ovolgelb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ovolgelb		Structured version Visualization version Unicode version

Theorem ovolgelb 23248

Description: The outer volume is the greatest lower bound on the sum of all interval coverings of

. (Contributed by Mario Carneiro, 15-Jun-2014.)

**Hypothesis**
Ref	Expression
ovolgelb.1

**Assertion**
Ref	Expression
ovolgelb	$/\$ $/\$ $/\$

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem ovolgelb Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . . . 6 $/\$ $/\$
2		simp3 1063	. . . . . 6 $/\$ $/\$
3	1, 2	ltaddrpd 11905	. . . . 5 $/\$ $/\$
4	2	rpred 11872	. . . . . . 7 $/\$ $/\$
5	1, 4	readdcld 10069	. . . . . 6 $/\$ $/\$
6	1, 5	ltnled 10184	. . . . 5 $/\$ $/\$
7	3, 6	mpbid 222	. . . 4 $/\$ $/\$
8		eqid 2622	. . . . . . . 8 $/\$ $/\$
9	8	ovolval 23242	. . . . . . 7 inf $/\$
10	9	3ad2ant1 1082	. . . . . 6 $/\$ $/\$ inf $/\$
11	10	breq2d 4665	. . . . 5 $/\$ $/\$ inf $/\$
12		ssrab2 3687	. . . . . . 7 $/\$
13	5	rexrd 10089	. . . . . . 7 $/\$ $/\$
14		infxrgelb 12165	. . . . . . 7 $/\$ $/\$ inf $/\$ $/\$
15	12, 13, 14	sylancr 695	. . . . . 6 $/\$ $/\$ inf $/\$ $/\$
16		eqeq1 2626	. . . . . . . . . . 11
17		ovolgelb.1	. . . . . . . . . . . . . 14
18	17	rneqi 5352	. . . . . . . . . . . . 13
19	18	supeq1i 8353	. . . . . . . . . . . 12
20	19	eqeq2i 2634	. . . . . . . . . . 11
21	16, 20	syl6bbr 278	. . . . . . . . . 10
22	21	anbi2d 740	. . . . . . . . 9 $/\$ $/\$
23	22	rexbidv 3052	. . . . . . . 8 $/\$ $/\$
24	23	ralrab 3368	. . . . . . 7 $/\$ $/\$
25		ralcom 3098	. . . . . . . 8 $/\$ $/\$
26		r19.23v 3023	. . . . . . . . 9 $/\$ $/\$
27	26	ralbii 2980	. . . . . . . 8 $/\$ $/\$
28		ancomst 468	. . . . . . . . . . . 12 $/\$ $/\$
29		impexp 462	. . . . . . . . . . . 12 $/\$
30	28, 29	bitri 264	. . . . . . . . . . 11 $/\$
31	30	ralbii 2980	. . . . . . . . . 10 $/\$
32		reex 10027	. . . . . . . . . . . . . . . . . 18
33	32, 32	xpex 6962	. . . . . . . . . . . . . . . . 17
34	33	inex2 4800	. . . . . . . . . . . . . . . 16
35		nnex 11026	. . . . . . . . . . . . . . . 16
36	34, 35	elmap 7886	. . . . . . . . . . . . . . 15
37		eqid 2622	. . . . . . . . . . . . . . . 16
38	37, 17	ovolsf 23241	. . . . . . . . . . . . . . 15
39	36, 38	sylbi 207	. . . . . . . . . . . . . 14
40		frn 6053	. . . . . . . . . . . . . 14
41	39, 40	syl 17	. . . . . . . . . . . . 13
42		icossxr 12258	. . . . . . . . . . . . 13
43	41, 42	syl6ss 3615	. . . . . . . . . . . 12
44		supxrcl 12145	. . . . . . . . . . . 12
45	43, 44	syl 17	. . . . . . . . . . 11
46		breq2 4657	. . . . . . . . . . . . 13
47	46	imbi2d 330	. . . . . . . . . . . 12
48	47	ceqsralv 3234	. . . . . . . . . . 11
49	45, 48	syl 17	. . . . . . . . . 10
50	31, 49	syl5bb 272	. . . . . . . . 9 $/\$
51	50	ralbiia 2979	. . . . . . . 8 $/\$
52	25, 27, 51	3bitr3i 290	. . . . . . 7 $/\$
53	24, 52	bitri 264	. . . . . 6 $/\$
54	15, 53	syl6rbb 277	. . . . 5 $/\$ $/\$ inf $/\$
55	11, 54	bitr4d 271	. . . 4 $/\$ $/\$
56	7, 55	mtbid 314	. . 3 $/\$ $/\$
57		rexanali 2998	. . 3 $/\$
58	56, 57	sylibr 224	. 2 $/\$ $/\$ $/\$
59		xrltnle 10105	. . . . . 6 $/\$
60		xrltle 11982	. . . . . 6 $/\$
61	59, 60	sylbird 250	. . . . 5 $/\$
62	45, 13, 61	syl2anr 495	. . . 4 $/\$ $/\$ $/\$
63	62	anim2d 589	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	reximdva 3017	. 2 $/\$ $/\$ $/\$ $/\$
65	58, 64	mpd 15	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

crab 2916

cin 3573

wss 3574

cuni 4436 class class class wbr 4653

cxp 5112

crn 5115

ccom 5118

wf 5884

cfv 5888 (class class class)co 6650

cmap 7857

csup 8346 infcinf 8347

cr 9935

cc0 9936

c1 9937

caddc 9939

cpnf 10071

cxr 10073

clt 10074

cle 10075

cmin 10266

cn 11020

crp 11832

cioo 12175

cico 12177

cseq 12801

cabs 13974

covol 23231

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-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-rp 11833 df-ico 12181 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-ovol 23233

This theorem is referenced by: ovolunlem2 23266 ovoliunlem3 23272 ovolscalem2 23282 ioombl1 23330 uniioombl 23357 mblfinlem3 33448 mblfinlem4 33449

W3C validator