lebnumii - Metamath Proof Explorer

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Theorem lebnumii 22765

Description: Specialize the Lebesgue number lemma lebnum 22763 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)

**Assertion**
Ref	Expression
lebnumii	$/\$

Distinct variable group:

Proof of Theorem lebnumii Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ii 22680	. . 3
2		cnmet 22575	. . . . 5
3		unitssre 12319	. . . . . 6
4		ax-resscn 9993	. . . . . 6
5	3, 4	sstri 3612	. . . . 5
6		metres2 22168	. . . . 5 $/\$
7	2, 5, 6	mp2an 708	. . . 4
8	7	a1i 11	. . 3 $/\$
9		iicmp 22689	. . . 4
10	9	a1i 11	. . 3 $/\$
11		simpl 473	. . 3 $/\$
12		simpr 477	. . 3 $/\$
13	1, 8, 10, 11, 12	lebnum 22763	. 2 $/\$
14		rpreccl 11857	. . . . . . . 8
15	14	adantl 482	. . . . . . 7 $/\$ $/\$
16	15	rpred 11872	. . . . . 6 $/\$ $/\$
17	15	rpge0d 11876	. . . . . 6 $/\$ $/\$
18		flge0nn0 12621	. . . . . 6 $/\$
19	16, 17, 18	syl2anc 693	. . . . 5 $/\$ $/\$
20		nn0p1nn 11332	. . . . 5
21	19, 20	syl 17	. . . 4 $/\$ $/\$
22		elfznn 12370	. . . . . . . . . . . 12
23	22	adantl 482	. . . . . . . . . . 11 $/\$ $/\$ $/\$
24	23	nnrpd 11870	. . . . . . . . . 10 $/\$ $/\$ $/\$
25	21	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$
26	25	nnrpd 11870	. . . . . . . . . 10 $/\$ $/\$ $/\$
27	24, 26	rpdivcld 11889	. . . . . . . . 9 $/\$ $/\$ $/\$
28	27	rpred 11872	. . . . . . . 8 $/\$ $/\$ $/\$
29	27	rpge0d 11876	. . . . . . . 8 $/\$ $/\$ $/\$
30		elfzle2 12345	. . . . . . . . . . 11
31	30	adantl 482	. . . . . . . . . 10 $/\$ $/\$ $/\$
32	25	nnred 11035	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
33	32	recnd 10068	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	33	mulid1d 10057	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	31, 34	breqtrrd 4681	. . . . . . . . 9 $/\$ $/\$ $/\$
36	23	nnred 11035	. . . . . . . . . 10 $/\$ $/\$ $/\$
37		1re 10039	. . . . . . . . . . 11
38	37	a1i 11	. . . . . . . . . 10 $/\$ $/\$ $/\$
39	25	nngt0d 11064	. . . . . . . . . 10 $/\$ $/\$ $/\$
40		ledivmul 10899	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	36, 38, 32, 39, 40	syl112anc 1330	. . . . . . . . 9 $/\$ $/\$ $/\$
42	35, 41	mpbird 247	. . . . . . . 8 $/\$ $/\$ $/\$
43		0re 10040	. . . . . . . . 9
44	43, 37	elicc2i 12239	. . . . . . . 8 $/\$ $/\$
45	28, 29, 42, 44	syl3anbrc 1246	. . . . . . 7 $/\$ $/\$ $/\$
46		oveq1 6657	. . . . . . . . . 10
47	46	sseq1d 3632	. . . . . . . . 9
48	47	rexbidv 3052	. . . . . . . 8
49	48	rspcv 3305	. . . . . . 7
50	45, 49	syl 17	. . . . . 6 $/\$ $/\$ $/\$
51		simplr 792	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
52	51	rpred 11872	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
53	28, 52	resubcld 10458	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
54	53	rexrd 10089	. . . . . . . . . . 11 $/\$ $/\$ $/\$
55	28, 52	readdcld 10069	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
56	55	rexrd 10089	. . . . . . . . . . 11 $/\$ $/\$ $/\$
57		nnm1nn0 11334	. . . . . . . . . . . . . . 15
58	23, 57	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
59	58	nn0red 11352	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
60	59, 25	nndivred 11069	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
61	36	recnd 10068	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
62	59	recnd 10068	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
63	25	nnne0d 11065	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
64	61, 62, 33, 63	divsubdird 10840	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
65		ax-1cn 9994	. . . . . . . . . . . . . . . 16
66		nncan 10310	. . . . . . . . . . . . . . . 16 $/\$
67	61, 65, 66	sylancl 694	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
68	67	oveq1d 6665	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
69	64, 68	eqtr3d 2658	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
70	51	rprecred 11883	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
71		flltp1 12601	. . . . . . . . . . . . . . 15
