iscau2 - Metamath Proof Explorer

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Theorem iscau2 23075

Description: Express the property "

is a Cauchy sequence of metric

," using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

**Assertion**
Ref	Expression
iscau2	$/\$ $/\$ $/\$

Distinct variable groups:

**Proof of Theorem iscau2**
Step	Hyp	Ref	Expression
1		iscau 23074	. 2 $/\$
2		elfvdm 6220	. . . . . . . . . 10
3		cnex 10017	. . . . . . . . . 10
4		elpmg 7873	. . . . . . . . . 10 $/\$ $/\$
5	2, 3, 4	sylancl 694	. . . . . . . . 9 $/\$
6	5	simprbda 653	. . . . . . . 8 $/\$
7		ffvresb 6394	. . . . . . . 8 $/\$
8	6, 7	syl 17	. . . . . . 7 $/\$ $/\$
9	8	rexbidv 3052	. . . . . 6 $/\$ $/\$
10	9	adantr 481	. . . . 5 $/\$ $/\$ $/\$
11		uzid 11702	. . . . . . . . . . 11
12	11	adantl 482	. . . . . . . . . 10 $/\$ $/\$
13		eleq1 2689	. . . . . . . . . . . 12
14		fveq2 6191	. . . . . . . . . . . . 13
15	14	eleq1d 2686	. . . . . . . . . . . 12
16	13, 15	anbi12d 747	. . . . . . . . . . 11 $/\$ $/\$
17	16	rspcv 3305	. . . . . . . . . 10 $/\$ $/\$
18	12, 17	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
19		n0i 3920	. . . . . . . . . . . 12
20		blf 22212	. . . . . . . . . . . . . . 15
21		fdm 6051	. . . . . . . . . . . . . . 15
22	20, 21	syl 17	. . . . . . . . . . . . . 14
23		ndmovg 6817	. . . . . . . . . . . . . . 15 $/\$ $/\$
24	23	ex 450	. . . . . . . . . . . . . 14 $/\$
25	22, 24	syl 17	. . . . . . . . . . . . 13 $/\$
26	25	con1d 139	. . . . . . . . . . . 12 $/\$
27		simpl 473	. . . . . . . . . . . 12 $/\$
28	19, 26, 27	syl56 36	. . . . . . . . . . 11
29	28	adantld 483	. . . . . . . . . 10 $/\$
30	29	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$
31	18, 30	syld 47	. . . . . . . 8 $/\$ $/\$ $/\$
32	14	eleq1d 2686	. . . . . . . . . . . 12
33	14	oveq1d 6665	. . . . . . . . . . . . 13
34	33	breq1d 4663	. . . . . . . . . . . 12
35	13, 32, 34	3anbi123d 1399	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36	35	rspcv 3305	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
37	12, 36	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp2 1062	. . . . . . . . 9 $/\$ $/\$
39	37, 38	syl6 35	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40		rpxr 11840	. . . . . . . . . . . . . . . 16
41		elbl 22193	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
42	40, 41	syl3an3 1361	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
43		xmetsym 22152	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
44	43	3expa 1265	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
45	44	3adantl3 1219	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
46	45	breq1d 4663	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
47	46	pm5.32da 673	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
48	42, 47	bitrd 268	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
49	48	3com23 1271	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
50	49	anbi2d 740	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
51		3anass 1042	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
52	50, 51	syl6bbr 278	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
53	52	ralbidv 2986	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
54	53	3expia 1267	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
55	54	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
56	31, 39, 55	pm5.21ndd 369	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
57	56	rexbidva 3049	. . . . . 6 $/\$ $/\$ $/\$ $/\$
58	57	adantlr 751	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
59	10, 58	bitrd 268	. . . 4 $/\$ $/\$ $/\$ $/\$
60	59	ralbidva 2985	. . 3 $/\$ $/\$ $/\$
61	60	pm5.32da 673	. 2 $/\$ $/\$ $/\$ $/\$
62	1, 61	bitrd 268	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1483

wcel 1990

wral 2912

wrex 2913

cvv 3200

wss 3574

c0 3915

cpw 4158 class class class wbr 4653

cxp 5112

cdm 5114

cres 5116

wfun 5882

wf 5884

cfv 5888 (class class class)co 6650

cpm 7858

cc 9934

cxr 10073

clt 10074

cz 11377

cuz 11687

crp 11832

cxmt 19731

cbl 19733

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: iscau3 23076 iscau4 23077 caun0 23079 caussi 23095

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