metcnp3 - Metamath Proof Explorer

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Theorem metcnp3 22345

Description: Two ways to express that

is continuous at

for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

**Hypotheses**
Ref	Expression
metcn.2
metcn.4

**Assertion**
Ref	Expression
metcnp3	$/\$ $/\$ $/\$

Distinct variable groups:

Proof of Theorem metcnp3 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metcn.2	. . . . 5
2	1	mopntopon 22244	. . . 4 TopOn
3	2	3ad2ant1 1082	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 22243	. . . 4
6	5	3ad2ant2 1083	. . 3 $/\$ $/\$
7	4	mopntopon 22244	. . . 4 TopOn
8	7	3ad2ant2 1083	. . 3 $/\$ $/\$ TopOn
9		simp3 1063	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 21057	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1101	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 792	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1102	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelrnd 6360	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 477	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 22218	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 11840	. . . . . . . . . 10
19	18	adantl 482	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 22222	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2690	. . . . . . . . . 10
23		sseq2 3627	. . . . . . . . . . . 12
24	23	anbi2d 740	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 3052	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 334	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 3305	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 44	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 1064	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 801	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 477	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 22298	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3610	. . . . . . . . . . . 12
37		imass2 5501	. . . . . . . . . . . 12
38	36, 37	syl11 33	. . . . . . . . . . 11
39	38	reximdv 3016	. . . . . . . . . 10
40	35, 39	syl5com 31	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 629	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 643	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 3030	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 47	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2969	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 1065	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 22230	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1268	. . . . . . . . 9 $/\$
49	46, 48	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 3073	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 766	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 762	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 11840	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 764	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 22305	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1326	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 794	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 22218	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1326	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3611	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 782	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 469	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2690	. . . . . . . . . . . . . . . . 17
64		imaeq2 5462	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3632	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 747	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 3309	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1324	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 643	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 3031	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 629	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 3031	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
73	50, 72	syl5 34	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73	expd 452	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
75	49, 74	syld 47	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	com23 86	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	exp4a 633	. . . . 5 $/\$ $/\$ $/\$ $/\$
78	77	ralrimdv 2968	. . . 4 $/\$ $/\$ $/\$ $/\$
79	45, 78	impbid 202	. . 3 $/\$ $/\$ $/\$ $/\$
80	79	pm5.32da 673	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	10, 80	bitrd 268	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

wral 2912

wrex 2913

wss 3574

crn 5115

cima 5117

wf 5884

cfv 5888 (class class class)co 6650

cxr 10073

crp 11832

ctg 16098

cxmt 19731

cbl 19733

cmopn 19736 TopOnctopon 20715

ccnp 21029

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 ax-pre-sup 10014

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-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 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-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cnp 21032

This theorem is referenced by: metcnp 22346

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