volivth - Metamath Proof Explorer

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Theorem volivth 23375

Description: The Intermediate Value Theorem for the Lebesgue volume function. For any positive

, there is a measurable subset of

whose volume is

. (Contributed by Mario Carneiro, 30-Aug-2014.)

**Assertion**
Ref	Expression
volivth	$/\$ $/\$

Distinct variable groups:

Proof of Theorem volivth Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 790	. . . 4 $/\$ $/\$
2		mnfxr 10096	. . . . . 6
3	2	a1i 11	. . . . 5 $/\$ $/\$
4		iccssxr 12256	. . . . . . 7
5		simpr 477	. . . . . . 7 $/\$
6	4, 5	sseldi 3601	. . . . . 6 $/\$
7	6	adantr 481	. . . . 5 $/\$ $/\$
8		iccssxr 12256	. . . . . . . 8
9		volf 23297	. . . . . . . . 9
10	9	ffvelrni 6358	. . . . . . . 8
11	8, 10	sseldi 3601	. . . . . . 7
12	11	adantr 481	. . . . . 6 $/\$
13	12	adantr 481	. . . . 5 $/\$ $/\$
14		0xr 10086	. . . . . . . . . 10
15		elicc1 12219	. . . . . . . . . 10 $/\$ $/\$ $/\$
16	14, 12, 15	sylancr 695	. . . . . . . . 9 $/\$ $/\$ $/\$
17	5, 16	mpbid 222	. . . . . . . 8 $/\$ $/\$ $/\$
18	17	simp2d 1074	. . . . . . 7 $/\$
19	18	adantr 481	. . . . . 6 $/\$ $/\$
20		mnflt0 11959	. . . . . . . 8
21		xrltletr 11988	. . . . . . . 8 $/\$ $/\$ $/\$
22	20, 21	mpani 712	. . . . . . 7 $/\$ $/\$
23	2, 14, 22	mp3an12 1414	. . . . . 6
24	7, 19, 23	sylc 65	. . . . 5 $/\$ $/\$
25		simpr 477	. . . . 5 $/\$ $/\$
26		xrre2 12001	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	3, 7, 13, 24, 25, 26	syl32anc 1334	. . . 4 $/\$ $/\$
28		volsup2 23373	. . . 4 $/\$ $/\$
29	1, 27, 25, 28	syl3anc 1326	. . 3 $/\$ $/\$
30		nnre 11027	. . . . . . 7
31	30	ad2antrl 764	. . . . . 6 $/\$ $/\$ $/\$ $/\$
32	31	renegcld 10457	. . . . 5 $/\$ $/\$ $/\$ $/\$
33	27	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$
34		0red 10041	. . . . . 6 $/\$ $/\$ $/\$ $/\$
35		nngt0 11049	. . . . . . . 8
36	35	ad2antrl 764	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
37	31	lt0neg2d 10598	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	36, 37	mpbid 222	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	32, 34, 31, 38, 36	lttrd 10198	. . . . 5 $/\$ $/\$ $/\$ $/\$
40		iccssre 12255	. . . . . 6 $/\$
41	32, 31, 40	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$
42		ax-resscn 9993	. . . . . . 7
43		ssid 3624	. . . . . . 7
44		cncfss 22702	. . . . . . 7 $/\$
45	42, 43, 44	mp2an 708	. . . . . 6
46	1	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
47		eqid 2622	. . . . . . . 8
48	47	volcn 23374	. . . . . . 7 $/\$
49	46, 32, 48	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$
50	45, 49	sseldi 3601	. . . . 5 $/\$ $/\$ $/\$ $/\$
51	41	sselda 3603	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52		cncff 22696	. . . . . . . 8
53	49, 52	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	ffvelrnda 6359	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
55	51, 54	syldan 487	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
56		oveq2 6658	. . . . . . . . . . . 12
57	56	ineq2d 3814	. . . . . . . . . . 11
58	57	fveq2d 6195	. . . . . . . . . 10
59		fvex 6201	. . . . . . . . . 10
60	58, 47, 59	fvmpt 6282	. . . . . . . . 9
61	32, 60	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
62		inss2 3834	. . . . . . . . . . . 12
63	32	rexrd 10089	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
64		iccid 12220	. . . . . . . . . . . . 13
65	63, 64	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
66	62, 65	syl5sseq 3653	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
67	32	snssd 4340	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
68	66, 67	sstrd 3613	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		ovolsn 23263	. . . . . . . . . . . 12
70	32, 69	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
71		ovolssnul 23255	. . . . . . . . . . 11 $/\$ $/\$
72	66, 67, 70, 71	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
73		nulmbl 23303	. . . . . . . . . 10 $/\$
74	68, 72, 73	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
75		mblvol 23298	. . . . . . . . 9
76	74, 75	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
77	61, 76, 72	3eqtrd 2660	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
78	19	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
79	77, 78	eqbrtrd 4675	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80		simprr 796	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
