cdleme0nex - Mathbox for Norm Megill

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Theorem cdleme0nex 35487

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 35408- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 34541, 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 1025	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simp12 1027	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
3	1, 2	jca 532	. . 3 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
4		simp3l 1024	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
5		simp13 1028	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
6		ralnex 2913	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
7	5, 6	sylibr 212	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
8		breq1 4456	. . . . . . . . . 10
9	8	notbid 294	. . . . . . . . 9
10		oveq2 6303	. . . . . . . . . 10 $.\/$ $.\/$
11		oveq2 6303	. . . . . . . . . 10 $.\/$ $.\/$
12	10, 11	eqeq12d 2489	. . . . . . . . 9 $.\/$ $.\/$ $.\/$ $.\/$
13	9, 12	anbi12d 710	. . . . . . . 8 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
14	13	notbid 294	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
15	14	rspcva 3217	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
16	4, 7, 15	syl2anc 661	. . . . 5 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
17		simp11 1026	. . . . . . . 8 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
18		hlcvl 34557	. . . . . . . 8
19	17, 18	syl 16	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simp21 1029	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simp22 1030	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simp23 1031	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
23		cdleme0nex.a	. . . . . . . 8
24		cdleme0nex.l	. . . . . . . 8
25		cdleme0nex.j	. . . . . . . 8 $.\/$
26	23, 24, 25	cvlsupr2 34541	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
27	19, 20, 21, 4, 22, 26	syl131anc 1241	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
28	27	anbi2d 703	. . . . 5 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$
29	16, 28	mtbid 300	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
30		ianor 488	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $/\$ $\/$ $/\$ $.\/$
31		df-3an 975	. . . . . . . 8 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
32	31	anbi2i 694	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
33		an12 795	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
34	32, 33	bitri 249	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
35	34	notbii 296	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
36		pm4.62 419	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $\/$ $/\$ $.\/$
37	30, 35, 36	3bitr4ri 278	. . . 4 $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
38	29, 37	sylibr 212	. . 3 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
39	3, 38	mt2d 117	. 2 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		neanior 2792	. . 3 $/\$ $\/$
41	40	con2bii 332	. 2 $\/$ $/\$
42	39, 41	sylibr 212	1 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $\/$ wo 368 $/\$ wa 369 $/\$ w3a 973

wceq 1379

wcel 1767

wne 2662

wral 2817

wrex 2818 class class class wbr 4453

cfv 5594 (class class class)co 6295

cple 14579

cjn 15448

catm 34461

clc 34463

chlt 34548

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1601 ax-4 1612 ax-5 1680 ax-6 1719 ax-7 1739 ax-8 1769 ax-9 1771 ax-10 1786 ax-11 1791 ax-12 1803 ax-13 1968 ax-ext 2445 ax-rep 4564 ax-sep 4574 ax-nul 4582 ax-pow 4631 ax-pr 4692 ax-un 6587

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1382 df-ex 1597 df-nf 1600 df-sb 1712 df-eu 2279 df-mo 2280 df-clab 2453 df-cleq 2459 df-clel 2462 df-nfc 2617 df-ne 2664 df-ral 2822 df-rex 2823 df-reu 2824 df-rab 2826 df-v 3120 df-sbc 3337 df-csb 3441 df-dif 3484 df-un 3486 df-in 3488 df-ss 3495 df-nul 3791 df-if 3946 df-pw 4018 df-sn 4034 df-pr 4036 df-op 4040 df-uni 4252 df-iun 4333 df-br 4454 df-opab 4512 df-mpt 4513 df-id 4801 df-xp 5011 df-rel 5012 df-cnv 5013 df-co 5014 df-dm 5015 df-rn 5016 df-res 5017 df-ima 5018 df-iota 5557 df-fun 5596 df-fn 5597 df-f 5598 df-f1 5599 df-fo 5600 df-f1o 5601 df-fv 5602 df-riota 6256 df-ov 6298 df-oprab 6299 df-poset 15450 df-plt 15462 df-lub 15478 df-glb 15479 df-join 15480 df-meet 15481 df-p0 15543 df-lat 15550 df-covers 34464 df-ats 34465 df-atl 34496 df-cvlat 34520 df-hlat 34549

This theorem is referenced by: cdleme18c 35490 cdleme18d 35492 cdlemg17b 35859 cdlemg17h 35865

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