Theorem climliminflimsup2 40041

Description: A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 40020). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
climliminflimsup2.1	|- ( ph -> M e. ZZ )
climliminflimsup2.2	|- Z = ( ZZ>= ` M )
climliminflimsup2.3	|- ( ph -> F : Z --> RR )

Assertion

Ref	Expression
climliminflimsup2	|- ( ph -> ( F e. dom ~~> <-> ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) )

Proof of Theorem climliminflimsup2

Step	Hyp	Ref	Expression
1		climliminflimsup2.1	. . 3 |- ( ph -> M e. ZZ )
2		climliminflimsup2.2	. . 3 |- Z = ( ZZ>= ` M )
3		climliminflimsup2.3	. . 3 |- ( ph -> F : Z --> RR )
4	1, 2, 3	climliminflimsup 40040	. 2 |- ( ph -> ( F e. dom ~~> <-> ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) )
5	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> M e. ZZ )
6	3	adantr 481	. . . . . . 7 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> F : Z --> RR )
7		simprl 794	. . . . . . 7 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> (liminf ` F ) e. RR )
8		simprr 796	. . . . . . 7 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> ( limsup ` F ) <_ (liminf ` F ) )
9	5, 2, 6, 7, 8	liminflimsupclim 40039	. . . . . 6 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> F e. dom ~~> )
10	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ F e. dom ~~> ) -> M e. ZZ )
11	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ F e. dom ~~> ) -> F : Z --> RR )
12		simpr 477	. . . . . . . 8 |- ( ( ph /\ F e. dom ~~> ) -> F e. dom ~~> )
13	10, 2, 11, 12	climliminflimsupd 40033	. . . . . . 7 |- ( ( ph /\ F e. dom ~~> ) -> (liminf ` F ) = ( limsup ` F ) )
14	13	eqcomd 2628	. . . . . 6 |- ( ( ph /\ F e. dom ~~> ) -> ( limsup ` F ) = (liminf ` F ) )
15	9, 14	syldan 487	. . . . 5 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> ( limsup ` F ) = (liminf ` F ) )
16	15, 7	eqeltrd 2701	. . . 4 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> ( limsup ` F ) e. RR )
17	16, 8	jca 554	. . 3 |- ( ( ph /\ ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) )
18		simpr 477	. . . . . . 7 |- ( ( ph /\ ( limsup ` F ) <_ (liminf ` F ) ) -> ( limsup ` F ) <_ (liminf ` F ) )
19	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( limsup ` F ) <_ (liminf ` F ) ) -> M e. ZZ )
20	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( limsup ` F ) <_ (liminf ` F ) ) -> F : Z --> RR )
21	19, 2, 20	liminfgelimsupuz 40020	. . . . . . 7 |- ( ( ph /\ ( limsup ` F ) <_ (liminf ` F ) ) -> ( ( limsup ` F ) <_ (liminf ` F ) <-> (liminf ` F ) = ( limsup ` F ) ) )
22	18, 21	mpbid 222	. . . . . 6 |- ( ( ph /\ ( limsup ` F ) <_ (liminf ` F ) ) -> (liminf ` F ) = ( limsup ` F ) )
23	22	adantrl 752	. . . . 5 |- ( ( ph /\ ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> (liminf ` F ) = ( limsup ` F ) )
24		simprl 794	. . . . 5 |- ( ( ph /\ ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> ( limsup ` F ) e. RR )
25	23, 24	eqeltrd 2701	. . . 4 |- ( ( ph /\ ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> (liminf ` F ) e. RR )
26		simprr 796	. . . 4 |- ( ( ph /\ ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> ( limsup ` F ) <_ (liminf ` F ) )
27	25, 26	jca 554	. . 3 |- ( ( ph /\ ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) -> ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) )
28	17, 27	impbida 877	. 2 |- ( ph -> ( ( (liminf ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) <-> ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) )
29	4, 28	bitrd 268	1 |- ( ph -> ( F e. dom ~~> <-> ( ( limsup ` F ) e. RR /\ ( limsup ` F ) <_ (liminf ` F ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   RRcr 9935   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   limsupclsp 14201   ~~> cli 14215  liminfclsi 39983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-ioo 12179  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-ceil 12594  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-liminf 39984

This theorem is referenced by:  climliminflimsup4  40043