cdleme0nex - Mathbox for Norm Megill

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Theorem cdleme0nex 35577

Description: Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p $\/$ q/0 (i.e. the sublattice from 0 to p $\/$ q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 35498- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 34630, our

$.\/$

is a shorter way to express

$/\$

$.\/$

. Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)

**Hypotheses**
Ref	Expression
cdleme0nex.l
cdleme0nex.j	$.\/$
cdleme0nex.a

**Assertion**
Ref	Expression
cdleme0nex	$/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Distinct variable groups:

$.\/$ ,

Allowed substitution hint:

(

)

**Proof of Theorem cdleme0nex**
Step	Hyp	Ref	Expression
1		simp3r 1090	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simp12 1092	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
3	1, 2	jca 554	. . 3 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
4		simp3l 1089	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
5		simp13 1093	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
6		ralnex 2992	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
7	5, 6	sylibr 224	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
8		breq1 4656	. . . . . . . . . 10
9	8	notbid 308	. . . . . . . . 9
10		oveq2 6658	. . . . . . . . . 10 $.\/$ $.\/$
11		oveq2 6658	. . . . . . . . . 10 $.\/$ $.\/$
12	10, 11	eqeq12d 2637	. . . . . . . . 9 $.\/$ $.\/$ $.\/$ $.\/$
13	9, 12	anbi12d 747	. . . . . . . 8 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
14	13	notbid 308	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
15	14	rspcva 3307	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
16	4, 7, 15	syl2anc 693	. . . . 5 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
17		simp11 1091	. . . . . . . 8 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
18		hlcvl 34646	. . . . . . . 8
19	17, 18	syl 17	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simp21 1094	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simp22 1095	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simp23 1096	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
23		cdleme0nex.a	. . . . . . . 8
24		cdleme0nex.l	. . . . . . . 8
25		cdleme0nex.j	. . . . . . . 8 $.\/$
26	23, 24, 25	cvlsupr2 34630	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
27	19, 20, 21, 4, 22, 26	syl131anc 1339	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
28	27	anbi2d 740	. . . . 5 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$
29	16, 28	mtbid 314	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
30		ianor 509	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $/\$ $\/$ $/\$ $.\/$
31		df-3an 1039	. . . . . . . 8 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
32	31	anbi2i 730	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
33		an12 838	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
34	32, 33	bitri 264	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
35	34	notbii 310	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
36		pm4.62 435	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $\/$ $/\$ $.\/$
37	30, 35, 36	3bitr4ri 293	. . . 4 $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
38	29, 37	sylibr 224	. . 3 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
39	3, 38	mt2d 131	. 2 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		neanior 2886	. . 3 $/\$ $\/$
41	40	con2bii 347	. 2 $\/$ $/\$
42	39, 41	sylibr 224	1 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cple 15948

cjn 16944

catm 34550

clc 34552

chlt 34637

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-lat 17046 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638

This theorem is referenced by: cdleme18c 35580 cdleme18d 35582 cdlemg17b 35950 cdlemg17h 35956

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