metcnp3 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > metcnp3		Unicode version

Theorem metcnp3 15376

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 15308	. . . 4 TopOn
3	2	3ad2ant1 1045	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 15307	. . . 4
6	5	3ad2ant2 1046	. . 3 $/\$ $/\$
7	4	mopntopon 15308	. . . 4 TopOn
8	7	3ad2ant2 1046	. . 3 $/\$ $/\$ TopOn
9		simp3 1026	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 15074	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1064	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1065	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelcdmd 5813	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 15281	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 9994	. . . . . . . . . 10
19	18	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 15285	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2296	. . . . . . . . . 10
23		sseq2 3262	. . . . . . . . . . . 12
24	23	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 2543	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 234	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 2917	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 1027	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 538	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 15348	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3245	. . . . . . . . . . . 12
37		imass2 5138	. . . . . . . . . . . 12
38	36, 37	syl11 31	. . . . . . . . . . 11
39	38	reximdv 2643	. . . . . . . . . 10
40	35, 39	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 363	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 375	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 2659	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 45	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2622	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 1028	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 15293	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1233	. . . . . . . . 9 $/\$
49	46, 48	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 2681	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 9994	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 490	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 15355	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 531	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 15281	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3246	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 508	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 268	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2296	. . . . . . . . . . . . . . . . 17
64		imaeq2 5097	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3267	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 473	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 2921	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1272	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 375	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 2660	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 363	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 2660	. . . . . . . . . 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 2621	. . . 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 1005

wceq 1398

wcel 2203

wral 2520

wrex 2521

wss 3211

crn 4750

cima 4752

wf 5348

cfv 5352 (class class class)co 6050

cxr 8307

crp 9986

ctg 13467

cxmet 14684

cbl 14686

cmopn 14689 TopOnctopon 14875

ccnp 15051

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-map 6884 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-xneg 10105 df-xadd 10106 df-seqfrec 10810 df-exp 10901 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-topgen 13473 df-psmet 14691 df-xmet 14692 df-bl 14694 df-mopn 14695 df-top 14863 df-topon 14876 df-bases 14908 df-cnp 15054

This theorem is referenced by: metcnp 15377

W3C validator