Theorem cdlemb2 29480

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 984	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
2		simp2ll 1027	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
3		simp2rl 1029	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp1r 985	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. H )
5		eqid 2258	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		cdlemb2.h	. . . 4 |- H = ( LHyp ` K )
7	5, 6	lhpbase 29437	. . 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 962	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
10		eqid 2258	. . . 4 |- ( 1. ` K ) = ( 1. ` K )
11		eqid 2258	. . . 4 |- ( <o ` K ) = ( <o ` K )
12	10, 11, 6	lhp1cvr 29438	. . 3 |- ( ( K e. HL /\ W e. H ) -> W ( <o ` K ) ( 1. ` K ) )
13	12	3ad2ant1 981	. 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 1028	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
15		simp2rr 1030	. 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 29233	. 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 1217	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 5   -> wi 6   /\ wa 360   /\ w3a 939   = wceq 1619   e. wcel 1621   =/= wne 2421   E.wrex 2519   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   joincjn 14041   1.cp1 14107   <o ccvr 28702   Atomscatm 28703   HLchlt 28790   LHypclh 29423

This theorem is referenced by:  cdlemd4  29640  cdlemd9  29645  cdleme25a  29792  cdleme25c  29794  cdleme25dN  29795  cdleme26ee  29799  cdlemefs32sn1aw  29853  cdleme43fsv1snlem  29859  cdleme41sn3a  29872  cdleme40m  29906  cdleme40n  29907  cdleme17d3  29935  cdlemeg46gfre  29971  cdleme50trn2  29990  cdlemb3  30045

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-lhyp 29427