Theorem 2atlt 34725

Description: Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)

Hypotheses

Ref	Expression
2atomslt.b	|- B = ( Base ` K )
2atomslt.s	|- .< = ( lt ` K )
2atomslt.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
2atlt	|- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) -> E. q e. A ( q =/= P /\ q .< X ) )

Distinct variable groups:   A, q   B, q   K, q   P, q   .< , q   X, q

Proof of Theorem 2atlt

Step	Hyp	Ref	Expression
1		2atomslt.b	. . . 4 |- B = ( Base ` K )
2		2atomslt.a	. . . 4 |- A = ( Atoms ` K )
3	1, 2	atbase 34576	. . 3 |- ( P e. A -> P e. B )
4		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
5		2atomslt.s	. . . 4 |- .< = ( lt ` K )
6		eqid 2622	. . . 4 |- ( join ` K ) = ( join ` K )
7	1, 4, 5, 6, 2	hlrelat 34688	. . 3 |- ( ( ( K e. HL /\ P e. B /\ X e. B ) /\ P .< X ) -> E. q e. A ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) )
8	3, 7	syl3anl2 1375	. 2 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) -> E. q e. A ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) )
9		simp3l 1089	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> P .< ( P ( join ` K ) q ) )
10		simp1l1 1154	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> K e. HL )
11		simp1l2 1155	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> P e. A )
12		simp2 1062	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> q e. A )
13		eqid 2622	. . . . . . . . . 10 |- ( <o ` K ) = ( <o ` K )
14	5, 6, 2, 13	atltcvr 34721	. . . . . . . . 9 |- ( ( K e. HL /\ ( P e. A /\ P e. A /\ q e. A ) ) -> ( P .< ( P ( join ` K ) q ) <-> P ( <o ` K ) ( P ( join ` K ) q ) ) )
15	10, 11, 11, 12, 14	syl13anc 1328	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( P .< ( P ( join ` K ) q ) <-> P ( <o ` K ) ( P ( join ` K ) q ) ) )
16	9, 15	mpbid 222	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> P ( <o ` K ) ( P ( join ` K ) q ) )
17	6, 13, 2	atcvr1 34703	. . . . . . . 8 |- ( ( K e. HL /\ P e. A /\ q e. A ) -> ( P =/= q <-> P ( <o ` K ) ( P ( join ` K ) q ) ) )
18	10, 11, 12, 17	syl3anc 1326	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( P =/= q <-> P ( <o ` K ) ( P ( join ` K ) q ) ) )
19	16, 18	mpbird 247	. . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> P =/= q )
20	19	necomd 2849	. . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> q =/= P )
21	5, 6, 2	atlt 34723	. . . . . . . . 9 |- ( ( K e. HL /\ q e. A /\ P e. A ) -> ( q .< ( q ( join ` K ) P ) <-> q =/= P ) )
22	10, 12, 11, 21	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( q .< ( q ( join ` K ) P ) <-> q =/= P ) )
23	20, 22	mpbird 247	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> q .< ( q ( join ` K ) P ) )
24		hllat 34650	. . . . . . . . 9 |- ( K e. HL -> K e. Lat )
25	10, 24	syl 17	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> K e. Lat )
26	11, 3	syl 17	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> P e. B )
27	1, 2	atbase 34576	. . . . . . . . 9 |- ( q e. A -> q e. B )
28	27	3ad2ant2 1083	. . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> q e. B )
29	1, 6	latjcom 17059	. . . . . . . 8 |- ( ( K e. Lat /\ P e. B /\ q e. B ) -> ( P ( join ` K ) q ) = ( q ( join ` K ) P ) )
30	25, 26, 28, 29	syl3anc 1326	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( P ( join ` K ) q ) = ( q ( join ` K ) P ) )
31	23, 30	breqtrrd 4681	. . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> q .< ( P ( join ` K ) q ) )
32		simp3r 1090	. . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( P ( join ` K ) q ) ( le ` K ) X )
33		hlpos 34652	. . . . . . . 8 |- ( K e. HL -> K e. Poset )
34	10, 33	syl 17	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> K e. Poset )
35	1, 6	latjcl 17051	. . . . . . . 8 |- ( ( K e. Lat /\ P e. B /\ q e. B ) -> ( P ( join ` K ) q ) e. B )
36	25, 26, 28, 35	syl3anc 1326	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( P ( join ` K ) q ) e. B )
37		simp1l3 1156	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> X e. B )
38	1, 4, 5	pltletr 16971	. . . . . . 7 |- ( ( K e. Poset /\ ( q e. B /\ ( P ( join ` K ) q ) e. B /\ X e. B ) ) -> ( ( q .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) -> q .< X ) )
39	34, 28, 36, 37, 38	syl13anc 1328	. . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( ( q .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) -> q .< X ) )
40	31, 32, 39	mp2and 715	. . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> q .< X )
41	20, 40	jca 554	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) /\ q e. A /\ ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) ) -> ( q =/= P /\ q .< X ) )
42	41	3exp 1264	. . 3 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) -> ( q e. A -> ( ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) -> ( q =/= P /\ q .< X ) ) ) )
43	42	reximdvai 3015	. 2 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) -> ( E. q e. A ( P .< ( P ( join ` K ) q ) /\ ( P ( join ` K ) q ) ( le ` K ) X ) -> E. q e. A ( q =/= P /\ q .< X ) ) )
44	8, 43	mpd 15	1 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ P .< X ) -> E. q e. A ( q =/= P /\ q .< X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   Posetcpo 16940   ltcplt 16941   joincjn 16944   Latclat 17045   <o ccvr 34549   Atomscatm 34550   HLchlt 34637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  cdlemb  35080  lhpexle1  35294