Theorem cdlemb2 33518

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 1029	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
2		simp2ll 1072	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
3		simp2rl 1074	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp1r 1030	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. H )
5		eqid 2428	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		cdlemb2.h	. . . 4 |- H = ( LHyp ` K )
7	5, 6	lhpbase 33475	. . 3 |- ( W e. H -> W e. ( Base ` K ) )
8	4, 7	syl 17	. 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 1007	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
10		eqid 2428	. . . 4 |- ( 1. ` K ) = ( 1. ` K )
11		eqid 2428	. . . 4 |- ( <o ` K ) = ( <o ` K )
12	10, 11, 6	lhp1cvr 33476	. . 3 |- ( ( K e. HL /\ W e. H ) -> W ( <o ` K ) ( 1. ` K ) )
13	12	3ad2ant1 1026	. 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 1073	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
15		simp2rr 1075	. 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 33271	. 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 1294	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 370   /\ w3a 982   = wceq 1437   e. wcel 1872   =/= wne 2599   E.wrex 2715   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   Basecbs 15064   lecple 15140   joincjn 16132   1.cp1 16227   <o ccvr 32740   Atomscatm 32741   HLchlt 32828   LHypclh 33461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-p1 16229  df-lat 16235  df-clat 16297  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-lhyp 33465

This theorem is referenced by:  cdlemd4  33679  cdlemd9  33684  cdleme25a  33832  cdleme25c  33834  cdleme25dN  33835  cdleme26ee  33839  cdlemefs32sn1aw  33893  cdleme43fsv1snlem  33899  cdleme41sn3a  33912  cdleme40m  33946  cdleme40n  33947  cdleme17d3  33975  cdlemeg46gfre  34011  cdleme50trn2  34030  cdlemb3  34085