metcnp3 - Intuitionistic Logic Explorer

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Theorem metcnp3 13052

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 12984	. . . 4 TopOn
3	2	3ad2ant1 1007	. . 3 $/\$ $/\$ TopOn
4		metcn.4	. . . . 5
5	4	mopnval 12983	. . . 4
6	5	3ad2ant2 1008	. . 3 $/\$ $/\$
7	4	mopntopon 12984	. . . 4 TopOn
8	7	3ad2ant2 1008	. . 3 $/\$ $/\$ TopOn
9		simp3 988	. . 3 $/\$ $/\$
10	3, 6, 8, 9	tgcnp 12750	. 2 $/\$ $/\$ $/\$ $/\$
11		simpll2 1026	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simplr 520	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		simpll3 1027	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	12, 13	ffvelrnd 5615	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		simpr 109	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16		blcntr 12957	. . . . . . . 8 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1227	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18		rpxr 9588	. . . . . . . . . 10
19	18	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
20		blelrn 12961	. . . . . . . . 9 $/\$ $/\$
21	11, 14, 19, 20	syl3anc 1227	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		eleq2 2228	. . . . . . . . . 10
23		sseq2 3161	. . . . . . . . . . . 12
24	23	anbi2d 460	. . . . . . . . . . 11 $/\$ $/\$
25	24	rexbidv 2465	. . . . . . . . . 10 $/\$ $/\$
26	22, 25	imbi12d 233	. . . . . . . . 9 $/\$ $/\$
27	26	rspcv 2821	. . . . . . . 8 $/\$ $/\$
28	21, 27	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	17, 28	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simpl1 989	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
31	30	ad2antrr 480	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 526	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	1	mopni2 13024	. . . . . . . . . . 11 $/\$ $/\$
35	31, 32, 33, 34	syl3anc 1227	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		sstr2 3144	. . . . . . . . . . . 12
37		imass2 4974	. . . . . . . . . . . 12
38	36, 37	syl11 31	. . . . . . . . . . 11
39	38	reximdv 2565	. . . . . . . . . 10
40	35, 39	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	expimpd 361	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expr 373	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	rexlimdv 2580	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
44	29, 43	syld 45	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	ralrimdva 2544	. . . 4 $/\$ $/\$ $/\$ $/\$
46		simpl2 990	. . . . . . . . 9 $/\$ $/\$ $/\$
47		blss 12969	. . . . . . . . . 10 $/\$ $/\$
48	47	3expib 1195	. . . . . . . . 9 $/\$
49	46, 48	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50		r19.29r 2602	. . . . . . . . . 10 $/\$ $/\$
51	30	ad3antrrr 484	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	13	ad2antrr 480	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		rpxr 9588	. . . . . . . . . . . . . . . . 17
54	53	ad2antrl 482	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	1	blopn 13031	. . . . . . . . . . . . . . . 16 $/\$ $/\$
56	51, 52, 54, 55	syl3anc 1227	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simprl 521	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		blcntr 12957	. . . . . . . . . . . . . . . 16 $/\$ $/\$
59	51, 52, 57, 58	syl3anc 1227	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60		sstr 3145	. . . . . . . . . . . . . . . . 17 $/\$
61	60	ad2ant2l 500	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	ancoms 266	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq2 2228	. . . . . . . . . . . . . . . . 17
64		imaeq2 4936	. . . . . . . . . . . . . . . . . 18
65	64	sseq1d 3166	. . . . . . . . . . . . . . . . 17
66	63, 65	anbi12d 465	. . . . . . . . . . . . . . . 16 $/\$ $/\$
67	66	rspcev 2825	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	56, 59, 62, 67	syl12anc 1225	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	expr 373	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	rexlimdva 2581	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	expimpd 361	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdva 2581	. . . . . . . . . 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 364	. . . . 5 $/\$ $/\$ $/\$ $/\$
78	77	ralrimdv 2543	. . . 4 $/\$ $/\$ $/\$ $/\$
79	45, 78	impbid 128	. . 3 $/\$ $/\$ $/\$ $/\$
80	79	pm5.32da 448	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	10, 80	bitrd 187	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 967

wceq 1342

wcel 2135

wral 2442

wrex 2443

wss 3111

crn 4599

cima 4601

wf 5178

cfv 5182 (class class class)co 5836

cxr 7923

crp 9580

ctg 12507

cxmet 12521

cbl 12523

cmopn 12526 TopOnctopon 12549

ccnp 12727

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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-map 6607 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-xneg 9699 df-xadd 9700 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-topgen 12513 df-psmet 12528 df-xmet 12529 df-bl 12531 df-mopn 12532 df-top 12537 df-topon 12550 df-bases 12582 df-cnp 12730

This theorem is referenced by: metcnp 13053

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