Theorem cdlemb2 29509

Description: Given two atoms not under the fiducial (reference) co-atom W, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)

Hypotheses

Ref	Expression
cdlemb2.l	|- .<_ = ( le ` K )
cdlemb2.j	|- .\/ = ( join ` K )
cdlemb2.a	|- A = ( Atoms ` K )
cdlemb2.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
cdlemb2	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )

Distinct variable groups:   A, r   .\/ , r   K, r   .<_ , r   P, r   Q, r   W, r

Allowed substitution hint:   H( r)

Proof of Theorem cdlemb2

Step	Hyp	Ref	Expression
1		simp1l 979	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
2		simp2ll 1022	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
3		simp2rl 1024	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp1r 980	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. H )
5		eqid 2284	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		cdlemb2.h	. . . 4 |- H = ( LHyp ` K )
7	5, 6	lhpbase 29466	. . 3 |- ( W e. H -> W e. ( Base ` K ) )
8	4, 7	syl 15	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. ( Base ` K ) )
9		simp3 957	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
10		eqid 2284	. . . 4 |- ( 1. ` K ) = ( 1. ` K )
11		eqid 2284	. . . 4 |- ( <o ` K ) = ( <o ` K )
12	10, 11, 6	lhp1cvr 29467	. . 3 |- ( ( K e. HL /\ W e. H ) -> W ( <o ` K ) ( 1. ` K ) )
13	12	3ad2ant1 976	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W ( <o ` K ) ( 1. ` K ) )
14		simp2lr 1023	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
15		simp2rr 1025	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W )
16		cdlemb2.l	. . 3 |- .<_ = ( le ` K )
17		cdlemb2.j	. . 3 |- .\/ = ( join ` K )
18		cdlemb2.a	. . 3 |- A = ( Atoms ` K )
19	5, 16, 17, 10, 11, 18	cdlemb 29262	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( W e. ( Base ` K ) /\ P =/= Q ) /\ ( W ( <o ` K ) ( 1. ` K ) /\ -. P .<_ W /\ -. Q .<_ W ) ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )
20	1, 2, 3, 8, 9, 13, 14, 15, 19	syl323anc 1212	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1623   e. wcel 1685   =/= wne 2447   E.wrex 2545   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   joincjn 14074   1.cp1 14140   <o ccvr 28731   Atomscatm 28732   HLchlt 28819   LHypclh 29452

This theorem is referenced by:  cdlemd4  29669  cdlemd9  29674  cdleme25a  29821  cdleme25c  29823  cdleme25dN  29824  cdleme26ee  29828  cdlemefs32sn1aw  29882  cdleme43fsv1snlem  29888  cdleme41sn3a  29901  cdleme40m  29935  cdleme40n  29936  cdleme17d3  29964  cdlemeg46gfre  30000  cdleme50trn2  30019  cdlemb3  30074

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-lhyp 29456