Theorem cdlemb2 33988

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 1012	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
2		simp2ll 1055	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
3		simp2rl 1057	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp1r 1013	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. H )
5		eqid 2451	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		cdlemb2.h	. . . 4 |- H = ( LHyp ` K )
7	5, 6	lhpbase 33945	. . 3 |- ( W e. H -> W e. ( Base ` K ) )
8	4, 7	syl 16	. 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 990	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
10		eqid 2451	. . . 4 |- ( 1. ` K ) = ( 1. ` K )
11		eqid 2451	. . . 4 |- ( <o ` K ) = ( <o ` K )
12	10, 11, 6	lhp1cvr 33946	. . 3 |- ( ( K e. HL /\ W e. H ) -> W ( <o ` K ) ( 1. ` K ) )
13	12	3ad2ant1 1009	. 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 1056	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
15		simp2rr 1058	. 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 33741	. 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 1249	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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2642   E.wrex 2794   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   Basecbs 14273   lecple 14344   joincjn 15213   1.cp1 15307   <o ccvr 33210   Atomscatm 33211   HLchlt 33298   LHypclh 33931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-poset 15215  df-plt 15227  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-p0 15308  df-p1 15309  df-lat 15315  df-clat 15377  df-oposet 33124  df-ol 33126  df-oml 33127  df-covers 33214  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299  df-lhyp 33935

This theorem is referenced by:  cdlemd4  34148  cdlemd9  34153  cdleme25a  34300  cdleme25c  34302  cdleme25dN  34303  cdleme26ee  34307  cdlemefs32sn1aw  34361  cdleme43fsv1snlem  34367  cdleme41sn3a  34380  cdleme40m  34414  cdleme40n  34415  cdleme17d3  34443  cdlemeg46gfre  34479  cdleme50trn2  34498  cdlemb3  34553