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Theorem xlimliminflimsup 44568
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 724 . . . . 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 724 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ 𝐹:π‘βŸΆβ„*)
6 simpr 485 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ (~~>*β€˜πΉ) ∈ ℝ)
7 xlimdm 44563 . . . . . . 7 (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*β€˜πΉ))
87biimpi 215 . . . . . 6 (𝐹 ∈ dom ~~>* β†’ 𝐹~~>*(~~>*β€˜πΉ))
98ad2antlr 725 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ 𝐹~~>*(~~>*β€˜πΉ))
102, 3, 5, 6, 9xlimxrre 44537 . . . 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 44103 . . . . . . 7 (𝑗 ∈ 𝑍 β†’ 𝑗 ∈ β„€)
1211ad2antlr 725 . . . . . 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 2732 . . . . . 6 (β„€β‰₯β€˜π‘—) = (β„€β‰₯β€˜π‘—)
14 simpr 485 . . . . . 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 44085 . . . . . . 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 481 . . . . . . . . 9 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝐹~~>*(~~>*β€˜πΉ))
173, 4fuzxrpmcn 44534 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))
1817ad3antrrr 728 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))
1911adantl 482 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) β†’ 𝑗 ∈ β„€)
2018, 19xlimres 44527 . . . . . . . . 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 481 . . . . . . 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 774 . . . . . . 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 44560 . . . . . 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 44507 . . . . 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 482 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝑗 ∈ β„€)
2717elexd 3494 . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ V)
2827adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝐹 ∈ V)
294fdmd 6728 . . . . . . . . . 10 (πœ‘ β†’ dom 𝐹 = 𝑍)
3026ssd 43759 . . . . . . . . . 10 (πœ‘ β†’ 𝑍 βŠ† β„€)
3129, 30eqsstrd 4020 . . . . . . . . 9 (πœ‘ β†’ dom 𝐹 βŠ† β„€)
3231adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
3326, 13, 28, 32liminfresuz2 44493 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))) = (lim infβ€˜πΉ))
3433eqcomd 2738 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim infβ€˜πΉ) = (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3534ad5ant14 756 . . . . 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 44415 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))) = (lim supβ€˜πΉ))
3736eqcomd 2738 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ (lim supβ€˜πΉ) = (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘—))))
3837ad5ant14 756 . . . . 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 2782 . . . 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 44529 . . 3 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
41 simpll 765 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ πœ‘)
428adantr 481 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*(~~>*β€˜πΉ))
43 simpr 485 . . . . . . . 8 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) = +∞)
4442, 43breqtrd 5174 . . . . . . 7 ((𝐹 ∈ dom ~~>* ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4544adantll 712 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
4617liminfcld 44476 . . . . . . . 8 (πœ‘ β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4746adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) ∈ ℝ*)
4817limsupcld 44396 . . . . . . . 8 (πœ‘ β†’ (lim supβ€˜πΉ) ∈ ℝ*)
4948adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
501, 3, 4liminflelimsupuz 44491 . . . . . . . 8 (πœ‘ β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
5150adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
5249pnfged 44174 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ≀ +∞)
531adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝑀 ∈ β„€)
544adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝐹:π‘βŸΆβ„*)
55 simpr 485 . . . . . . . . 9 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ 𝐹~~>*+∞)
5653, 3, 54, 55xlimpnfliminf 44566 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) = +∞)
5752, 56breqtrrd 5176 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
5847, 49, 51, 57xrletrid 13133 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*+∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
5941, 45, 58syl2anc 584 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ (~~>*β€˜πΉ) = +∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
6059adantlr 713 . . . 4 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ (~~>*β€˜πΉ) = +∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
