metcnp3 - Intuitionistic Logic Explorer

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Theorem metcnp3 14307

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 14239	. . . 4 TopOn
3	2	3ad2ant1 1019	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 14238	. . . 4
6	5	3ad2ant2 1020	. . 3 $/\$ $/\$
7	4	mopntopon 14239	. . . 4 TopOn
8	7	3ad2ant2 1020	. . 3 $/\$ $/\$ TopOn
9		simp3 1000	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 14005	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1038	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1039	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelcdmd 5665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 14212	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1248	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 9675	. . . . . . . . . 10
19	18	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 14216	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1248	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2251	. . . . . . . . . 10
23		sseq2 3191	. . . . . . . . . . . 12
24	23	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 2488	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 234	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 2849	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 1001	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 14279	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1248	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3174	. . . . . . . . . . . 12
37		imass2 5016	. . . . . . . . . . . 12
38	36, 37	syl11 31	. . . . . . . . . . 11
39	38	reximdv 2588	. . . . . . . . . 10
40	35, 39	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 363	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 375	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 2603	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 45	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2567	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 1002	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 14224	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1207	. . . . . . . . 9 $/\$
49	46, 48	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 2625	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 9675	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 490	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 14286	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1248	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 529	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 14212	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1248	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3175	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 508	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 268	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2251	. . . . . . . . . . . . . . . . 17
64		imaeq2 4978	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3196	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 473	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 2853	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1246	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 375	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 2604	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 363	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 2604	. . . . . . . . . 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 2566	. . . 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 979

wceq 1363

wcel 2158

wral 2465

wrex 2466

wss 3141

crn 4639

cima 4641

wf 5224

cfv 5228 (class class class)co 5888

cxr 8005

crp 9667

ctg 12721

cxmet 13722

cbl 13724

cmopn 13727 TopOnctopon 13806

ccnp 13982

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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943 ax-arch 7944 ax-caucvg 7945

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-frec 6406 df-map 6664 df-sup 6997 df-inf 6998 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-n0 9191 df-z 9268 df-uz 9543 df-q 9634 df-rp 9668 df-xneg 9786 df-xadd 9787 df-seqfrec 10460 df-exp 10534 df-cj 10865 df-re 10866 df-im 10867 df-rsqrt 11021 df-abs 11022 df-topgen 12727 df-psmet 13729 df-xmet 13730 df-bl 13732 df-mopn 13733 df-top 13794 df-topon 13807 df-bases 13839 df-cnp 13985

This theorem is referenced by: metcnp 14308

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