caussi - Metamath Proof Explorer

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Theorem caussi 23095

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

**Assertion**
Ref	Expression
caussi

Proof of Theorem caussi Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . . . . . 9
2		xpss2 5229	. . . . . . . . 9
3	1, 2	ax-mp 5	. . . . . . . 8
4		sstr 3611	. . . . . . . 8 $/\$
5	3, 4	mpan2 707	. . . . . . 7
6	5	anim2i 593	. . . . . 6 $/\$ $/\$
7	6	a1i 11	. . . . 5 $/\$ $/\$
8		elfvdm 6220	. . . . . . 7
9		inex1g 4801	. . . . . . 7
10	8, 9	syl 17	. . . . . 6
11		cnex 10017	. . . . . 6
12		elpmg 7873	. . . . . 6 $/\$ $/\$
13	10, 11, 12	sylancl 694	. . . . 5 $/\$
14		elpmg 7873	. . . . . 6 $/\$ $/\$
15	8, 11, 14	sylancl 694	. . . . 5 $/\$
16	7, 13, 15	3imtr4d 283	. . . 4
17		uzid 11702	. . . . . . . . . 10
18	17	adantl 482	. . . . . . . . 9 $/\$
19		simp2 1062	. . . . . . . . . 10 $/\$ $/\$
20	19	ralimi 2952	. . . . . . . . 9 $/\$ $/\$
21		fveq2 6191	. . . . . . . . . . 11
22	21	eleq1d 2686	. . . . . . . . . 10
23	22	rspcva 3307	. . . . . . . . 9 $/\$
24	18, 20, 23	syl2an 494	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25		inss2 3834	. . . . . . . . . . . . . 14
26		simpr 477	. . . . . . . . . . . . . 14 $/\$ $/\$
27	25, 26	sseldi 3601	. . . . . . . . . . . . 13 $/\$ $/\$
28	25	a1i 11	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
29	28	sselda 3603	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
30		simplr 792	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
31	29, 30	ovresd 6801	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
32	31	breq1d 4663	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
33	32	biimpd 219	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
34	33	imdistanda 729	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
35	1	a1i 11	. . . . . . . . . . . . . . . 16 $/\$ $/\$
36	35	sseld 3602	. . . . . . . . . . . . . . 15 $/\$ $/\$
37	36	anim1d 588	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
38	34, 37	syld 47	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
39	27, 38	syldan 487	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
40	39	anim2d 589	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		3anass 1042	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
42		3anass 1042	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
43	40, 41, 42	3imtr4g 285	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	43	ralimdv 2963	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	44	impancom 456	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	24, 45	mpd 15	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	46	ex 450	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
48	47	reximdva 3017	. . . . 5 $/\$ $/\$ $/\$ $/\$
49	48	ralimdv 2963	. . . 4 $/\$ $/\$ $/\$ $/\$
50	16, 49	anim12d 586	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51		xmetres 22169	. . . 4
52		iscau2 23075	. . . 4 $/\$ $/\$ $/\$
53	51, 52	syl 17	. . 3 $/\$ $/\$ $/\$
54		iscau2 23075	. . 3 $/\$ $/\$ $/\$
55	50, 53, 54	3imtr4d 283	. 2
56	55	ssrdv 3609	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wcel 1990

wral 2912

wrex 2913

cvv 3200

cin 3573

wss 3574 class class class wbr 4653

cxp 5112

cdm 5114

cres 5116

wfun 5882

cfv 5888 (class class class)co 6650

cpm 7858

cc 9934

clt 10074

cz 11377

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

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-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 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-neg 10269 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-psmet 19738 df-xmet 19739 df-bl 19741 df-cau 23054

This theorem is referenced by: (None)

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