4atex - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atex		Structured version Visualization version Unicode version

Theorem 4atex 35362

Description: Whenever there are at least 4 atoms under

$.\/$

(specifically,

, and

$.\/$

$./\$

), there are also at least 4 atoms under

$.\/$

. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p $\/$ q/0 and hence p $\/$ s/0 contains at least four atoms..." Note that by cvlsupr2 34630, our

$.\/$

is a shorter way to express

$/\$

$.\/$

. (Contributed by NM, 27-May-2013.)

**Hypotheses**
Ref	Expression
4that.l
4that.j	$.\/$
4that.a
4that.h

**Assertion**
Ref	Expression
4atex	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Distinct variable groups:

$.\/$ ,

Allowed substitution hint:

(

)

**Proof of Theorem 4atex**
Step	Hyp	Ref	Expression
1		simp21l 1178	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
2	1	ad2antrr 762	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$
3		simp21r 1179	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
4	3	ad2antrr 762	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$
5		oveq1 6657	. . . . . 6 $.\/$ $.\/$
6	5	eqcoms 2630	. . . . 5 $.\/$ $.\/$
7	6	adantl 482	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
8		breq1 4656	. . . . . . 7
9	8	notbid 308	. . . . . 6
10		oveq2 6658	. . . . . . 7 $.\/$ $.\/$
11		oveq2 6658	. . . . . . 7 $.\/$ $.\/$
12	10, 11	eqeq12d 2637	. . . . . 6 $.\/$ $.\/$ $.\/$ $.\/$
13	9, 12	anbi12d 747	. . . . 5 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
14	13	rspcev 3309	. . . 4 $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
15	2, 4, 7, 14	syl12anc 1324	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
16		simpl3r 1117	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
17	16	ad2antrr 762	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
18		oveq1 6657	. . . . . . . . . 10 $.\/$ $.\/$
19	18	eqeq2d 2632	. . . . . . . . 9 $.\/$ $.\/$ $.\/$ $.\/$
20	19	anbi2d 740	. . . . . . . 8 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
21	20	rexbidv 3052	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
22		breq1 4656	. . . . . . . . . 10
23	22	notbid 308	. . . . . . . . 9
24		oveq2 6658	. . . . . . . . . 10 $.\/$ $.\/$
25		oveq2 6658	. . . . . . . . . 10 $.\/$ $.\/$
26	24, 25	eqeq12d 2637	. . . . . . . . 9 $.\/$ $.\/$ $.\/$ $.\/$
27	23, 26	anbi12d 747	. . . . . . . 8 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
28	27	cbvrexv 3172	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
29	21, 28	syl6rbbr 279	. . . . . 6 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
30	29	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
31	17, 30	mpbid 222	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
32		simp22l 1180	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
33	32	ad3antrrr 766	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
34		simp22r 1181	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
35	34	ad3antrrr 766	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
36		simp3l 1089	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
37	36	necomd 2849	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
38	37	ad3antrrr 766	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
39		simpr 477	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
40	39	necomd 2849	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
41		simpllr 799	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$
42		simp1l 1085	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
43		hlcvl 34646	. . . . . . . . . 10
44	42, 43	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
45	44	ad3antrrr 766	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
46		simp23 1096	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
47	46	ad3antrrr 766	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
48	1	ad3antrrr 766	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
49		simplr 792	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
50		4that.l	. . . . . . . . 9
51		4that.j	. . . . . . . . 9 $.\/$
52		4that.a	. . . . . . . . 9
53	50, 51, 52	cvlatexch1 34623	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
54	45, 47, 33, 48, 49, 53	syl131anc 1339	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
55	41, 54	mpd 15	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$
56	49	necomd 2849	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$
57	52, 50, 51	cvlsupr2 34630	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
58	45, 48, 47, 33, 56, 57	syl131anc 1339	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
59	38, 40, 55, 58	mpbir3and 1245	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
60		breq1 4656	. . . . . . . 8
61	60	notbid 308	. . . . . . 7
62		oveq2 6658	. . . . . . . 8 $.\/$ $.\/$
63		oveq2 6658	. . . . . . . 8 $.\/$ $.\/$
64	62, 63	eqeq12d 2637	. . . . . . 7 $.\/$ $.\/$ $.\/$ $.\/$
65	61, 64	anbi12d 747	. . . . . 6 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
66	65	rspcev 3309	. . . . 5 $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
67	33, 35, 59, 66	syl12anc 1324	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
68	31, 67	pm2.61dane 2881	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
69	15, 68	pm2.61dane 2881	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
70		simpl1 1064	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$
71		simpl2 1065	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$
72		simpl3l 1116	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
73		simpr 477	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
74		simpl3r 1117	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
75		4that.h	. . . 4
76	50, 51, 52, 75	4atexlem7 35361	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
77	70, 71, 72, 73, 74, 76	syl113anc 1338	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
78	69, 77	pm2.61dan 832	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1483

wcel 1990

wne 2794

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cple 15948

cjn 16944

catm 34550

clc 34552

chlt 34637

clh 35270

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-reu 2919 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lhyp 35274

This theorem is referenced by: 4atex2 35363

W3C validator