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Theorem xlimliminflimsup 44878
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 722 . . . . 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 722 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ 𝐹:π‘βŸΆβ„*)
6 simpr 483 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ (~~>*β€˜πΉ) ∈ ℝ)
7 xlimdm 44873 . . . . . . 7 (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*β€˜πΉ))
87biimpi 215 . . . . . 6 (𝐹 ∈ dom ~~>* β†’ 𝐹~~>*(~~>*β€˜πΉ))
98ad2antlr 723 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ 𝐹~~>*(~~>*β€˜πΉ))
102, 3, 5, 6, 9xlimxrre 44847 . . . 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 44413 . . . . . . 7 (𝑗 ∈ 𝑍 β†’ 𝑗 ∈ β„€)
1211ad2antlr 723 . . . . . 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 2730 . . . . . 6 (β„€β‰₯β€˜π‘—) = (β„€β‰₯β€˜π‘—)
14 simpr 483 . . . . . 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 44395 . . . . . . 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 479 . . . . . . . . 9 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝐹~~>*(~~>*β€˜πΉ))
173, 4fuzxrpmcn 44844 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))
1817ad3antrrr 726 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))
1911adantl 480 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝑗 ∈ β„€)
2018, 19xlimres 44837 . . . . . . . . 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 479 . . . . . . 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 772 . . . . . . 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 44870 . . . . . 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 44817 . . . . 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 480 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝑗 ∈ β„€)
2717elexd 3493 . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ V)
2827adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝐹 ∈ V)
294fdmd 6729 . . . . . . . . . 10 (πœ‘ β†’ dom 𝐹 = 𝑍)
3026ssd 44072 . . . . . . . . . 10 (πœ‘ β†’ 𝑍 βŠ† β„€)
3129, 30eqsstrd 4021 . . . . . . . . 9 (πœ‘ β†’ dom 𝐹 βŠ† β„€)
3231adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
3326, 13, 28, 32liminfresuz2 44803 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))) = (lim infβ€˜πΉ))
3433eqcomd 2736 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim infβ€˜πΉ) = (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3534ad5ant14 754 . . . . 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 44725 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))) = (lim supβ€˜πΉ))
3736eqcomd 2736 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim supβ€˜πΉ) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3837ad5ant14 754 . . . . 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 2780 . . . 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 44839 . . 3 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
41 simpll 763 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ πœ‘)
428adantr 479 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*(~~>*β€˜πΉ))
43 simpr 483 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) = +∞)
4442, 43breqtrd 5175 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4544adantll 710 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4617liminfcld 44786 . . . . . . . 8 (πœ‘ β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4746adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4817limsupcld 44706 . . . . . . . 8 (πœ‘ β†’ (lim supβ€˜πΉ) ∈ ℝ*)
4948adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
501, 3, 4liminflelimsupuz 44801 . . . . . . . 8 (πœ‘ β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
5150adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
5249pnfged 44484 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ≀ +∞)
531adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝑀 ∈ β„€)
544adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝐹:π‘βŸΆβ„*)
55 simpr 483 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝐹~~>*+∞)
5653, 3, 54, 55xlimpnfliminf 44876 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) = +∞)
5752, 56breqtrrd 5177 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
5847, 49, 51, 57xrletrid 13140 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
5941, 45, 58syl2anc 582 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
6059adantlr 711 . . . 4 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ (~~>*β€˜πΉ) = +∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
61 simplll 771 . . . . 5 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ πœ‘)
628ad2antrr 722 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*(~~>*β€˜πΉ))
63 xlimcl 44838 . . . . . . . . . 10 (𝐹~~>*(~~>*β€˜πΉ) β†’ (~~>*β€˜πΉ) ∈ ℝ*)
648, 63syl 17 . . . . . . . . 9 (𝐹 ∈ dom ~~>* β†’ (~~>*β€˜πΉ) ∈ ℝ*)
6564ad2antrr 722 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) ∈ ℝ*)
66 simplr 765 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ Β¬ (~~>*β€˜πΉ) ∈ ℝ)
67 neqne 2946 . . . . . . . . 9 (Β¬ (~~>*β€˜πΉ) = +∞ β†’ (~~>*β€˜πΉ) β‰  +∞)
6867adantl 480 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) β‰  +∞)
6965, 66, 68xrnpnfmnf 44485 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) = -∞)
7062, 69breqtrd 5175 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*-∞)
7170adantlll 714 . . . . 5 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*-∞)
7246adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) ∈ ℝ*)
7348adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
7450adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
751adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝑀 ∈ β„€)
764adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝐹:π‘βŸΆβ„*)
77 simpr 483 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝐹~~>*-∞)
7875, 3, 76, 77xlimmnflimsup 44872 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) = -∞)
7972mnfled 13121 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ -∞ ≀ (lim infβ€˜πΉ))
8078, 79eqbrtrd 5171 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
8172, 73, 74, 80xrletrid 13140 . . . . 5 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8261, 71, 81syl2anc 582 . . . 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 809 . . 3 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8440, 83pm2.61dan 809 . 2 ((πœ‘ ∧ 𝐹 ∈ dom ~~>*) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8527adantr 479 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ V)
86 mnfxr 11277 . . . . . 6 -∞ ∈ ℝ*
8786a1i 11 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ -∞ ∈ ℝ*)
88 simpr 483 . . . . . 6 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ (lim supβ€˜πΉ) = -∞)
891adantr 479 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝑀 ∈ β„€)
904adantr 479 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹:π‘βŸΆβ„*)
