Theorem cdlemg35 35509

Description: TODO: Fix comment. TODO: should we have a more general version of hlsupr 34182 to avoid the =/= conditions? (Contributed by NM, 31-May-2013.)

Hypotheses

Ref	Expression
cdlemg35.l	|- .<_ = ( le ` K )
cdlemg35.j	|- .\/ = ( join ` K )
cdlemg35.m	|- ./\ = ( meet ` K )
cdlemg35.a	|- A = ( Atoms ` K )
cdlemg35.h	|- H = ( LHyp ` K )
cdlemg35.t	|- T = ( ( LTrn ` K ) ` W )
cdlemg35.r	|- R = ( ( trL ` K ) ` W )

Assertion

Ref	Expression
cdlemg35	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) )

Distinct variable groups:   v, A   v, F   v, G   v, H   v, K   v, .<_   v, P   v, R   v, T   v, W

Allowed substitution hints:   .\/ ( v)   ./\ ( v)

Proof of Theorem cdlemg35

Step	Hyp	Ref	Expression
1		simp1l 1020	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. HL )
2		simp1 996	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) )
3		simp21 1029	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) )
4		simp22 1030	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> F e. T )
5		simp31 1032	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` P ) =/= P )
6		cdlemg35.l	. . . . 5 |- .<_ = ( le ` K )
7		cdlemg35.a	. . . . 5 |- A = ( Atoms ` K )
8		cdlemg35.h	. . . . 5 |- H = ( LHyp ` K )
9		cdlemg35.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
10		cdlemg35.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
11	6, 7, 8, 9, 10	trlat 34965	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
12	2, 3, 4, 5, 11	syl112anc 1232	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) e. A )
13		simp23 1031	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> G e. T )
14		simp32 1033	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G ` P ) =/= P )
15	6, 7, 8, 9, 10	trlat 34965	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( G e. T /\ ( G ` P ) =/= P ) ) -> ( R ` G ) e. A )
16	2, 3, 13, 14, 15	syl112anc 1232	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) e. A )
17		simp33 1034	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` G ) )
18		cdlemg35.j	. . . 4 |- .\/ = ( join ` K )
19	6, 18, 7	hlsupr 34182	. . 3 |- ( ( ( K e. HL /\ ( R ` F ) e. A /\ ( R ` G ) e. A ) /\ ( R ` F ) =/= ( R ` G ) ) -> E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) )
20	1, 12, 16, 17, 19	syl31anc 1231	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) )
21		eqid 2467	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
22		simp11l 1107	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> K e. HL )
23		hllat 34160	. . . . . . 7 |- ( K e. HL -> K e. Lat )
24	22, 23	syl 16	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> K e. Lat )
25	21, 7	atbase 34086	. . . . . . 7 |- ( v e. A -> v e. ( Base ` K ) )
26	25	3ad2ant2 1018	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v e. ( Base ` K ) )
27		simp11 1026	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
28		simp122 1129	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> F e. T )
29	21, 8, 9, 10	trlcl 34960	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
30	27, 28, 29	syl2anc 661	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` F ) e. ( Base ` K ) )
31		simp123 1130	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> G e. T )
32	21, 8, 9, 10	trlcl 34960	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
33	27, 31, 32	syl2anc 661	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` G ) e. ( Base ` K ) )
34	21, 18	latjcl 15534	. . . . . . 7 |- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) )
35	24, 30, 33, 34	syl3anc 1228	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) )
36		simp11r 1108	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> W e. H )
37	21, 8	lhpbase 34794	. . . . . . 7 |- ( W e. H -> W e. ( Base ` K ) )
38	36, 37	syl 16	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> W e. ( Base ` K ) )
39		simp33 1034	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v .<_ ( ( R ` F ) .\/ ( R ` G ) ) )
40	6, 8, 9, 10	trlle 34980	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
41	27, 28, 40	syl2anc 661	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` F ) .<_ W )
42	6, 8, 9, 10	trlle 34980	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
43	27, 31, 42	syl2anc 661	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` G ) .<_ W )
44	21, 6, 18	latjle12 15545	. . . . . . . 8 |- ( ( K e. Lat /\ ( ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` G ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W ) )
45	24, 30, 33, 38, 44	syl13anc 1230	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` G ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W ) )
46	41, 43, 45	mpbi2and 919	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W )
47	21, 6, 24, 26, 35, 38, 39, 46	lattrd 15541	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v .<_ W )
48		simp31 1032	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v =/= ( R ` F ) )
49		simp32 1033	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v =/= ( R ` G ) )
50	47, 48, 49	jca32 535	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) )
51	50	3expia 1198	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A ) -> ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) -> ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) )
52	51	reximdva 2938	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) )
53	20, 52	mpd 15	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   Latclat 15528   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955

This theorem is referenced by:  cdlemg36  35510