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Theorem causs 21863

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 21847	. . . . . . . . 9 $/\$
2		elfvdm 5898	. . . . . . . . . . 11
3		cnex 9590	. . . . . . . . . . 11
4		elpmg 7453	. . . . . . . . . . 11 $/\$ $/\$
5	2, 3, 4	sylancl 662	. . . . . . . . . 10 $/\$
6	5	biimpa 484	. . . . . . . . 9 $/\$ $/\$
7	1, 6	syldan 470	. . . . . . . 8 $/\$ $/\$
8	7	simprd 463	. . . . . . 7 $/\$
9		rnss 5241	. . . . . . 7
10	8, 9	syl 16	. . . . . 6 $/\$
11		rnxpss 5446	. . . . . 6
12	10, 11	syl6ss 3511	. . . . 5 $/\$
13	12	adantlr 714	. . . 4 $/\$ $/\$
14		frn 5743	. . . . 5
15	14	ad2antlr 726	. . . 4 $/\$ $/\$
16	13, 15	ssind 3718	. . 3 $/\$ $/\$
17	16	ex 434	. 2 $/\$
18		xmetres 20993	. . . . . . . . 9
19		caufpm 21847	. . . . . . . . 9 $/\$
20	18, 19	sylan 471	. . . . . . . 8 $/\$
21		inex1g 4599	. . . . . . . . . . 11
22	2, 21	syl 16	. . . . . . . . . 10
23		elpmg 7453	. . . . . . . . . 10 $/\$ $/\$
24	22, 3, 23	sylancl 662	. . . . . . . . 9 $/\$
25	24	biimpa 484	. . . . . . . 8 $/\$ $/\$
26	20, 25	syldan 470	. . . . . . 7 $/\$ $/\$
27	26	simprd 463	. . . . . 6 $/\$
28		rnss 5241	. . . . . 6
29	27, 28	syl 16	. . . . 5 $/\$
30		rnxpss 5446	. . . . 5
31	29, 30	syl6ss 3511	. . . 4 $/\$
32	31	ex 434	. . 3
33	32	adantr 465	. 2 $/\$
34		ffn 5737	. . . 4
35		df-f 5598	. . . . 5 $/\$
36	35	simplbi2 625	. . . 4
37	34, 36	syl 16	. . 3
38		inss2 3715	. . . . . . . . 9
39	38	a1i 11	. . . . . . . 8
40		fss 5745	. . . . . . . 8 $/\$
41	39, 40	sylan2 474	. . . . . . 7 $/\$
42	41	ancoms 453	. . . . . 6 $/\$
43		ffvelrn 6030	. . . . . . . . . . . 12 $/\$
44	43	adantr 465	. . . . . . . . . . 11 $/\$ $/\$
45		eluznn 11177	. . . . . . . . . . . . 13 $/\$
46		ffvelrn 6030	. . . . . . . . . . . . 13 $/\$
47	45, 46	sylan2 474	. . . . . . . . . . . 12 $/\$ $/\$
48	47	anassrs 648	. . . . . . . . . . 11 $/\$ $/\$
49	44, 48	ovresd 6442	. . . . . . . . . 10 $/\$ $/\$
50	49	breq1d 4466	. . . . . . . . 9 $/\$ $/\$
51	50	ralbidva 2893	. . . . . . . 8 $/\$
52	51	rexbidva 2965	. . . . . . 7
53	52	ralbidv 2896	. . . . . 6
54	42, 53	syl 16	. . . . 5 $/\$
55		nnuz 11141	. . . . . 6
56	18	adantr 465	. . . . . 6 $/\$
57		1zzd 10916	. . . . . 6 $/\$
58		eqidd 2458	. . . . . 6 $/\$ $/\$
59		eqidd 2458	. . . . . 6 $/\$ $/\$
60		simpr 461	. . . . . 6 $/\$
61	55, 56, 57, 58, 59, 60	iscauf 21845	. . . . 5 $/\$
62		simpl 457	. . . . . 6 $/\$
63		id 22	. . . . . . 7
64		inss1 3714	. . . . . . . 8
65	64	a1i 11	. . . . . . 7
66		fss 5745	. . . . . . 7 $/\$
67	63, 65, 66	syl2anr 478	. . . . . 6 $/\$
68	55, 62, 57, 58, 59, 67	iscauf 21845	. . . . 5 $/\$
69	54, 61, 68	3bitr4rd 286	. . . 4 $/\$
70	69	ex 434	. . 3
71	37, 70	sylan9r 658	. 2 $/\$
72	17, 33, 71	pm5.21ndd 354	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 184 $/\$ wa 369

wcel 1819

wral 2807

wrex 2808

cvv 3109

cin 3470

wss 3471 class class class wbr 4456

cxp 5006

cdm 5008

crn 5009

cres 5010

wfun 5588

wfn 5589

wf 5590

cfv 5594 (class class class)co 6296

cpm 7439

cc 9507

c1 9510

clt 9645

cn 10556

cuz 11106

crp 11245

cxmt 18530

cca 21818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1619 ax-4 1632 ax-5 1705 ax-6 1748 ax-7 1791 ax-8 1821 ax-9 1823 ax-10 1838 ax-11 1843 ax-12 1855 ax-13 2000 ax-ext 2435 ax-sep 4578 ax-nul 4586 ax-pow 4634 ax-pr 4695 ax-un 6591 ax-cnex 9565 ax-resscn 9566 ax-1cn 9567 ax-icn 9568 ax-addcl 9569 ax-addrcl 9570 ax-mulcl 9571 ax-mulrcl 9572 ax-mulcom 9573 ax-addass 9574 ax-mulass 9575 ax-distr 9576 ax-i2m1 9577 ax-1ne0 9578 ax-1rid 9579 ax-rnegex 9580 ax-rrecex 9581 ax-cnre 9582 ax-pre-lttri 9583 ax-pre-lttrn 9584 ax-pre-ltadd 9585 ax-pre-mulgt0 9586

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1614 df-nf 1618 df-sb 1741 df-eu 2287 df-mo 2288 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3431 df-dif 3474 df-un 3476 df-in 3478 df-ss 3485 df-pss 3487 df-nul 3794 df-if 3945 df-pw 4017 df-sn 4033 df-pr 4035 df-tp 4037 df-op 4039 df-uni 4252 df-iun 4334 df-br 4457 df-opab 4516 df-mpt 4517 df-tr 4551 df-eprel 4800 df-id 4804 df-po 4809 df-so 4810 df-fr 4847 df-we 4849 df-ord 4890 df-on 4891 df-lim 4892 df-suc 4893 df-xp 5014 df-rel 5015 df-cnv 5016 df-co 5017 df-dm 5018 df-rn 5019 df-res 5020 df-ima 5021 df-iota 5557 df-fun 5596 df-fn 5597 df-f 5598 df-f1 5599 df-fo 5600 df-f1o 5601 df-fv 5602 df-riota 6258 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6700 df-1st 6799 df-2nd 6800 df-recs 7060 df-rdg 7094 df-er 7329 df-map 7440 df-pm 7441 df-en 7536 df-dom 7537 df-sdom 7538 df-pnf 9647 df-mnf 9648 df-xr 9649 df-ltxr 9650 df-le 9651 df-sub 9826 df-neg 9827 df-div 10228 df-nn 10557 df-2 10615 df-z 10886 df-uz 11107 df-rp 11246 df-xneg 11343 df-xadd 11344 df-psmet 18538 df-xmet 18539 df-bl 18541 df-cau 21821

This theorem is referenced by: minvecolem4a 25920 hhsscms 26322

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