81	7	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
82		iccmbl 23334	. . . . . . . . . . . 12 $/\$
83	32, 31, 82	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
84		inmbl 23310	. . . . . . . . . . 11 $/\$
85	46, 83, 84	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
86	9	ffvelrni 6358	. . . . . . . . . . 11
87	8, 86	sseldi 3601	. . . . . . . . . 10
88	85, 87	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
89		xrltle 11982	. . . . . . . . 9 $/\$
90	81, 88, 89	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
91	80, 90	mpd 15	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
92		oveq2 6658	. . . . . . . . . . 11
93	92	ineq2d 3814	. . . . . . . . . 10
94	93	fveq2d 6195	. . . . . . . . 9
95		fvex 6201	. . . . . . . . 9
96	94, 47, 95	fvmpt 6282	. . . . . . . 8
97	31, 96	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
98	91, 97	breqtrrd 4681	. . . . . 6 $/\$ $/\$ $/\$ $/\$
99	79, 98	jca 554	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
100	32, 31, 33, 39, 41, 50, 55, 99	ivthle 23225	. . . 4 $/\$ $/\$ $/\$ $/\$
101	41	sselda 3603	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
102		oveq2 6658	. . . . . . . . . . 11
103	102	ineq2d 3814	. . . . . . . . . 10
104	103	fveq2d 6195	. . . . . . . . 9
105		fvex 6201	. . . . . . . . 9
106	104, 47, 105	fvmpt 6282	. . . . . . . 8
107	101, 106	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
108	107	eqeq1d 2624	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
109	46	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
110	32	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
111	101	adantrr 753	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
112		iccmbl 23334	. . . . . . . . . 10 $/\$
113	110, 111, 112	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
114		inmbl 23310	. . . . . . . . 9 $/\$
115	109, 113, 114	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
116		inss1 3833	. . . . . . . . 9
117	116	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
118		simprr 796	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
119		sseq1 3626	. . . . . . . . . 10
120		fveq2 6191	. . . . . . . . . . 11
121	120	eqeq1d 2624	. . . . . . . . . 10
122	119, 121	anbi12d 747	. . . . . . . . 9 $/\$ $/\$
123	122	rspcev 3309	. . . . . . . 8 $/\$ $/\$ $/\$
124	115, 117, 118, 123	syl12anc 1324	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
125	124	expr 643	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
126	108, 125	sylbid 230	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
127	126	rexlimdva 3031	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
128	100, 127	mpd 15	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
129	29, 128	rexlimddv 3035	. 2 $/\$ $/\$ $/\$
130		simpll 790	. . 3 $/\$ $/\$
131		ssid 3624	. . . 4
132	131	a1i 11	. . 3 $/\$ $/\$
133		simpr 477	. . . 4 $/\$ $/\$
134	133	eqcomd 2628	. . 3 $/\$ $/\$
135		sseq1 3626	. . . . 5
136		fveq2 6191	. . . . . 6
137	136	eqeq1d 2624	. . . . 5
138	135, 137	anbi12d 747	. . . 4 $/\$ $/\$
139	138	rspcev 3309	. . 3 $/\$ $/\$ $/\$
140	130, 132, 134, 139	syl12anc 1324	. 2 $/\$ $/\$ $/\$
141	17	simp3d 1075	. . 3 $/\$
142		xrleloe 11977	. . . 4 $/\$ $\/$
143	6, 12, 142	syl2anc 693	. . 3 $/\$ $\/$
144	141, 143	mpbid 222	. 2 $/\$ $\/$
145	129, 140, 144	mpjaodan 827	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wrex 2913

cin 3573

wss 3574

csn 4177 class class class wbr 4653

cmpt 4729

cdm 5114

wf 5884

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

cle 10075

cneg 10267

cn 11020

cicc 12178

ccncf 22679

covol 23231

cvol 23232

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-cc 9257 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-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-disj 4621 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-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 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-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-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-rlim 14220 df-sum 14417 df-rest 16083 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-cmp 21190 df-cncf 22681 df-ovol 23233 df-vol 23234

This theorem is referenced by: itg2const2 23508

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