61 simplll 773 . . . . 5 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ πœ‘)
628ad2antrr 724 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*(~~>*β€˜πΉ))
63 xlimcl 44528 . . . . . . . . . 10 (𝐹~~>*(~~>*β€˜πΉ) β†’ (~~>*β€˜πΉ) ∈ ℝ*)
648, 63syl 17 . . . . . . . . 9 (𝐹 ∈ dom ~~>* β†’ (~~>*β€˜πΉ) ∈ ℝ*)
6564ad2antrr 724 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) ∈ ℝ*)
66 simplr 767 . . . . . . . 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 482 . . . . . . . 8 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) β‰  +∞)
6965, 66, 68xrnpnfmnf 44175 . . . . . . 7 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ (~~>*β€˜πΉ) = -∞)
7062, 69breqtrd 5174 . . . . . 6 (((𝐹 ∈ dom ~~>* ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*-∞)
7170adantlll 716 . . . . 5 ((((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) ∧ Β¬ (~~>*β€˜πΉ) = +∞) β†’ 𝐹~~>*-∞)
7246adantr 481 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) ∈ ℝ*)
7348adantr 481 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
7450adantr 481 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) ≀ (lim supβ€˜πΉ))
751adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝑀 ∈ β„€)
764adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝐹:π‘βŸΆβ„*)
77 simpr 485 . . . . . . . 8 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ 𝐹~~>*-∞)
7875, 3, 76, 77xlimmnflimsup 44562 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) = -∞)
7972mnfled 13114 . . . . . . 7 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ -∞ ≀ (lim infβ€˜πΉ))
8078, 79eqbrtrd 5170 . . . . . 6 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
8172, 73, 74, 80xrletrid 13133 . . . . 5 ((πœ‘ ∧ 𝐹~~>*-∞) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8261, 71, 81syl2anc 584 . . . 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 811 . . 3 (((πœ‘ ∧ 𝐹 ∈ dom ~~>*) ∧ Β¬ (~~>*β€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8440, 83pm2.61dan 811 . 2 ((πœ‘ ∧ 𝐹 ∈ dom ~~>*) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
8527adantr 481 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ V)
86 mnfxr 11270 . . . . . 6 -∞ ∈ ℝ*
8786a1i 11 . . . . 5 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ -∞ ∈ ℝ*)
88 simpr 485 . . . . . 6 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ (lim supβ€˜πΉ) = -∞)
891adantr 481 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝑀 ∈ β„€)
904adantr 481 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹:π‘βŸΆβ„*)
9189, 3, 90xlimmnflimsup2 44558 . . . . . 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 5912 . . . 4 ((πœ‘ ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ dom ~~>*)
9493adantlr 713 . . 3 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) = -∞) β†’ 𝐹 ∈ dom ~~>*)
951ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝑀 ∈ β„€)
964ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹:π‘βŸΆβ„*)
97 simpr 485 . . . . . . . 8 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ)
9897renepnfd 11264 . . . . . . 7 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) β‰  +∞)
99 simplr 767 . . . . . . . . 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 2833 . . . . . . . 8 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) ∈ ℝ)
101100renemnfd 11265 . . . . . . 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 44523 . . . . . 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 44103 . . . . . . . . 9 (π‘š ∈ 𝑍 β†’ π‘š ∈ β„€)
104103ad2antlr 725 . . . . . . . 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 2732 . . . . . . . 8 (β„€β‰₯β€˜π‘š) = (β„€β‰₯β€˜π‘š)
106 simpr 485 . . . . . . . 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 773 . . . . . . . . . . 11 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ πœ‘)
108 simpl 483 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 13 (((lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ)
110108, 109eqeltrd 2833 . . . . . . . . . . . 12 (((lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim infβ€˜πΉ) ∈ ℝ)
111110ad4ant23 751 . . . . . . . . . . 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 485 . . . . . . . . . . 11 ((((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ 𝑍)
1131033ad2ant3 1135 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ β„€)
114273ad2ant1 1133 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ 𝐹 ∈ V)
115313ad2ant1 1133 . . . . . . . . . . . . 13 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
116113, 105, 114, 115liminfresuz2 44493 . . . . . . . . . . . 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 1137 . . . . . . . . . . . 12 ((πœ‘ ∧ (lim infβ€˜πΉ) ∈ ℝ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜πΉ) ∈ ℝ)
