cdleme0nex - Mathbox for Norm Megill

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Theorem cdleme0nex 30772

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 30693- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 29826, 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 986	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simp12 988	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
3	1, 2	jca 519	. . 3 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
4		simp3l 985	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
5		simp13 989	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
6		ralnex 2676	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
7	5, 6	sylibr 204	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
8		breq1 4175	. . . . . . . . . 10
9	8	notbid 286	. . . . . . . . 9
10		oveq2 6048	. . . . . . . . . 10 $.\/$ $.\/$
11		oveq2 6048	. . . . . . . . . 10 $.\/$ $.\/$
12	10, 11	eqeq12d 2418	. . . . . . . . 9 $.\/$ $.\/$ $.\/$ $.\/$
13	9, 12	anbi12d 692	. . . . . . . 8 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
14	13	notbid 286	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
15	14	rspcva 3010	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
16	4, 7, 15	syl2anc 643	. . . . 5 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
17		simp11 987	. . . . . . . 8 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
18		hlcvl 29842	. . . . . . . 8
19	17, 18	syl 16	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simp21 990	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simp22 991	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simp23 992	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
23		cdleme0nex.a	. . . . . . . 8
24		cdleme0nex.l	. . . . . . . 8
25		cdleme0nex.j	. . . . . . . 8 $.\/$
26	23, 24, 25	cvlsupr2 29826	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
27	19, 20, 21, 4, 22, 26	syl131anc 1197	. . . . . 6 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
28	27	anbi2d 685	. . . . 5 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$
29	16, 28	mtbid 292	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
30		ianor 475	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $/\$ $\/$ $/\$ $.\/$
31		df-3an 938	. . . . . . . 8 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
32	31	anbi2i 676	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
33		an12 773	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
34	32, 33	bitri 241	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
35	34	notbii 288	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
36		pm4.62 409	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $\/$ $/\$ $.\/$
37	30, 35, 36	3bitr4ri 270	. . . 4 $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
38	29, 37	sylibr 204	. . 3 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
39	3, 38	mt2d 111	. 2 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		neanior 2652	. . 3 $/\$ $\/$
41	40	con2bii 323	. 2 $\/$ $/\$
42	39, 41	sylibr 204	1 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 177 $\/$ wo 358 $/\$ wa 359 $/\$ w3a 936

wceq 1649

wcel 1721

wne 2567

wral 2666

wrex 2667 class class class wbr 4172

cfv 5413 (class class class)co 6040

cple 13491

cjn 14356

catm 29746

clc 29748

chlt 29833

This theorem is referenced by: cdleme18c 30775 cdleme18d 30777 cdlemg17b 31144 cdlemg17h 31150

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660

This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-op 3783 df-uni 3976 df-iun 4055 df-br 4173 df-opab 4227 df-mpt 4228 df-id 4458 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-1st 6308 df-2nd 6309 df-undef 6502 df-riota 6508 df-poset 14358 df-plt 14370 df-lub 14386 df-join 14388 df-lat 14430 df-covers 29749 df-ats 29750 df-atl 29781 df-cvlat 29805 df-hlat 29834

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