climuni - Metamath Proof Explorer

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Theorem climuni 14283

Description: An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

**Assertion**
Ref	Expression
climuni	$/\$

Proof of Theorem climuni Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1z 11407	. 2
2		nnuz 11723	. . . . . . 7
3		1zzd 11408	. . . . . . 7 $/\$ $/\$
4		climcl 14230	. . . . . . . . . . 11
5	4	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$
6		climcl 14230	. . . . . . . . . . 11
7	6	3ad2ant2 1083	. . . . . . . . . 10 $/\$ $/\$
8	5, 7	subcld 10392	. . . . . . . . 9 $/\$ $/\$
9		simp3 1063	. . . . . . . . . 10 $/\$ $/\$
10	5, 7, 9	subne0d 10401	. . . . . . . . 9 $/\$ $/\$
11	8, 10	absrpcld 14187	. . . . . . . 8 $/\$ $/\$
12	11	rphalfcld 11884	. . . . . . 7 $/\$ $/\$
13		eqidd 2623	. . . . . . 7 $/\$ $/\$ $/\$
14		simp1 1061	. . . . . . 7 $/\$ $/\$
15	2, 3, 12, 13, 14	climi 14241	. . . . . 6 $/\$ $/\$ $/\$
16		simp2 1062	. . . . . . 7 $/\$ $/\$
17	2, 3, 12, 13, 16	climi 14241	. . . . . 6 $/\$ $/\$ $/\$
18	2	rexanuz2 14089	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	15, 17, 18	sylanbrc 698	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnz 11399	. . . . . . . . 9
21		uzid 11702	. . . . . . . . 9
22		ne0i 3921	. . . . . . . . 9
23		r19.2z 4060	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	23	ex 450	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	20, 21, 22, 24	4syl 19	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26		simpr 477	. . . . . . . . . . . . . . 15 $/\$ $/\$
27		simpll 790	. . . . . . . . . . . . . . 15 $/\$ $/\$
28	26, 27	abssubd 14192	. . . . . . . . . . . . . 14 $/\$ $/\$
29	28	breq1d 4663	. . . . . . . . . . . . 13 $/\$ $/\$
30		simplr 792	. . . . . . . . . . . . . . . 16 $/\$ $/\$
31		subcl 10280	. . . . . . . . . . . . . . . . . 18 $/\$
32	31	adantr 481	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
33	32	abscld 14175	. . . . . . . . . . . . . . . 16 $/\$ $/\$
34		abs3lem 14078	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
35	27, 30, 26, 33, 34	syl22anc 1327	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
36	33	ltnrd 10171	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37	36	pm2.21d 118	. . . . . . . . . . . . . . 15 $/\$ $/\$
38	35, 37	syld 47	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
39	38	expd 452	. . . . . . . . . . . . 13 $/\$ $/\$
40	29, 39	sylbid 230	. . . . . . . . . . . 12 $/\$ $/\$
41	40	impr 649	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42	41	adantld 483	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	42	expimpd 629	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44	43	rexlimdvw 3034	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
45	25, 44	sylan9r 690	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	rexlimdva 3031	. . . . . 6 $/\$ $/\$ $/\$ $/\$
47	5, 7, 46	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
48	19, 47	mpd 15	. . . 4 $/\$ $/\$
49	48	3expia 1267	. . 3 $/\$
50	49	necon4ad 2813	. 2 $/\$
51	1, 50	mpi 20	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

c0 3915 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

c1 9937

clt 10074

cmin 10266

cdiv 10684

cn 11020

c2 11070

cz 11377

cuz 11687

cabs 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: fclim 14284 climeu 14286 summolem2 14447 summo 14448 prodmolem2 14665 prodmo 14666 ef0 14821 efcj 14822 efaddlem 14823 ioombl1lem4 23329 mbflimlem 23434 itg2i1fseq 23522 itg2addlem 23525 plyeq0lem 23966 ulmuni 24146 leibpi 24669 lgamp1 24783 lgam1 24790 sumnnodd 39862 climfveq 39901 climfveqf 39912 climfv 39923 climlimsupcex 40001 climliminflimsupd 40033 stirlinglem15 40305 fouriersw 40448 sge0isum 40644 vonioolem2 40895 vonicclem2 40898

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