Theorem climisp 39978

Description: If a sequence converges to an isolated point (w.r.t. the standard topology on the complex numbers) then the sequence eventually becomes that point. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
climisp.m	|- ( ph -> M e. ZZ )
climisp.z	|- Z = ( ZZ>= ` M )
climisp.f	|- ( ph -> F : Z --> CC )
climisp.c	|- ( ph -> F ~~> A )
climisp.x	|- ( ph -> X e. RR+ )
climisp.l	|- ( ( ph /\ k e. Z /\ ( F ` k ) =/= A ) -> X <_ ( abs ` ( ( F ` k ) - A ) ) )

Assertion

Ref	Expression
climisp	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( F ` k ) = A )

Distinct variable groups:   A, j, k   j, F, k   j, M   j, X, k   j, Z, k   ph, j, k

Allowed substitution hint:   M( k)

Proof of Theorem climisp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ k ( ph /\ j e. Z )
2		nfra1 2941	. . . 4 |- F/ k A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X )
3	1, 2	nfan 1828	. . 3 |- F/ k ( ( ph /\ j e. Z ) /\ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) )
4		simplll 798	. . . 4 |- ( ( ( ( ph /\ j e. Z ) /\ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) /\ k e. ( ZZ>= ` j ) ) -> ph )
5		climisp.z	. . . . . 6 |- Z = ( ZZ>= ` M )
6	5	uztrn2 11705	. . . . 5 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
7	6	ad4ant24 1298	. . . 4 |- ( ( ( ( ph /\ j e. Z ) /\ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
8		rspa 2930	. . . . . 6 |- ( ( A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) /\ k e. ( ZZ>= ` j ) ) -> ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) )
9	8	simprd 479	. . . . 5 |- ( ( A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) /\ k e. ( ZZ>= ` j ) ) -> ( abs ` ( ( F ` k ) - A ) ) < X )
10	9	adantll 750	. . . 4 |- ( ( ( ( ph /\ j e. Z ) /\ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) /\ k e. ( ZZ>= ` j ) ) -> ( abs ` ( ( F ` k ) - A ) ) < X )
11		simpl3 1066	. . . . 5 |- ( ( ( ph /\ k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < X ) /\ -. ( F ` k ) = A ) -> ( abs ` ( ( F ` k ) - A ) ) < X )
12		neqne 2802	. . . . . . 7 |- ( -. ( F ` k ) = A -> ( F ` k ) =/= A )
13		climisp.x	. . . . . . . . . 10 |- ( ph -> X e. RR+ )
14	13	rpred 11872	. . . . . . . . 9 |- ( ph -> X e. RR )
15	14	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ k e. Z ) /\ ( F ` k ) =/= A ) -> X e. RR )
16		climisp.f	. . . . . . . . . . . 12 |- ( ph -> F : Z --> CC )
17	16	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
18		climisp.c	. . . . . . . . . . . . . 14 |- ( ph -> F ~~> A )
19	5	fvexi 6202	. . . . . . . . . . . . . . . . 17 |- Z e. _V
20	19	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ph -> Z e. _V )
21	16, 20	fexd 39296	. . . . . . . . . . . . . . 15 |- ( ph -> F e. _V )
22		eqidd 2623	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
23	21, 22	clim 14225	. . . . . . . . . . . . . 14 |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) )
24	18, 23	mpbid 222	. . . . . . . . . . . . 13 |- ( ph -> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
25	24	simpld 475	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
26	25	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> A e. CC )
27	17, 26	subcld 10392	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) - A ) e. CC )
28	27	abscld 14175	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) e. RR )
29	28	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ k e. Z ) /\ ( F ` k ) =/= A ) -> ( abs ` ( ( F ` k ) - A ) ) e. RR )
30		climisp.l	. . . . . . . . 9 |- ( ( ph /\ k e. Z /\ ( F ` k ) =/= A ) -> X <_ ( abs ` ( ( F ` k ) - A ) ) )
31	30	3expa 1265	. . . . . . . 8 |- ( ( ( ph /\ k e. Z ) /\ ( F ` k ) =/= A ) -> X <_ ( abs ` ( ( F ` k ) - A ) ) )
32	15, 29, 31	lensymd 10188	. . . . . . 7 |- ( ( ( ph /\ k e. Z ) /\ ( F ` k ) =/= A ) -> -. ( abs ` ( ( F ` k ) - A ) ) < X )
33	12, 32	sylan2 491	. . . . . 6 |- ( ( ( ph /\ k e. Z ) /\ -. ( F ` k ) = A ) -> -. ( abs ` ( ( F ` k ) - A ) ) < X )
34	33	3adantl3 1219	. . . . 5 |- ( ( ( ph /\ k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < X ) /\ -. ( F ` k ) = A ) -> -. ( abs ` ( ( F ` k ) - A ) ) < X )
35	11, 34	condan 835	. . . 4 |- ( ( ph /\ k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < X ) -> ( F ` k ) = A )
36	4, 7, 10, 35	syl3anc 1326	. . 3 |- ( ( ( ( ph /\ j e. Z ) /\ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) /\ k e. ( ZZ>= ` j ) ) -> ( F ` k ) = A )
37	3, 36	ralrimia 39315	. 2 |- ( ( ( ph /\ j e. Z ) /\ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) -> A. k e. ( ZZ>= ` j ) ( F ` k ) = A )
38		breq2 4657	. . . . . 6 |- ( x = X -> ( ( abs ` ( ( F ` k ) - A ) ) < x <-> ( abs ` ( ( F ` k ) - A ) ) < X ) )
39	38	anbi2d 740	. . . . 5 |- ( x = X -> ( ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) )
40	39	rexralbidv 3058	. . . 4 |- ( x = X -> ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) )
41	24	simprd 479	. . . 4 |- ( ph -> A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) )
42	40, 41, 13	rspcdva 3316	. . 3 |- ( ph -> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) )
43		climisp.m	. . . 4 |- ( ph -> M e. ZZ )
44	5	rexuz3 14088	. . . 4 |- ( M e. ZZ -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) )
45	43, 44	syl 17	. . 3 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) ) )
46	42, 45	mpbird 247	. 2 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < X ) )
47	37, 46	reximddv3 39343	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( F ` k ) = A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   abscabs 13974   ~~> cli 14215

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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by: (None)