Theorem climrescn 39980

Description: A sequence converging w.r.t. the standard topology on the complex numbers, eventually becomes a sequence of complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
climrescn.m	|- ( ph -> M e. ZZ )
climrescn.z	|- Z = ( ZZ>= ` M )
climrescn.f	|- ( ph -> F Fn Z )
climrescn.c	|- ( ph -> F e. dom ~~> )

Assertion

Ref	Expression
climrescn	|- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> CC )

Distinct variable groups:   j, F   j, Z

Allowed substitution hints:   ph( j)   M( j)

Proof of Theorem climrescn

Dummy variables i k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . . 6 |- F/ k ( ph /\ i e. Z )
2		nfra1 2941	. . . . . 6 |- F/ k A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 )
3	1, 2	nfan 1828	. . . . 5 |- F/ k ( ( ph /\ i e. Z ) /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) )
4		climrescn.z	. . . . . . . . . 10 |- Z = ( ZZ>= ` M )
5	4	uztrn2 11705	. . . . . . . . 9 |- ( ( i e. Z /\ k e. ( ZZ>= ` i ) ) -> k e. Z )
6	5	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ i e. Z ) /\ k e. ( ZZ>= ` i ) ) -> k e. Z )
7		climrescn.f	. . . . . . . . . 10 |- ( ph -> F Fn Z )
8	7	fndmd 39441	. . . . . . . . 9 |- ( ph -> dom F = Z )
9	8	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ i e. Z ) /\ k e. ( ZZ>= ` i ) ) -> dom F = Z )
10	6, 9	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ph /\ i e. Z ) /\ k e. ( ZZ>= ` i ) ) -> k e. dom F )
11	10	adantlr 751	. . . . . 6 |- ( ( ( ( ph /\ i e. Z ) /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) /\ k e. ( ZZ>= ` i ) ) -> k e. dom F )
12		rspa 2930	. . . . . . . . 9 |- ( ( A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) /\ k e. ( ZZ>= ` i ) ) -> ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) )
13	12	adantll 750	. . . . . . . 8 |- ( ( ( i e. Z /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) /\ k e. ( ZZ>= ` i ) ) -> ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) )
14	13	simpld 475	. . . . . . 7 |- ( ( ( i e. Z /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) /\ k e. ( ZZ>= ` i ) ) -> ( F ` k ) e. CC )
15	14	adantlll 754	. . . . . 6 |- ( ( ( ( ph /\ i e. Z ) /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) /\ k e. ( ZZ>= ` i ) ) -> ( F ` k ) e. CC )
16	11, 15	jca 554	. . . . 5 |- ( ( ( ( ph /\ i e. Z ) /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) /\ k e. ( ZZ>= ` i ) ) -> ( k e. dom F /\ ( F ` k ) e. CC ) )
17	3, 16	ralrimia 39315	. . . 4 |- ( ( ( ph /\ i e. Z ) /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) -> A. k e. ( ZZ>= ` i ) ( k e. dom F /\ ( F ` k ) e. CC ) )
18		fnfun 5988	. . . . . 6 |- ( F Fn Z -> Fun F )
19		ffvresb 6394	. . . . . 6 |- ( Fun F -> ( ( F |` ( ZZ>= ` i ) ) : ( ZZ>= ` i ) --> CC <-> A. k e. ( ZZ>= ` i ) ( k e. dom F /\ ( F ` k ) e. CC ) ) )
20	7, 18, 19	3syl 18	. . . . 5 |- ( ph -> ( ( F |` ( ZZ>= ` i ) ) : ( ZZ>= ` i ) --> CC <-> A. k e. ( ZZ>= ` i ) ( k e. dom F /\ ( F ` k ) e. CC ) ) )
21	20	ad2antrr 762	. . . 4 |- ( ( ( ph /\ i e. Z ) /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) -> ( ( F |` ( ZZ>= ` i ) ) : ( ZZ>= ` i ) --> CC <-> A. k e. ( ZZ>= ` i ) ( k e. dom F /\ ( F ` k ) e. CC ) ) )
22	17, 21	mpbird 247	. . 3 |- ( ( ( ph /\ i e. Z ) /\ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) -> ( F |` ( ZZ>= ` i ) ) : ( ZZ>= ` i ) --> CC )
23		breq2 4657	. . . . . . 7 |- ( x = 1 -> ( ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < x <-> ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) )
24	23	anbi2d 740	. . . . . 6 |- ( x = 1 -> ( ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < x ) <-> ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) )
25	24	rexralbidv 3058	. . . . 5 |- ( x = 1 -> ( E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < x ) <-> E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) )
26		climrescn.c	. . . . . . . 8 |- ( ph -> F e. dom ~~> )
27		climdm 14285	. . . . . . . 8 |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
28	26, 27	sylib 208	. . . . . . 7 |- ( ph -> F ~~> ( ~~> ` F ) )
29		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
30	26, 29	clim 14225	. . . . . . 7 |- ( ph -> ( F ~~> ( ~~> ` F ) <-> ( ( ~~> ` F ) e. CC /\ A. x e. RR+ E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < x ) ) ) )
31	28, 30	mpbid 222	. . . . . 6 |- ( ph -> ( ( ~~> ` F ) e. CC /\ A. x e. RR+ E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < x ) ) )
32	31	simprd 479	. . . . 5 |- ( ph -> A. x e. RR+ E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < x ) )
33		1rp 11836	. . . . . 6 |- 1 e. RR+
34	33	a1i 11	. . . . 5 |- ( ph -> 1 e. RR+ )
35	25, 32, 34	rspcdva 3316	. . . 4 |- ( ph -> E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) )
36		climrescn.m	. . . . 5 |- ( ph -> M e. ZZ )
37	4	rexuz3 14088	. . . . 5 |- ( M e. ZZ -> ( E. i e. Z A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) <-> E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) )
38	36, 37	syl 17	. . . 4 |- ( ph -> ( E. i e. Z A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) <-> E. i e. ZZ A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) ) )
39	35, 38	mpbird 247	. . 3 |- ( ph -> E. i e. Z A. k e. ( ZZ>= ` i ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( ~~> ` F ) ) ) < 1 ) )
40	22, 39	reximddv3 39343	. 2 |- ( ph -> E. i e. Z ( F |` ( ZZ>= ` i ) ) : ( ZZ>= ` i ) --> CC )
41		fveq2 6191	. . . . 5 |- ( j = i -> ( ZZ>= ` j ) = ( ZZ>= ` i ) )
42	41	reseq2d 5396	. . . 4 |- ( j = i -> ( F |` ( ZZ>= ` j ) ) = ( F |` ( ZZ>= ` i ) ) )
43	42, 41	feq12d 6033	. . 3 |- ( j = i -> ( ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> CC <-> ( F |` ( ZZ>= ` i ) ) : ( ZZ>= ` i ) --> CC ) )
44	43	cbvrexv 3172	. 2 |- ( E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> CC <-> E. i e. Z ( F |` ( ZZ>= ` i ) ) : ( ZZ>= ` i ) --> CC )
45	40, 44	sylibr 224	1 |- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> CC )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   < clt 10074   - 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-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:  climxlim2  40072