metcnp3 - Intuitionistic Logic Explorer

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Theorem metcnp3 14831

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 14763	. . . 4 TopOn
3	2	3ad2ant1 1020	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 14762	. . . 4
6	5	3ad2ant2 1021	. . 3 $/\$ $/\$
7	4	mopntopon 14763	. . . 4 TopOn
8	7	3ad2ant2 1021	. . 3 $/\$ $/\$ TopOn
9		simp3 1001	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 14529	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1039	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1040	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelcdmd 5701	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 14736	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 9753	. . . . . . . . . 10
19	18	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 14740	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1249	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2260	. . . . . . . . . 10
23		sseq2 3208	. . . . . . . . . . . 12
24	23	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 2498	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 234	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 2864	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 1002	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 14803	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3191	. . . . . . . . . . . 12
37		imass2 5046	. . . . . . . . . . . 12
38	36, 37	syl11 31	. . . . . . . . . . 11
39	38	reximdv 2598	. . . . . . . . . 10
40	35, 39	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 363	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 375	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 2613	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 45	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2577	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 1003	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 14748	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1208	. . . . . . . . 9 $/\$
49	46, 48	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 2635	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 9753	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 490	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 14810	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1249	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 529	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 14736	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1249	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3192	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 508	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 268	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2260	. . . . . . . . . . . . . . . . 17
64		imaeq2 5006	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3213	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 473	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 2868	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1247	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 375	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 2614	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 363	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 2614	. . . . . . . . . 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 2576	. . . 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 980

wceq 1364

wcel 2167

wral 2475

wrex 2476

wss 3157

crn 4665

cima 4667

wf 5255

cfv 5259 (class class class)co 5925

cxr 8077

crp 9745

ctg 12956

cxmet 14168

cbl 14170

cmopn 14173 TopOnctopon 14330

ccnp 14506

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-map 6718 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-xneg 9864 df-xadd 9865 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-topgen 12962 df-psmet 14175 df-xmet 14176 df-bl 14178 df-mopn 14179 df-top 14318 df-topon 14331 df-bases 14363 df-cnp 14509

This theorem is referenced by: metcnp 14832

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