72	70, 71	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
73		rpgt0 11844	. . . . . . . . . . . . . . . 16
74	73	ad2antlr 763	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
75		ltdiv23 10914	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
76	38, 52, 74, 32, 39, 75	syl122anc 1335	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
77	72, 76	mpbid 222	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
78	69, 77	eqbrtrd 4675	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79	28, 60, 52, 78	ltsub23d 10632	. . . . . . . . . . 11 $/\$ $/\$ $/\$
80	28, 51	ltaddrpd 11905	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81		iccssioo 12242	. . . . . . . . . . 11 $/\$ $/\$ $/\$
82	54, 56, 79, 80, 81	syl22anc 1327	. . . . . . . . . 10 $/\$ $/\$ $/\$
83		0red 10041	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	58	nn0ge0d 11354	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
85		divge0 10892	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
86	59, 84, 32, 39, 85	syl22anc 1327	. . . . . . . . . . 11 $/\$ $/\$ $/\$
87		iccss 12241	. . . . . . . . . . 11 $/\$ $/\$ $/\$
88	83, 38, 86, 42, 87	syl22anc 1327	. . . . . . . . . 10 $/\$ $/\$ $/\$
89	82, 88	ssind 3837	. . . . . . . . 9 $/\$ $/\$ $/\$
90		eqid 2622	. . . . . . . . . . . . 13
91	90	rexmet 22594	. . . . . . . . . . . 12
92	91	a1i 11	. . . . . . . . . . 11 $/\$ $/\$ $/\$
93		sseqin2 3817	. . . . . . . . . . . . 13
94	3, 93	mpbi 220	. . . . . . . . . . . 12
95	45, 94	syl6eleqr 2712	. . . . . . . . . . 11 $/\$ $/\$ $/\$
96		rpxr 11840	. . . . . . . . . . . 12
97	96	ad2antlr 763	. . . . . . . . . . 11 $/\$ $/\$ $/\$
98		xpss12 5225	. . . . . . . . . . . . . . 15 $/\$
99	3, 3, 98	mp2an 708	. . . . . . . . . . . . . 14
100		resabs1 5427	. . . . . . . . . . . . . 14
101	99, 100	ax-mp 5	. . . . . . . . . . . . 13
102	101	eqcomi 2631	. . . . . . . . . . . 12
103	102	blres 22236	. . . . . . . . . . 11 $/\$ $/\$
104	92, 95, 97, 103	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$
105	90	bl2ioo 22595	. . . . . . . . . . . 12 $/\$
106	28, 52, 105	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$
107	106	ineq1d 3813	. . . . . . . . . 10 $/\$ $/\$ $/\$
108	104, 107	eqtrd 2656	. . . . . . . . 9 $/\$ $/\$ $/\$
109	89, 108	sseqtr4d 3642	. . . . . . . 8 $/\$ $/\$ $/\$
110		sstr2 3610	. . . . . . . 8
111	109, 110	syl 17	. . . . . . 7 $/\$ $/\$ $/\$
112	111	reximdv 3016	. . . . . 6 $/\$ $/\$ $/\$
113	50, 112	syld 47	. . . . 5 $/\$ $/\$ $/\$
114	113	ralrimdva 2969	. . . 4 $/\$ $/\$
115		oveq2 6658	. . . . . 6
116		oveq2 6658	. . . . . . . . 9
117		oveq2 6658	. . . . . . . . 9
118	116, 117	oveq12d 6668	. . . . . . . 8
119	118	sseq1d 3632	. . . . . . 7
120	119	rexbidv 3052	. . . . . 6
121	115, 120	raleqbidv 3152	. . . . 5
122	121	rspcev 3309	. . . 4 $/\$
123	21, 114, 122	syl6an 568	. . 3 $/\$ $/\$
124	123	rexlimdva 3031	. 2 $/\$
125	13, 124	mpd 15	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

wrex 2913

cin 3573

wss 3574

cuni 4436 class class class wbr 4653

cxp 5112

cres 5116

ccom 5118

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

cxr 10073

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

cn0 11292

crp 11832

cioo 12175

cicc 12178

cfz 12326

cfl 12591

cabs 13974

cxmt 19731

cme 19732

cbl 19733

ccmp 21189

cii 22678

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-inf2 8538 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 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-int 4476 df-iun 4522 df-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 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-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 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-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-cn 21031 df-cnp 21032 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-ii 22680

This theorem is referenced by: cvmliftlem15 31280

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