118116, 117eqeltrd 2833 . . . . . . . . . . 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 1371 . . . . . . . . . 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 481 . . . . . . . . 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 1137 . . . . . . . . . . 11 ((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ) ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
122103adantl 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ π‘š ∈ β„€)
12327adantr 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ 𝐹 ∈ V)
12431adantr 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ dom 𝐹 βŠ† β„€)
125122, 105, 123, 124liminfresuz2 44493 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (lim infβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim infβ€˜πΉ))
1261253adant2 1131 . . . . . . . . . . 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 44415 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (lim supβ€˜(𝐹 β†Ύ (β„€β‰₯β€˜π‘š))) = (lim supβ€˜πΉ))
1281273adant2 1131 . . . . . . . . . . 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 2782 . . . . . . . . . 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 1365 . . . . . . . . 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 44516 . . . . . . . . 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 711 . . . . . . . 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 44557 . . . . . . 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 730 . . . . . . . 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 44565 . . . . . . 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 44529 . . . . 5 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim supβ€˜πΉ) ∈ ℝ) β†’ 𝐹 ∈ dom ~~>*)
138137adantlr 713 . . . 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 765 . . . . 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 774 . . . . . 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 724 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) ∈ ℝ*)
142 simpr 485 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ Β¬ (lim supβ€˜πΉ) ∈ ℝ)
143 simplr 767 . . . . . . . 8 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) β‰  -∞)
144141, 142, 143xrnmnfpnf 43762 . . . . . . 7 (((πœ‘ ∧ (lim supβ€˜πΉ) β‰  -∞) ∧ Β¬ (lim supβ€˜πΉ) ∈ ℝ) β†’ (lim supβ€˜πΉ) = +∞)
145144adantllr 717 . . . . . 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 2772 . . . . 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 481 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ V)
148 pnfxr 11267 . . . . . . . 8 +∞ ∈ ℝ*
149148a1i 11 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ +∞ ∈ ℝ*)
1501, 3, 4xlimpnfliminf2 44567 . . . . . . . 8 (πœ‘ β†’ (𝐹~~>*+∞ ↔ (lim infβ€˜πΉ) = +∞))
151150biimpar 478 . . . . . . 7 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹~~>*+∞)
152147, 149, 151breldmd 5912 . . . . . 6 ((πœ‘ ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ dom ~~>*)
153152adantlr 713 . . . . 5 (((πœ‘ ∧ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)) ∧ (lim infβ€˜πΉ) = +∞) β†’ 𝐹 ∈ dom ~~>*)
154139, 146, 153syl2anc 584 . . . 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 811 . . 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 799 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474   βŠ† wss 3948   class class class wbr 5148  dom cdm 5676   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑pm cpm 8820  β„‚cc 11107  β„cr 11108  +∞cpnf 11244  -∞cmnf 11245  β„*cxr 11246   ≀ cle 11248  β„€cz 12557  β„€β‰₯cuz 12821  lim supclsp 15413   ⇝ cli 15427  lim infclsi 44457  ~~>*clsxlim 44524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fi 9405  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-q 12932  df-rp 12974  df-xneg 13091  df-xadd 13092  df-xmul 13093  df-ioo 13327  df-ioc 13328  df-ico 13329  df-icc 13330  df-fz 13484  df-fzo 13627  df-fl 13756  df-ceil 13757  df-seq 13966  df-exp 14027  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-limsup 15414  df-clim 15431  df-rlim 15432  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-mulr 17210  df-starv 17211  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-rest 17367  df-topn 17368  df-topgen 17388  df-ordt 17446  df-ps 18518  df-tsr 18519  df-psmet 20935  df-xmet 20936  df-met 20937  df-bl 20938  df-mopn 20939  df-cnfld 20944  df-top 22395  df-topon 22412  df-topsp 22434  df-bases 22448  df-lm 22732  df-haus 22818  df-xms 23825  df-ms 23826  df-liminf 44458  df-xlim 44525
This theorem is referenced by:  xlimlimsupleliminf  44569
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