Theorem cdlemb 33359

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 1038	. . 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 1039	. . . 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 1040	. . . 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 1034	. . . 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 1035	. . . 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 1044	. . . 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 1045	. . . 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 2451	. . . . 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 33041	. . . 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 1291	. . 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 32929	. . . . . 6 |- ( K e. HL -> K e. Lat )
18	1, 17	syl 17	. . . . 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 32855	. . . . . . 7 |- ( P e. A -> P e. B )
20	2, 19	syl 17	. . . . . 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 32855	. . . . . . 7 |- ( Q e. A -> Q e. B )
22	3, 21	syl 17	. . . . . 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 16297	. . . . . 6 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
24	18, 20, 22, 23	syl3anc 1268	. . . . 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 16323	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ X e. B ) -> ( ( P .\/ Q ) ( meet ` K ) X ) .<_ X )
26	18, 24, 4, 25	syl3anc 1268	. . . 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 2451	. . . . 5 |- ( lt ` K ) = ( lt ` K )
28	8, 9, 27, 12, 13, 14	1cvratlt 33039	. . . 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 1276	. . 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 33004	. . 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 1271	. 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 1083	. . . 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 1084	. . . 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 764	. . . 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 1090	. . . . 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 775	. . . . . . . 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 1061	. . . . . . . . 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 16207	. . . . . . . . 9 |- ( ( K e. HL /\ u e. A /\ X e. B ) -> ( u ( lt ` K ) X -> u .<_ X ) )
39	32, 34, 37, 38	syl3anc 1268	. . . . . . . 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 4405	. . . . . . 7 |- ( P = u -> ( P .<_ X <-> u .<_ X ) )
42	40, 41	syl5ibrcom 226	. . . . . 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 2638	. . . . 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 32951	. . . 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 1271	. . 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 2451	. . . . . . . 8 |- ( ( P .\/ Q ) ( meet ` K ) X ) = ( ( P .\/ Q ) ( meet ` K ) X )
48	8, 9, 10, 12, 13, 14, 27, 11, 47	cdlemblem 33358	. . . . . . 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 1207	. . . . . 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 611	. . . . 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 431	. . . 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 2859	. . 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 2883	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 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Basecbs 15121   lecple 15197   ltcplt 16186   joincjn 16189   meetcmee 16190   1.cp1 16284   Latclat 16291   <o ccvr 32828   Atomscatm 32829   HLchlt 32916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32742  df-ol 32744  df-oml 32745  df-covers 32832  df-ats 32833  df-atl 32864  df-cvlat 32888  df-hlat 32917

This theorem is referenced by:  cdlemb2  33606