Theorem cdlemb 34590

Description: Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)

Hypotheses

Ref	Expression
cdlemb.b	|- B = ( Base ` K )
cdlemb.l	|- .<_ = ( le ` K )
cdlemb.j	|- .\/ = ( join ` K )
cdlemb.u	|- .1. = ( 1. ` K )
cdlemb.c	|- C = ( <o ` K )
cdlemb.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
cdlemb	|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> E. r e. A ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) )

Distinct variable groups:   A, r   B, r   C, r   .\/ , r   K, r   .<_ , r   P, r   Q, r   .1. , r   X, r

Proof of Theorem cdlemb

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11 1026	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> K e. HL )
2		simp12 1027	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> P e. A )
3		simp13 1028	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> Q e. A )
4		simp2l 1022	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> X e. B )
5		simp2r 1023	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> P =/= Q )
6		simp31 1032	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> X C .1. )
7		simp32 1033	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> -. P .<_ X )
8		cdlemb.b	. . . . 5 |- B = ( Base ` K )
9		cdlemb.l	. . . . 5 |- .<_ = ( le ` K )
10		cdlemb.j	. . . . 5 |- .\/ = ( join ` K )
11		eqid 2467	. . . . 5 |- ( meet ` K ) = ( meet ` K )
12		cdlemb.u	. . . . 5 |- .1. = ( 1. ` K )
13		cdlemb.c	. . . . 5 |- C = ( <o ` K )
14		cdlemb.a	. . . . 5 |- A = ( Atoms ` K )
15	8, 9, 10, 11, 12, 13, 14	1cvrat 34272	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ( meet ` K ) X ) e. A )
16	1, 2, 3, 4, 5, 6, 7, 15	syl133anc 1251	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ( meet ` K ) X ) e. A )
17		hllat 34160	. . . . . 6 |- ( K e. HL -> K e. Lat )
18	1, 17	syl 16	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> K e. Lat )
19	8, 14	atbase 34086	. . . . . . 7 |- ( P e. A -> P e. B )
20	2, 19	syl 16	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> P e. B )
21	8, 14	atbase 34086	. . . . . . 7 |- ( Q e. A -> Q e. B )
22	3, 21	syl 16	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> Q e. B )
23	8, 10	latjcl 15531	. . . . . 6 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
24	18, 20, 22, 23	syl3anc 1228	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> ( P .\/ Q ) e. B )
25	8, 9, 11	latmle2 15557	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ X e. B ) -> ( ( P .\/ Q ) ( meet ` K ) X ) .<_ X )
26	18, 24, 4, 25	syl3anc 1228	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ( meet ` K ) X ) .<_ X )
27		eqid 2467	. . . . 5 |- ( lt ` K ) = ( lt ` K )
28	8, 9, 27, 12, 13, 14	1cvratlt 34270	. . . 4 |- ( ( ( K e. HL /\ ( ( P .\/ Q ) ( meet ` K ) X ) e. A /\ X e. B ) /\ ( X C .1. /\ ( ( P .\/ Q ) ( meet ` K ) X ) .<_ X ) ) -> ( ( P .\/ Q ) ( meet ` K ) X ) ( lt ` K ) X )
29	1, 16, 4, 6, 26, 28	syl32anc 1236	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ( meet ` K ) X ) ( lt ` K ) X )
30	8, 27, 14	2atlt 34235	. . 3 |- ( ( ( K e. HL /\ ( ( P .\/ Q ) ( meet ` K ) X ) e. A /\ X e. B ) /\ ( ( P .\/ Q ) ( meet ` K ) X ) ( lt ` K ) X ) -> E. u e. A ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) )
31	1, 16, 4, 29, 30	syl31anc 1231	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> E. u e. A ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) )
32		simpl11 1071	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> K e. HL )
33		simpl12 1072	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> P e. A )
34		simprl 755	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> u e. A )
35		simpl32 1078	. . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> -. P .<_ X )
36		simprrr 764	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> u ( lt ` K ) X )
37		simpl2l 1049	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> X e. B )
38	9, 27	pltle 15441	. . . . . . . . 9 |- ( ( K e. HL /\ u e. A /\ X e. B ) -> ( u ( lt ` K ) X -> u .<_ X ) )
39	32, 34, 37, 38	syl3anc 1228	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> ( u ( lt ` K ) X -> u .<_ X ) )
40	36, 39	mpd 15	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> u .<_ X )
41		breq1 4450	. . . . . . 7 |- ( P = u -> ( P .<_ X <-> u .<_ X ) )
42	40, 41	syl5ibrcom 222	. . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> ( P = u -> P .<_ X ) )
43	42	necon3bd 2679	. . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> ( -. P .<_ X -> P =/= u ) )
44	35, 43	mpd 15	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> P =/= u )
45	9, 10, 14	hlsupr 34182	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ u e. A ) /\ P =/= u ) -> E. r e. A ( r =/= P /\ r =/= u /\ r .<_ ( P .\/ u ) ) )
46	32, 33, 34, 44, 45	syl31anc 1231	. . 3 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> E. r e. A ( r =/= P /\ r =/= u /\ r .<_ ( P .\/ u ) ) )
47		eqid 2467	. . . . . . . 8 |- ( ( P .\/ Q ) ( meet ` K ) X ) = ( ( P .\/ Q ) ( meet ` K ) X )
48	8, 9, 10, 12, 13, 14, 27, 11, 47	cdlemblem 34589	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) /\ ( r e. A /\ ( r =/= P /\ r =/= u /\ r .<_ ( P .\/ u ) ) ) ) -> ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) )
49	48	3exp 1195	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> ( ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) -> ( ( r e. A /\ ( r =/= P /\ r =/= u /\ r .<_ ( P .\/ u ) ) ) -> ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) ) ) )
50	49	exp4a 606	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> ( ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) -> ( r e. A -> ( ( r =/= P /\ r =/= u /\ r .<_ ( P .\/ u ) ) -> ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) ) ) ) )
51	50	imp 429	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> ( r e. A -> ( ( r =/= P /\ r =/= u /\ r .<_ ( P .\/ u ) ) -> ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) ) ) )
52	51	reximdvai 2935	. . 3 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> ( E. r e. A ( r =/= P /\ r =/= u /\ r .<_ ( P .\/ u ) ) -> E. r e. A ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) ) )
53	46, 52	mpd 15	. 2 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) /\ ( u e. A /\ ( u =/= ( ( P .\/ Q ) ( meet ` K ) X ) /\ u ( lt ` K ) X ) ) ) -> E. r e. A ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) )
54	31, 53	rexlimddv 2959	1 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P =/= Q ) /\ ( X C .1. /\ -. P .<_ X /\ -. Q .<_ X ) ) -> E. r e. A ( -. r .<_ X /\ -. r .<_ ( P .\/ Q ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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 14483   lecple 14555   ltcplt 15421   joincjn 15424   meetcmee 15425   1.cp1 15518   Latclat 15525   <o ccvr 34059   Atomscatm 34060   HLchlt 34147

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-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148

This theorem is referenced by:  cdlemb2  34837