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Theorem causs 20831

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 20815	. . . . . . . . 9 $/\$
2		elfvdm 5737	. . . . . . . . . . 11
3		cnex 9384	. . . . . . . . . . 11
4		elpmg 7249	. . . . . . . . . . 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 5089	. . . . . . 7
10	8, 9	syl 16	. . . . . 6 $/\$
11		rnxpss 5291	. . . . . 6
12	10, 11	syl6ss 3389	. . . . 5 $/\$
13	12	adantlr 714	. . . 4 $/\$ $/\$
14		frn 5586	. . . . 5
15	14	ad2antlr 726	. . . 4 $/\$ $/\$
16	13, 15	ssind 3595	. . 3 $/\$ $/\$
17	16	ex 434	. 2 $/\$
18		xmetres 19961	. . . . . . . . 9
19		caufpm 20815	. . . . . . . . 9 $/\$
20	18, 19	sylan 471	. . . . . . . 8 $/\$
21		inex1g 4456	. . . . . . . . . . 11
22	2, 21	syl 16	. . . . . . . . . 10
23		elpmg 7249	. . . . . . . . . 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 5089	. . . . . 6
29	27, 28	syl 16	. . . . 5 $/\$
30		rnxpss 5291	. . . . 5
31	29, 30	syl6ss 3389	. . . 4 $/\$
32	31	ex 434	. . 3
33	32	adantr 465	. 2 $/\$
34		ffn 5580	. . . 4
35		df-f 5443	. . . . 5 $/\$
36	35	simplbi2 625	. . . 4
37	34, 36	syl 16	. . 3
38		inss2 3592	. . . . . . . . 9
39	38	a1i 11	. . . . . . . 8
40		fss 5588	. . . . . . . 8 $/\$
41	39, 40	sylan2 474	. . . . . . 7 $/\$
42	41	ancoms 453	. . . . . 6 $/\$
43		ffvelrn 5862	. . . . . . . . . . . 12 $/\$
44	43	adantr 465	. . . . . . . . . . 11 $/\$ $/\$
45		eluznn 10946	. . . . . . . . . . . . 13 $/\$
46		ffvelrn 5862	. . . . . . . . . . . . 13 $/\$
47	45, 46	sylan2 474	. . . . . . . . . . . 12 $/\$ $/\$
48	47	anassrs 648	. . . . . . . . . . 11 $/\$ $/\$
49	44, 48	ovresd 6252	. . . . . . . . . 10 $/\$ $/\$
50	49	breq1d 4323	. . . . . . . . 9 $/\$ $/\$
51	50	ralbidva 2752	. . . . . . . 8 $/\$
52	51	rexbidva 2753	. . . . . . 7
53	52	ralbidv 2756	. . . . . 6
54	42, 53	syl 16	. . . . 5 $/\$
55		nnuz 10917	. . . . . 6
56	18	adantr 465	. . . . . 6 $/\$
57		1zzd 10698	. . . . . 6 $/\$
58		eqidd 2444	. . . . . 6 $/\$ $/\$
59		eqidd 2444	. . . . . 6 $/\$ $/\$
60		simpr 461	. . . . . 6 $/\$
61	55, 56, 57, 58, 59, 60	iscauf 20813	. . . . 5 $/\$
62		simpl 457	. . . . . 6 $/\$
63		id 22	. . . . . . 7
64		inss1 3591	. . . . . . . 8
65	64	a1i 11	. . . . . . 7
66		fss 5588	. . . . . . 7 $/\$
67	63, 65, 66	syl2anr 478	. . . . . 6 $/\$
68	55, 62, 57, 58, 59, 67	iscauf 20813	. . . . 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 1756

wral 2736

wrex 2737

cvv 2993

cin 3348

wss 3349 class class class wbr 4313

cxp 4859

cdm 4861

crn 4862

cres 4863

wfun 5433

wfn 5434

wf 5435

cfv 5439 (class class class)co 6112

cpm 7236

cc 9301

c1 9304

clt 9439

cn 10343

cuz 10882

crp 11012

cxmt 17823

cca 20786

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1591 ax-4 1602 ax-5 1670 ax-6 1708 ax-7 1728 ax-8 1758 ax-9 1760 ax-10 1775 ax-11 1780 ax-12 1792 ax-13 1943 ax-ext 2423 ax-sep 4434 ax-nul 4442 ax-pow 4491 ax-pr 4552 ax-un 6393 ax-cnex 9359 ax-resscn 9360 ax-1cn 9361 ax-icn 9362 ax-addcl 9363 ax-addrcl 9364 ax-mulcl 9365 ax-mulrcl 9366 ax-mulcom 9367 ax-addass 9368 ax-mulass 9369 ax-distr 9370 ax-i2m1 9371 ax-1ne0 9372 ax-1rid 9373 ax-rnegex 9374 ax-rrecex 9375 ax-cnre 9376 ax-pre-lttri 9377 ax-pre-lttrn 9378 ax-pre-ltadd 9379 ax-pre-mulgt0 9380

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1372 df-ex 1587 df-nf 1590 df-sb 1701 df-eu 2257 df-mo 2258 df-clab 2430 df-cleq 2436 df-clel 2439 df-nfc 2577 df-ne 2622 df-nel 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 2995 df-sbc 3208 df-csb 3310 df-dif 3352 df-un 3354 df-in 3356 df-ss 3363 df-pss 3365 df-nul 3659 df-if 3813 df-pw 3883 df-sn 3899 df-pr 3901 df-tp 3903 df-op 3905 df-uni 4113 df-iun 4194 df-br 4314 df-opab 4372 df-mpt 4373 df-tr 4407 df-eprel 4653 df-id 4657 df-po 4662 df-so 4663 df-fr 4700 df-we 4702 df-ord 4743 df-on 4744 df-lim 4745 df-suc 4746 df-xp 4867 df-rel 4868 df-cnv 4869 df-co 4870 df-dm 4871 df-rn 4872 df-res 4873 df-ima 4874 df-iota 5402 df-fun 5441 df-fn 5442 df-f 5443 df-f1 5444 df-fo 5445 df-f1o 5446 df-fv 5447 df-riota 6073 df-ov 6115 df-oprab 6116 df-mpt2 6117 df-om 6498 df-1st 6598 df-2nd 6599 df-recs 6853 df-rdg 6887 df-er 7122 df-map 7237 df-pm 7238 df-en 7332 df-dom 7333 df-sdom 7334 df-pnf 9441 df-mnf 9442 df-xr 9443 df-ltxr 9444 df-le 9445 df-sub 9618 df-neg 9619 df-div 10015 df-nn 10344 df-2 10401 df-z 10668 df-uz 10883 df-rp 11013 df-xneg 11110 df-xadd 11111 df-psmet 17831 df-xmet 17832 df-bl 17834 df-cau 20789

This theorem is referenced by: minvecolem4a 24300 hhsscms 24702

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