Theorem cdlemb 33443

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 1018	. . 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 1019	. . . 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 1020	. . . 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 1014	. . . 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 1015	. . . 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 1024	. . . 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 1025	. . . 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 2443	. . . . 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 33125	. . . 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 1241	. . 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 33013	. . . . . 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 32939	. . . . . . 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 32939	. . . . . . 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 15226	. . . . . 6 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
24	18, 20, 22, 23	syl3anc 1218	. . . . 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 15252	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ X e. B ) -> ( ( P .\/ Q ) ( meet ` K ) X ) .<_ X )
26	18, 24, 4, 25	syl3anc 1218	. . . 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 2443	. . . . 5 |- ( lt ` K ) = ( lt ` K )
28	8, 9, 27, 12, 13, 14	1cvratlt 33123	. . . 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 1226	. . 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 33088	. . 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 1221	. 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 1063	. . . 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 1064	. . . 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 1070	. . . . 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 1041	. . . . . . . . 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 15136	. . . . . . . . 9 |- ( ( K e. HL /\ u e. A /\ X e. B ) -> ( u ( lt ` K ) X -> u .<_ X ) )
39	32, 34, 37, 38	syl3anc 1218	. . . . . . . 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 4300	. . . . . . 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 2650	. . . . 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 33035	. . . 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 1221	. . 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 2443	. . . . . . . 8 |- ( ( P .\/ Q ) ( meet ` K ) X ) = ( ( P .\/ Q ) ( meet ` K ) X )
48	8, 9, 10, 12, 13, 14, 27, 11, 47	cdlemblem 33442	. . . . . . 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 1186	. . . . . 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 2831	. . 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 2850	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 965   = wceq 1369   e. wcel 1756   =/= wne 2611   E.wrex 2721   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   ltcplt 15116   joincjn 15119   meetcmee 15120   1.cp1 15213   Latclat 15220   <o ccvr 32912   Atomscatm 32913   HLchlt 33000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001

This theorem is referenced by:  cdlemb2  33690