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Theorem causs 23096

Description: Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

**Assertion**
Ref	Expression
causs	$/\$

Proof of Theorem causs Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caufpm 23080	. . . . . . . . 9 $/\$
2		elfvdm 6220	. . . . . . . . . . 11
3		cnex 10017	. . . . . . . . . . 11
4		elpmg 7873	. . . . . . . . . . 11 $/\$ $/\$
5	2, 3, 4	sylancl 694	. . . . . . . . . 10 $/\$
6	5	biimpa 501	. . . . . . . . 9 $/\$ $/\$
7	1, 6	syldan 487	. . . . . . . 8 $/\$ $/\$
8	7	simprd 479	. . . . . . 7 $/\$
9		rnss 5354	. . . . . . 7
10	8, 9	syl 17	. . . . . 6 $/\$
11		rnxpss 5566	. . . . . 6
12	10, 11	syl6ss 3615	. . . . 5 $/\$
13	12	adantlr 751	. . . 4 $/\$ $/\$
14		frn 6053	. . . . 5
15	14	ad2antlr 763	. . . 4 $/\$ $/\$
16	13, 15	ssind 3837	. . 3 $/\$ $/\$
17	16	ex 450	. 2 $/\$
18		xmetres 22169	. . . . . . . . 9
19		caufpm 23080	. . . . . . . . 9 $/\$
20	18, 19	sylan 488	. . . . . . . 8 $/\$
21		inex1g 4801	. . . . . . . . . . 11
22	2, 21	syl 17	. . . . . . . . . 10
23		elpmg 7873	. . . . . . . . . 10 $/\$ $/\$
24	22, 3, 23	sylancl 694	. . . . . . . . 9 $/\$
25	24	biimpa 501	. . . . . . . 8 $/\$ $/\$
26	20, 25	syldan 487	. . . . . . 7 $/\$ $/\$
27	26	simprd 479	. . . . . 6 $/\$
28		rnss 5354	. . . . . 6
29	27, 28	syl 17	. . . . 5 $/\$
30		rnxpss 5566	. . . . 5
31	29, 30	syl6ss 3615	. . . 4 $/\$
32	31	ex 450	. . 3
33	32	adantr 481	. 2 $/\$
34		ffn 6045	. . . 4
35		df-f 5892	. . . . 5 $/\$
36	35	simplbi2 655	. . . 4
37	34, 36	syl 17	. . 3
38		inss2 3834	. . . . . . . . 9
39	38	a1i 11	. . . . . . . 8
40		fss 6056	. . . . . . . 8 $/\$
41	39, 40	sylan2 491	. . . . . . 7 $/\$
42	41	ancoms 469	. . . . . 6 $/\$
43		ffvelrn 6357	. . . . . . . . . . . 12 $/\$
44	43	adantr 481	. . . . . . . . . . 11 $/\$ $/\$
45		eluznn 11758	. . . . . . . . . . . . 13 $/\$
46		ffvelrn 6357	. . . . . . . . . . . . 13 $/\$
47	45, 46	sylan2 491	. . . . . . . . . . . 12 $/\$ $/\$
48	47	anassrs 680	. . . . . . . . . . 11 $/\$ $/\$
49	44, 48	ovresd 6801	. . . . . . . . . 10 $/\$ $/\$
50	49	breq1d 4663	. . . . . . . . 9 $/\$ $/\$
51	50	ralbidva 2985	. . . . . . . 8 $/\$
52	51	rexbidva 3049	. . . . . . 7
53	52	ralbidv 2986	. . . . . 6
54	42, 53	syl 17	. . . . 5 $/\$
55		nnuz 11723	. . . . . 6
56	18	adantr 481	. . . . . 6 $/\$
57		1zzd 11408	. . . . . 6 $/\$
58		eqidd 2623	. . . . . 6 $/\$ $/\$
59		eqidd 2623	. . . . . 6 $/\$ $/\$
60		simpr 477	. . . . . 6 $/\$
61	55, 56, 57, 58, 59, 60	iscauf 23078	. . . . 5 $/\$
62		simpl 473	. . . . . 6 $/\$
63		id 22	. . . . . . 7
64		inss1 3833	. . . . . . . 8
65	64	a1i 11	. . . . . . 7
66		fss 6056	. . . . . . 7 $/\$
67	63, 65, 66	syl2anr 495	. . . . . 6 $/\$
68	55, 62, 57, 58, 59, 67	iscauf 23078	. . . . 5 $/\$
69	54, 61, 68	3bitr4rd 301	. . . 4 $/\$
70	69	ex 450	. . 3
71	37, 70	sylan9r 690	. 2 $/\$
72	17, 33, 71	pm5.21ndd 369	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wcel 1990

wral 2912

wrex 2913

cvv 3200

cin 3573

wss 3574 class class class wbr 4653

cxp 5112

cdm 5114

crn 5115

cres 5116

wfun 5882

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cpm 7858

cc 9934

c1 9937

clt 10074

cn 11020

cuz 11687

crp 11832

cxmt 19731

cca 23051

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

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-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 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-z 11378 df-uz 11688 df-rp 11833 df-xneg 11946 df-xadd 11947 df-psmet 19738 df-xmet 19739 df-bl 19741 df-cau 23054

This theorem is referenced by: minvecolem4a 27733 hhsscms 28136

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