9189, 3, 90xlimmnflimsup2 44868 . . . . . 6 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ (𝐹~~>*-∞ ↔ (lim supβ€˜πΉ) = -∞))
9288, 91mpbird 256 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹~~>*-∞)
9385, 87, 92breldmd 5913 . . . 4 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ dom ~~>*)
9493adantlr 711 . . 3 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ dom ~~>*)
951ad2antrr 722 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝑀 ∈ β„€)
964ad2antrr 722 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹:π‘βŸΆβ„*)
97 simpr 483 . . . . . . . 8 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ)
9897renepnfd 11271 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) β‰  +∞)
99 simplr 765 . . . . . . . . 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 2831 . . . . . . . 8 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) ∈ ℝ)
101100renemnfd 11272 . . . . . . 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 44833 . . . . . 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 44413 . . . . . . . . 9 (π‘š ∈ 𝑍 β†’ π‘š ∈ β„€)
104103ad2antlr 723 . . . . . . . 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 2730 . . . . . . . 8 (β„€β‰₯β€˜π‘š) = (β„€β‰₯β€˜π‘š)
106 simpr 483 . . . . . . . 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 771 . . . . . . . . . . 11 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ πœ‘)
108 simpl 481 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 13 (((lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ)
110108, 109eqeltrd 2831 . . . . . . . . . . . 12 (((lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) ∈ ℝ)
111110ad4ant23 749 . . . . . . . . . . 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 483 . . . . . . . . . . 11 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ 𝑍)
1131033ad2ant3 1133 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ β„€)
114273ad2ant1 1131 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ 𝐹 ∈ V)
115313ad2ant1 1131 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
116113, 105, 114, 115liminfresuz2 44803 . . . . . . . . . . . 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 1135 . . . . . . . . . . . 12 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜πΉ) ∈ ℝ)
118116, 117eqeltrd 2831 . . . . . . . . . . 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 1369 . . . . . . . . . 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 479 . . . . . . . . 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 1135 . . . . . . . . . . 11 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
122103adantl 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ β„€)
12327adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ 𝐹 ∈ V)
12431adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
125122, 105, 123, 124liminfresuz2 44803 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim infβ€˜πΉ))
1261253adant2 1129 . . . . . . . . . . 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 44725 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim supβ€˜πΉ))
1281273adant2 1129 . . . . . . . . . . 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 2780 . . . . . . . . . 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 1363 . . . . . . . . 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 44826 . . . . . . . . 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 709 . . . . . . . 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 44867 . . . . . . 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 728 . . . . . . . 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 44875 . . . . . . 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 256 . . . . . 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 44839 . . . . 5 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹 ∈ dom ~~>*)
138137adantlr 711 . . . 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 763 . . . . 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 772 . . . . . 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 722 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
142 simpr 483 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ Β¬ (lim supβ€˜πΉ) ∈ ℝ)
143 simplr 765 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) β‰  -∞)
144141, 142, 143xrnmnfpnf 44075 . . . . . . 7 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) = +∞)
145144adantllr 715 . . . . . 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 2770 . . . . 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 479 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ V)
148 pnfxr 11274 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 44877 . . . . . . . 8 (πœ‘ β†’ (𝐹~~>*+∞ ↔ (lim infβ€˜πΉ) = +∞))
151150biimpar 476 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
152147, 149, 151breldmd 5913 . . . . . 6 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ dom ~~>*)
153152adantlr 711 . . . . 5 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ dom ~~>*)
154139, 146, 153syl2anc 582 . . . 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 809 . . 3 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) β‰  -∞) β†’ 𝐹 ∈ dom ~~>*)
15694, 155pm2.61dane 3027 . 2 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) β†’ 𝐹 ∈ dom ~~>*)
15784, 156impbida 797 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472   βŠ† wss 3949   class class class wbr 5149  dom cdm 5677   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413   ↑pm cpm 8825  β„‚cc 11112  β„cr 11113  +∞cpnf 11251  -∞cmnf 11252  β„*cxr 11253   ≀ cle 11255  β„€cz 12564  β„€β‰₯cuz 12828  lim supclsp 15420   ⇝ cli 15434  lim infclsi 44767  ~~>*clsxlim 44834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fi 9410  df-sup 9441  df-inf 9442  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-div 11878  df-nn 12219  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12479  df-z 12565  df-dec 12684  df-uz 12829  df-q 12939  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ioo 13334  df-ioc 13335  df-ico 13336  df-icc 13337  df-fz 13491  df-fzo 13634  df-fl 13763  df-ceil 13764  df-seq 13973  df-exp 14034  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15421  df-clim 15438  df-rlim 15439  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17151  df-plusg 17216  df-mulr 17217  df-starv 17218  df-tset 17222  df-ple 17223  df-ds 17225  df-unif 17226  df-rest 17374  df-topn 17375  df-topgen 17395  df-ordt 17453  df-ps 18525  df-tsr 18526  df-psmet 21138  df-xmet 21139  df-met 21140  df-bl 21141  df-mopn 21142  df-cnfld 21147  df-top 22618  df-topon 22635  df-topsp 22657  df-bases 22671  df-lm 22955  df-haus 23041  df-xms 24048  df-ms 24049  df-liminf 44768  df-xlim 44835
This theorem is referenced by:  xlimlimsupleliminf  44879
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