iccntr - Metamath Proof Explorer

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Theorem iccntr 22624

Description: The interior of a closed interval in the standard topology on

is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)

**Assertion**
Ref	Expression
iccntr	$/\$

Proof of Theorem iccntr Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexr 10085	. . . . . . . 8
2		rexr 10085	. . . . . . . 8
3		icc0 12223	. . . . . . . 8 $/\$
4	1, 2, 3	syl2an 494	. . . . . . 7 $/\$
5	4	biimpar 502	. . . . . 6 $/\$ $/\$
6	5	fveq2d 6195	. . . . 5 $/\$ $/\$
7		retop 22565	. . . . . . 7
8		ntr0 20885	. . . . . . 7
9	7, 8	ax-mp 5	. . . . . 6
10		0ss 3972	. . . . . 6
11	9, 10	eqsstri 3635	. . . . 5
12	6, 11	syl6eqss 3655	. . . 4 $/\$ $/\$
13		iccssre 12255	. . . . . . 7 $/\$
14		uniretop 22566	. . . . . . . 8
15	14	ntrss2 20861	. . . . . . 7 $/\$
16	7, 13, 15	sylancr 695	. . . . . 6 $/\$
17	16	adantr 481	. . . . 5 $/\$ $/\$
18	1, 2	anim12i 590	. . . . . 6 $/\$ $/\$
19		uncom 3757	. . . . . . . 8
20		prunioo 12301	. . . . . . . 8 $/\$ $/\$
21	19, 20	syl5eq 2668	. . . . . . 7 $/\$ $/\$
22	21	3expa 1265	. . . . . 6 $/\$ $/\$
23	18, 22	sylan 488	. . . . 5 $/\$ $/\$
24	17, 23	sseqtr4d 3642	. . . 4 $/\$ $/\$
25		simpr 477	. . . 4 $/\$
26		simpl 473	. . . 4 $/\$
27	12, 24, 25, 26	ltlecasei 10145	. . 3 $/\$
28	14	ntropn 20853	. . . . . . . . 9 $/\$
29	7, 13, 28	sylancr 695	. . . . . . . 8 $/\$
30		eqid 2622	. . . . . . . . . 10
31	30	rexmet 22594	. . . . . . . . 9
32		eqid 2622	. . . . . . . . . . 11
33	30, 32	tgioo 22599	. . . . . . . . . 10
34	33	mopni2 22298	. . . . . . . . 9 $/\$ $/\$
35	31, 34	mp3an1 1411	. . . . . . . 8 $/\$
36	29, 35	sylan 488	. . . . . . 7 $/\$ $/\$
37	26	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$
38		rphalfcl 11858	. . . . . . . . . . . 12
39	38	adantl 482	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	37, 39	ltsubrpd 11904	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	39	rpred 11872	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
42	37, 41	resubcld 10458	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	42, 37	ltnled 10184	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 222	. . . . . . . . 9 $/\$ $/\$ $/\$
45		rpre 11839	. . . . . . . . . . . . . . 15
46	45	adantl 482	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
47		rphalflt 11860	. . . . . . . . . . . . . . 15
48	47	adantl 482	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
49	41, 46, 37, 48	ltsub2dd 10640	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
50	37, 46	readdcld 10069	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
51		ltaddrp 11867	. . . . . . . . . . . . . . 15 $/\$
52	37, 51	sylancom 701	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
53	42, 37, 50, 40, 52	lttrd 10198	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
54	37, 46	resubcld 10458	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
55	54	rexrd 10089	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
56	50	rexrd 10089	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
57		elioo2 12216	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
58	55, 56, 57	syl2anc 693	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
59	42, 49, 53, 58	mpbir3and 1245	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
60	30	bl2ioo 22595	. . . . . . . . . . . . 13 $/\$
61	37, 46, 60	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
62	59, 61	eleqtrrd 2704	. . . . . . . . . . 11 $/\$ $/\$ $/\$
63		ssel 3597	. . . . . . . . . . 11
64	62, 63	syl5com 31	. . . . . . . . . 10 $/\$ $/\$ $/\$
65	16	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$
66	65	sseld 3602	. . . . . . . . . 10 $/\$ $/\$ $/\$
67		elicc2 12238	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
68		simp2 1062	. . . . . . . . . . . 12 $/\$ $/\$
69	67, 68	syl6bi 243	. . . . . . . . . . 11 $/\$
70	69	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ $/\$
71	64, 66, 70	3syld 60	. . . . . . . . 9 $/\$ $/\$ $/\$
72	44, 71	mtod 189	. . . . . . . 8 $/\$ $/\$ $/\$
73	72	nrexdv 3001	. . . . . . 7 $/\$ $/\$
74	36, 73	pm2.65da 600	. . . . . 6 $/\$
75	33	mopni2 22298	. . . . . . . . 9 $/\$ $/\$
76	31, 75	mp3an1 1411	. . . . . . . 8 $/\$
77	29, 76	sylan 488	. . . . . . 7 $/\$ $/\$
78	25	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$
79	38	adantl 482	. . . . . . . . . . 11 $/\$ $/\$ $/\$
80	78, 79	ltaddrpd 11905	. . . . . . . . . 10 $/\$ $/\$ $/\$
81	79	rpred 11872	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
82	78, 81	readdcld 10069	. . . . . . . . . . 11 $/\$ $/\$ $/\$
83	78, 82	ltnled 10184	. . . . . . . . . 10 $/\$ $/\$ $/\$
84	80, 83	mpbid 222	. . . . . . . . 9 $/\$ $/\$ $/\$
85	45	adantl 482	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
86	78, 85	resubcld 10458	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
87		ltsubrp 11866	. . . . . . . . . . . . . . 15 $/\$
88	78, 87	sylancom 701	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
89	86, 78, 82, 88, 80	lttrd 10198	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
90	47	adantl 482	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
91	81, 85, 78, 90	ltadd2dd 10196	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
92	86	rexrd 10089	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
93	78, 85	readdcld 10069	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
94	93	rexrd 10089	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
95		elioo2 12216	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
96	92, 94, 95	syl2anc 693	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
97	82, 89, 91, 96	mpbir3and 1245	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
98	30	bl2ioo 22595	. . . . . . . . . . . . 13 $/\$
99	78, 85, 98	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
100	97, 99	eleqtrrd 2704	. . . . . . . . . . 11 $/\$ $/\$ $/\$
101		ssel 3597	. . . . . . . . . . 11
102	100, 101	syl5com 31	. . . . . . . . . 10 $/\$ $/\$ $/\$
103	16	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$
104	103	sseld 3602	. . . . . . . . . 10 $/\$ $/\$ $/\$
105		elicc2 12238	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
106		simp3 1063	. . . . . . . . . . . 12 $/\$ $/\$
107	105, 106	syl6bi 243	. . . . . . . . . . 11 $/\$
108	107	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ $/\$
109	102, 104, 108	3syld 60	. . . . . . . . 9 $/\$ $/\$ $/\$
110	84, 109	mtod 189	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	nrexdv 3001	. . . . . . 7 $/\$ $/\$
112	77, 111	pm2.65da 600	. . . . . 6 $/\$
113		eleq1 2689	. . . . . . . 8
114	113	notbid 308	. . . . . . 7
115		eleq1 2689	. . . . . . . 8
116	115	notbid 308	. . . . . . 7
117	114, 116	ralprg 4234	. . . . . 6 $/\$ $/\$
118	74, 112, 117	mpbir2and 957	. . . . 5 $/\$
119		disjr 4018	. . . . 5
120	118, 119	sylibr 224	. . . 4 $/\$
121		disjssun 4036	. . . 4
122	120, 121	syl 17	. . 3 $/\$
123	27, 122	mpbid 222	. 2 $/\$
124		iooretop 22569	. . . 4
125		ioossicc 12259	. . . 4
126	14	ssntr 20862	. . . 4 $/\$ $/\$ $/\$
127	124, 125, 126	mpanr12 721	. . 3 $/\$
128	7, 13, 127	sylancr 695	. 2 $/\$
129	123, 128	eqssd 3620	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913

