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Theorem metcnp3 15179

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 15111	. . . 4 TopOn
3	2	3ad2ant1 1042	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 15110	. . . 4
6	5	3ad2ant2 1043	. . 3 $/\$ $/\$
7	4	mopntopon 15111	. . . 4 TopOn
8	7	3ad2ant2 1043	. . 3 $/\$ $/\$ TopOn
9		simp3 1023	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 14877	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1061	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1062	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelcdmd 5770	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 15084	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1271	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 9853	. . . . . . . . . 10
19	18	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 15088	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1271	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2293	. . . . . . . . . 10
23		sseq2 3248	. . . . . . . . . . . 12
24	23	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 2531	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 234	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 2903	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 1024	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 15151	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1271	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3231	. . . . . . . . . . . 12
37		imass2 5103	. . . . . . . . . . . 12
38	36, 37	syl11 31	. . . . . . . . . . 11
39	38	reximdv 2631	. . . . . . . . . 10
40	35, 39	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 363	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 375	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 2647	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 45	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2610	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 1025	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 15096	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1230	. . . . . . . . 9 $/\$
49	46, 48	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 2669	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 9853	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 490	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 15158	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1271	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 529	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 15084	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1271	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3232	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 508	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 268	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2293	. . . . . . . . . . . . . . . . 17
64		imaeq2 5063	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3253	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 473	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 2907	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1269	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 375	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 2648	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 363	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 2648	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
73	50, 72	syl5 32	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73	expd 258	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
75	49, 74	syld 45	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	com23 78	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	exp4a 366	. . . . 5 $/\$ $/\$ $/\$ $/\$
78	77	ralrimdv 2609	. . . 4 $/\$ $/\$ $/\$ $/\$
79	45, 78	impbid 129	. . 3 $/\$ $/\$ $/\$ $/\$
80	79	pm5.32da 452	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	10, 80	bitrd 188	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1002

wceq 1395

wcel 2200

wral 2508

wrex 2509

wss 3197

crn 4719

cima 4721

wf 5313

cfv 5317 (class class class)co 6000

cxr 8176

crp 9845

ctg 13282

cxmet 14494

cbl 14496

cmopn 14499 TopOnctopon 14678

ccnp 14854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-map 6795 df-sup 7147 df-inf 7148 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-xneg 9964 df-xadd 9965 df-seqfrec 10665 df-exp 10756 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-topgen 13288 df-psmet 14501 df-xmet 14502 df-bl 14504 df-mopn 14505 df-top 14666 df-topon 14679 df-bases 14711 df-cnp 14857

This theorem is referenced by: metcnp 15180

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