ovoliunlem3 - Metamath Proof Explorer

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Theorem ovoliunlem3 23272

Description: Lemma for ovoliun 23273. (Contributed by Mario Carneiro, 12-Jun-2014.)

**Hypotheses**
Ref	Expression
ovoliun.t
ovoliun.g
ovoliun.a	$/\$
ovoliun.v	$/\$
ovoliun.r
ovoliun.b

**Assertion**
Ref	Expression
ovoliunlem3

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem ovoliunlem3 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . 4
2		nfcsb1v 3549	. . . 4
3		csbeq1a 3542	. . . 4
4	1, 2, 3	cbviun 4557	. . 3
5	4	fveq2i 6194	. 2
6		ovoliun.a	. . . . . 6 $/\$
7		ovoliun.v	. . . . . 6 $/\$
8		ovoliun.b	. . . . . . 7
9		2nn 11185	. . . . . . . . 9
10		nnnn0 11299	. . . . . . . . 9
11		nnexpcl 12873	. . . . . . . . 9 $/\$
12	9, 10, 11	sylancr 695	. . . . . . . 8
13	12	nnrpd 11870	. . . . . . 7
14		rpdivcl 11856	. . . . . . 7 $/\$
15	8, 13, 14	syl2an 494	. . . . . 6 $/\$
16		eqid 2622	. . . . . . 7
17	16	ovolgelb 23248	. . . . . 6 $/\$ $/\$ $/\$
18	6, 7, 15, 17	syl3anc 1326	. . . . 5 $/\$ $/\$
19	18	ralrimiva 2966	. . . 4 $/\$
20		ovex 6678	. . . . 5
21		nnenom 12779	. . . . 5
22		coeq2 5280	. . . . . . . . 9
23	22	rneqd 5353	. . . . . . . 8
24	23	unieqd 4446	. . . . . . 7
25	24	sseq2d 3633	. . . . . 6
26		coeq2 5280	. . . . . . . . . 10
27	26	seqeq3d 12809	. . . . . . . . 9
28	27	rneqd 5353	. . . . . . . 8
29	28	supeq1d 8352	. . . . . . 7
30	29	breq1d 4663	. . . . . 6
31	25, 30	anbi12d 747	. . . . 5 $/\$ $/\$
32	20, 21, 31	axcc4 9261	. . . 4 $/\$ $/\$ $/\$
33	19, 32	syl 17	. . 3 $/\$ $/\$
34		xpnnen 14939	. . . . . . 7
35	34	ensymi 8006	. . . . . 6
36		bren 7964	. . . . . 6
37	35, 36	mpbi 220	. . . . 5
38		ovoliun.t	. . . . . . . 8
39		ovoliun.g	. . . . . . . . 9
40		nfcv 2764	. . . . . . . . . 10
41		nfcv 2764	. . . . . . . . . . 11
42	41, 2	nffv 6198	. . . . . . . . . 10
43	3	fveq2d 6195	. . . . . . . . . 10
44	40, 42, 43	cbvmpt 4749	. . . . . . . . 9
45	39, 44	eqtri 2644	. . . . . . . 8
46		simpll 790	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
47	6	ralrimiva 2966	. . . . . . . . . . 11
48		nfv 1843	. . . . . . . . . . . 12
49		nfcv 2764	. . . . . . . . . . . . 13
50	2, 49	nfss 3596	. . . . . . . . . . . 12
51	3	sseq1d 3632	. . . . . . . . . . . 12
52	48, 50, 51	cbvral 3167	. . . . . . . . . . 11
53	47, 52	sylib 208	. . . . . . . . . 10
54	53	r19.21bi 2932	. . . . . . . . 9 $/\$
55	46, 54	sylan 488	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
56	7	ralrimiva 2966	. . . . . . . . . . 11
57	40	nfel1 2779	. . . . . . . . . . . 12
58	42	nfel1 2779	. . . . . . . . . . . 12
59	43	eleq1d 2686	. . . . . . . . . . . 12
60	57, 58, 59	cbvral 3167	. . . . . . . . . . 11
61	56, 60	sylib 208	. . . . . . . . . 10
62	61	r19.21bi 2932	. . . . . . . . 9 $/\$
63	46, 62	sylan 488	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
64		ovoliun.r	. . . . . . . . 9
65	64	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
66	8	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
67		eqid 2622	. . . . . . . 8
68		eqid 2622	. . . . . . . 8
69		eqid 2622	. . . . . . . 8
70		simplr 792	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
71		simprl 794	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
72		simprr 796	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
73		nfv 1843	. . . . . . . . . . . 12 $/\$
74		nfcv 2764	. . . . . . . . . . . . . 14
75	2, 74	nfss 3596	. . . . . . . . . . . . 13
76		nfcv 2764	. . . . . . . . . . . . . 14
77		nfcv 2764	. . . . . . . . . . . . . 14
78		nfcv 2764	. . . . . . . . . . . . . . 15
79		nfcv 2764	. . . . . . . . . . . . . . 15
80	42, 78, 79	nfov 6676	. . . . . . . . . . . . . 14
81	76, 77, 80	nfbr 4699	. . . . . . . . . . . . 13
82	75, 81	nfan 1828	. . . . . . . . . . . 12 $/\$
83		fveq2 6191	. . . . . . . . . . . . . . . . 17
84	83	coeq2d 5284	. . . . . . . . . . . . . . . 16
85	84	rneqd 5353	. . . . . . . . . . . . . . 15
86	85	unieqd 4446	. . . . . . . . . . . . . 14
87	3, 86	sseq12d 3634	. . . . . . . . . . . . 13
88	83	coeq2d 5284	. . . . . . . . . . . . . . . . 17
89	88	seqeq3d 12809	. . . . . . . . . . . . . . . 16
90	89	rneqd 5353	. . . . . . . . . . . . . . 15
91	90	supeq1d 8352	. . . . . . . . . . . . . 14
92		oveq2 6658	. . . . . . . . . . . . . . . 16
93	92	oveq2d 6666	. . . . . . . . . . . . . . 15
94	43, 93	oveq12d 6668	. . . . . . . . . . . . . 14
95	91, 94	breq12d 4666	. . . . . . . . . . . . 13
96	87, 95	anbi12d 747	. . . . . . . . . . . 12 $/\$ $/\$
97	73, 82, 96	cbvral 3167	. . . . . . . . . . 11 $/\$ $/\$
98	72, 97	sylib 208	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	r19.21bi 2932	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
100	99	simpld 475	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
101	99	simprd 479	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
102	38, 45, 55, 63, 65, 66, 67, 68, 69, 70, 71, 100, 101	ovoliunlem2 23271	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
103	102	exp31 630	. . . . . 6 $/\$ $/\$
104	103	exlimdv 1861	. . . . 5 $/\$ $/\$
105	37, 104	mpi 20	. . . 4 $/\$ $/\$
106	105	exlimdv 1861	. . 3 $/\$ $/\$
107	33, 106	mpd 15	. 2
108	5, 107	syl5eqbr 4688	1

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wex 1704

wcel 1990

wral 2912

wrex 2913

csb 3533

cin 3573

wss 3574

cuni 4436

ciun 4520 class class class wbr 4653

cmpt 4729

cxp 5112

crn 5115

ccom 5118

wf 5884

wf1o 5887

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cmap 7857

cen 7952

csup 8346

cr 9935

c1 9937

caddc 9939

cxr 10073

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

c2 11070

cn0 11292

crp 11832

cioo 12175

cseq 12801

cexp 12860

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-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-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-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 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-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 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-ioo 12179 df-ico 12181 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-ovol 23233

This theorem is referenced by: ovoliun 23273

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