Theorem cdlemb 30276

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 987	. . 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 988	. . . 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 989	. . . 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 983	. . . 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 984	. . . 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 993	. . . 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 994	. . . 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 2404	. . . . 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 29958	. . . 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 1207	. . 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 29846	. . . . . 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 29772	. . . . . . 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 29772	. . . . . . 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 14434	. . . . . 6 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
24	18, 20, 22, 23	syl3anc 1184	. . . . 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 14461	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ X e. B ) -> ( ( P .\/ Q ) ( meet ` K ) X ) .<_ X )
26	18, 24, 4, 25	syl3anc 1184	. . . 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 2404	. . . . 5 |- ( lt ` K ) = ( lt ` K )
28	8, 9, 27, 12, 13, 14	1cvratlt 29956	. . . 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 1192	. . 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 29921	. . 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 1187	. 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 1032	. . . 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 1033	. . . 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 733	. . . 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 1039	. . . . 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 742	. . . . . . . 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 1010	. . . . . . . . 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 14373	. . . . . . . . 9 |- ( ( K e. HL /\ u e. A /\ X e. B ) -> ( u ( lt ` K ) X -> u .<_ X ) )
39	32, 34, 37, 38	syl3anc 1184	. . . . . . . 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 4175	. . . . . . 7 |- ( P = u -> ( P .<_ X <-> u .<_ X ) )
42	40, 41	syl5ibrcom 214	. . . . . 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 2604	. . . . 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 29868	. . . 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 1187	. . 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 2404	. . . . . . . 8 |- ( ( P .\/ Q ) ( meet ` K ) X ) = ( ( P .\/ Q ) ( meet ` K ) X )
48	8, 9, 10, 12, 13, 14, 27, 11, 47	cdlemblem 30275	. . . . . . 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 1152	. . . . . 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 590	. . . . 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 419	. . . 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 2776	. . 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 2794	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 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   ltcplt 14353   joincjn 14356   meetcmee 14357   1.cp1 14422   Latclat 14429   <o ccvr 29745   Atomscatm 29746   HLchlt 29833

This theorem is referenced by:  cdlemb2  30523

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834