Theorem cdlemb2 36162

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 1018	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
2		simp2ll 1061	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
3		simp2rl 1063	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp1r 1019	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. H )
5		eqid 2454	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		cdlemb2.h	. . . 4 |- H = ( LHyp ` K )
7	5, 6	lhpbase 36119	. . 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 996	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
10		eqid 2454	. . . 4 |- ( 1. ` K ) = ( 1. ` K )
11		eqid 2454	. . . 4 |- ( <o ` K ) = ( <o ` K )
12	10, 11, 6	lhp1cvr 36120	. . 3 |- ( ( K e. HL /\ W e. H ) -> W ( <o ` K ) ( 1. ` K ) )
13	12	3ad2ant1 1015	. 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 1062	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
15		simp2rr 1064	. 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 35915	. 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 1256	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   1.cp1 15867   <o ccvr 35384   Atomscatm 35385   HLchlt 35472   LHypclh 36105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-lhyp 36109

This theorem is referenced by:  cdlemd4  36323  cdlemd9  36328  cdleme25a  36476  cdleme25c  36478  cdleme25dN  36479  cdleme26ee  36483  cdlemefs32sn1aw  36537  cdleme43fsv1snlem  36543  cdleme41sn3a  36556  cdleme40m  36590  cdleme40n  36591  cdleme17d3  36619  cdlemeg46gfre  36655  cdleme50trn2  36674  cdlemb3  36729