 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) ∈ ℝ)
120119adantr 482 . . . . . . . . 9 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) ∈ ℝ)
121 simp2 1138 . . . . . . . . . . 11 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
122103adantl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ β„€)
12327adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ 𝐹 ∈ V)
12431adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
125122, 105, 123, 124liminfresuz2 44118 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim infβ€˜πΉ))
1261253adant2 1132 . . . . . . . . . . 11 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim infβ€˜πΉ))
127122, 105, 123, 124limsupresuz2 44040 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim supβ€˜πΉ))
1281273adant2 1132 . . . . . . . . . . 11 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ π‘š ∈ 𝑍) β†’ (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim supβ€˜πΉ))
129121, 126, 1283eqtr4d 2783 . . . . . . . . . 10 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))))
130129ad5ant124 1366 . . . . . . . . 9 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))))
131104, 105, 106climliminflimsup3 44141 . . . . . . . . 9 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ ((𝐹 β†Ύ (β„€β‰₯β€˜π‘š)) ∈ dom ⇝ ↔ ((lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) ∈ ℝ ∧ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))))))
132120, 130, 131mpbir2and 712 . . . . . . . 8 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)) ∈ dom ⇝ )
133104, 105, 106, 132dmclimxlim 44182 . . . . . . 7 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)) ∈ dom ~~>*)
13417ad4antr 731 . . . . . . . 8 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))
135134, 104xlimresdm 44190 . . . . . . 7 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ (𝐹 ∈ dom ~~>* ↔ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)) ∈ dom ~~>*))
136133, 135mpbird 257 . . . . . 6 (((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) ∧ (𝐹 β†Ύ (β„€β‰₯β€˜π‘š)):(β„€β‰₯β€˜π‘š)βŸΆβ„) β†’ 𝐹 ∈ dom ~~>*)
137102, 136rexlimddv2 44154 . . . . 5 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹 ∈ dom ~~>*)
138137adantlr 714 . . . 4 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹 ∈ dom ~~>*)
139 simpll 766 . . . . 5 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)))
140 simpllr 775 . . . . . 6 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
14148ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
142 simpr 486 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ Β¬ (lim supβ€˜πΉ) ∈ ℝ)
143 simplr 768 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) β‰  -∞)
144141, 142, 143xrnmnfpnf 43385 . . . . . . 7 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) = +∞)
145144adantllr 718 . . . . . 6 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) = +∞)
146140, 145eqtrd 2773 . . . . 5 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = +∞)
14727adantr 482 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ V)
148 pnfxr 11217 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 44192 . . . . . . . 8 (πœ‘ β†’ (𝐹~~>*+∞ ↔ (lim infβ€˜πΉ) = +∞))
151150biimpar 479 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
152147, 149, 151breldmd 5872 . . . . . 6 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ dom ~~>*)
153152adantlr 714 . . . . 5 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ dom ~~>*)
154139, 146, 153syl2anc 585 . . . 4 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹 ∈ dom ~~>*)
155138, 154pm2.61dan 812 . . 3 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) β†’ 𝐹 ∈ dom ~~>*)
15694, 155pm2.61dane 3029 . 2 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) β†’ 𝐹 ∈ dom ~~>*)
15784, 156impbida 800 1 (πœ‘ β†’ (𝐹 ∈ dom ~~>* ↔ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  Vcvv 3447   βŠ† wss 3914   class class class wbr 5109  dom cdm 5637   β†Ύ cres 5639  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑pm cpm 8772  β„‚cc 11057  β„cr 11058  +∞cpnf 11194  -∞cmnf 11195  β„*cxr 11196   ≀ cle 11198  β„€cz 12507  β„€β‰₯cuz 12771  lim supclsp 15361   ⇝ cli 15375  lim infclsi 44082  ~~>*clsxlim 44149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fi 9355  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-ioo 13277  df-ioc 13278  df-ico 13279  df-icc 13280  df-fz 13434  df-fzo 13577  df-fl 13706  df-ceil 13707  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-limsup 15362  df-clim 15379  df-rlim 15380  df-struct 17027  df-slot 17062  df-ndx 17074  df-base 17092  df-plusg 17154  df-mulr 17155  df-starv 17156  df-tset 17160  df-ple 17161  df-ds 17163  df-unif 17164  df-rest 17312  df-topn 17313  df-topgen 17333  df-ordt 17391  df-ps 18463  df-tsr 18464  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-cnfld 20820  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-lm 22603  df-haus 22689  df-xms 23696  df-ms 23697  df-liminf 44083  df-xlim 44150
This theorem is referenced by:  xlimlimsupleliminf  44194
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