cun 3572

cin 3573

wss 3574

c0 3915

cpr 4179 class class class wbr 4653

cxp 5112

crn 5115

cres 5116

ccom 5118

cfv 5888 (class class class)co 6650

cr 9935

caddc 9939

cxr 10073

clt 10074

cle 10075

cmin 10266

cdiv 10684

c2 11070

crp 11832

cioo 12175

cicc 12178

cabs 13974

ctg 16098

cxmt 19731

cbl 19733

cmopn 19736

ctop 20698

cnt 20821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-ntr 20824

This theorem is referenced by: rolle 23753 cmvth 23754 mvth 23755 dvlip 23756 dvlipcn 23757 dvlip2 23758 c1liplem1 23759 dvgt0lem1 23765 dvle 23770 lhop1lem 23776 dvcnvrelem1 23780 dvcvx 23783 dvfsumabs 23786 ftc1cn 23806 ftc2 23807 ftc2ditglem 23808 itgparts 23810 itgsubstlem 23811 efcvx 24203 pige3 24269 logccv 24409 lgamgulmlem2 24756 ftc2re 30676 ftc1cnnc 33484 ftc2nc 33494 areacirc 33505 itgpowd 37800 lhe4.4ex1a 38528 dvmptresicc 40134 dvbdfbdioolem1 40143 itgsin0pilem1 40165 itgsinexplem1 40169 itgcoscmulx 40185 itgiccshift 40196 itgperiod 40197 itgsbtaddcnst 40198 dirkeritg 40319 fourierdlem39 40363 fourierdlem73 40396 etransclem46 40497

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