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Theorem xlimliminflimsup 44578
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 44573 . . . . . . 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 44547 . . . 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 44113 . . . . . . 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 44095 . . . . . . 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 44544 . . . . . . . . . . 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 44537 . . . . . . . . 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 44570 . . . . . 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 44517 . . . . 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 3495 . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ V)
2827adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝐹 ∈ V)
294fdmd 6729 . . . . . . . . . 10 (πœ‘ β†’ dom 𝐹 = 𝑍)
3026ssd 43769 . . . . . . . . . 10 (πœ‘ β†’ 𝑍 βŠ† β„€)
3129, 30eqsstrd 4021 . . . . . . . . 9 (πœ‘ β†’ dom 𝐹 βŠ† β„€)
3231adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
3326, 13, 28, 32liminfresuz2 44503 . . . . . . 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 44425 . . . . . . 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 44539 . . 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 5175 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4544adantll 713 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4617liminfcld 44486 . . . . . . . 8 (πœ‘ β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4746adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4817limsupcld 44406 . . . . . . . 8 (πœ‘ β†’ (lim supβ€˜πΉ) ∈ ℝ*)
4948adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
501, 3, 4liminflelimsupuz 44501 . . . . . . . 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 44184 . . . . . . . 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 44576 . . . . . . . 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 13134 . . . . . 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 44538 . . . . . . . . . 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 2949 . . . . . . . . 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 44185 . . . . . . 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 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 44572 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) = -∞)
7972mnfled 13115 . . . . . . 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 13134 . . . . 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 11271 . . . . . 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 44568 . . . . . 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 5913 . . . 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 11265 . . . . . . 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 11266 . . . . . . 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 44533 . . . . . 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 44113 . . . . . . . . 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 44503 . . . . . . . . . . . 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 44503 . . . . . . . . . . . 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 44425 . . . . . . . . . . . 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 44526 . . . . . . . . 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 44567 . . . . . . 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 44575 . . . . . . 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 44539 . . . . 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 43772 . . . . . . 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 11268 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 44577 . . . . . . . 8 (πœ‘ β†’ (𝐹~~>*+∞ ↔ (lim infβ€˜πΉ) = +∞))
151150biimpar 479 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
152147, 149, 151breldmd 5913 . . . . . 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 3030 . 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 2941  Vcvv 3475   βŠ† wss 3949   class class class wbr 5149  dom cdm 5677   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑pm cpm 8821  β„‚cc 11108  β„cr 11109  +∞cpnf 11245  -∞cmnf 11246  β„*cxr 11247   ≀ cle 11249  β„€cz 12558  β„€β‰₯cuz 12822  lim supclsp 15414   ⇝ cli 15428  lim infclsi 44467  ~~>*clsxlim 44534
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ioc 13329  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-ceil 13758  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-limsup 15415  df-clim 15432  df-rlim 15433  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-starv 17212  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-rest 17368  df-topn 17369  df-topgen 17389  df-ordt 17447  df-ps 18519  df-tsr 18520  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-lm 22733  df-haus 22819  df-xms 23826  df-ms 23827  df-liminf 44468  df-xlim 44535
This theorem is referenced by:  xlimlimsupleliminf  44579
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