metcnp3 - Intuitionistic Logic Explorer

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Theorem metcnp3 12680

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 12612	. . . 4 TopOn
3	2	3ad2ant1 1002	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 12611	. . . 4
6	5	3ad2ant2 1003	. . 3 $/\$ $/\$
7	4	mopntopon 12612	. . . 4 TopOn
8	7	3ad2ant2 1003	. . 3 $/\$ $/\$ TopOn
9		simp3 983	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 12378	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1021	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 519	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1022	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelrnd 5556	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 109	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 12585	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1216	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 9449	. . . . . . . . . 10
19	18	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 12589	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1216	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2203	. . . . . . . . . 10
23		sseq2 3121	. . . . . . . . . . . 12
24	23	anbi2d 459	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 2438	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 233	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 2785	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 984	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 479	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 525	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 12652	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1216	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3104	. . . . . . . . . . . 12
37		imass2 4915	. . . . . . . . . . . 12
38	36, 37	syl11 31	. . . . . . . . . . 11
39	38	reximdv 2533	. . . . . . . . . 10
40	35, 39	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 360	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 372	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 2548	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 45	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2512	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 985	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 12597	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1184	. . . . . . . . 9 $/\$
49	46, 48	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 2570	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 483	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 479	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 9449	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 481	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 12659	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1216	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 520	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 12585	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1216	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3105	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 499	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 266	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2203	. . . . . . . . . . . . . . . . 17
64		imaeq2 4877	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3126	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 464	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 2789	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1214	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 372	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 2549	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 360	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 2549	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
73	50, 72	syl5 32	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73	expd 256	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
75	49, 74	syld 45	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	com23 78	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	exp4a 363	. . . . 5 $/\$ $/\$ $/\$ $/\$
78	77	ralrimdv 2511	. . . 4 $/\$ $/\$ $/\$ $/\$
79	45, 78	impbid 128	. . 3 $/\$ $/\$ $/\$ $/\$
80	79	pm5.32da 447	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	10, 80	bitrd 187	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 962

wceq 1331

wcel 1480

wral 2416

wrex 2417

wss 3071

crn 4540

cima 4542

wf 5119

cfv 5123 (class class class)co 5774

cxr 7799

crp 9441

ctg 12135

cxmet 12149

cbl 12151

cmopn 12154 TopOnctopon 12177

ccnp 12355

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-topgen 12141 df-psmet 12156 df-xmet 12157 df-bl 12159 df-mopn 12160 df-top 12165 df-topon 12178 df-bases 12210 df-cnp 12358

This theorem is referenced by: metcnp 12681

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