metcnp3 - Intuitionistic Logic Explorer

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Theorem metcnp3 15098

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 15030	. . . 4 TopOn
3	2	3ad2ant1 1021	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 15029	. . . 4
6	5	3ad2ant2 1022	. . 3 $/\$ $/\$
7	4	mopntopon 15030	. . . 4 TopOn
8	7	3ad2ant2 1022	. . 3 $/\$ $/\$ TopOn
9		simp3 1002	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 14796	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1040	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1041	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelcdmd 5739	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 15003	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 9818	. . . . . . . . . 10
19	18	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 15007	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1250	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2271	. . . . . . . . . 10
23		sseq2 3225	. . . . . . . . . . . 12
24	23	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 2509	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 234	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 2880	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 1003	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 15070	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1250	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3208	. . . . . . . . . . . 12
37		imass2 5077	. . . . . . . . . . . 12
38	36, 37	syl11 31	. . . . . . . . . . 11
39	38	reximdv 2609	. . . . . . . . . 10
40	35, 39	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 363	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 375	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 2624	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 45	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2588	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 1004	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 15015	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1209	. . . . . . . . 9 $/\$
49	46, 48	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 2646	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 9818	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 490	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 15077	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1250	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 529	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 15003	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1250	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3209	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 508	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 268	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2271	. . . . . . . . . . . . . . . . 17
64		imaeq2 5037	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3230	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 473	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 2884	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1248	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 375	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 2625	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 363	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 2625	. . . . . . . . . 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 2587	. . . 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 981

wceq 1373

wcel 2178

wral 2486

wrex 2487

wss 3174

crn 4694

cima 4696

wf 5286

cfv 5290 (class class class)co 5967

cxr 8141

crp 9810

ctg 13201

cxmet 14413

cbl 14415

cmopn 14418 TopOnctopon 14597

ccnp 14773

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-frec 6500 df-map 6760 df-sup 7112 df-inf 7113 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-xneg 9929 df-xadd 9930 df-seqfrec 10630 df-exp 10721 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-topgen 13207 df-psmet 14420 df-xmet 14421 df-bl 14423 df-mopn 14424 df-top 14585 df-topon 14598 df-bases 14630 df-cnp 14776

This theorem is referenced by: metcnp 15099

W3C validator