Theorem lnnat 34713

Description: A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)

Hypotheses

Ref	Expression
lnnat.j	|- .\/ = ( join ` K )
lnnat.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
lnnat	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> -. ( P .\/ Q ) e. A ) )

Proof of Theorem lnnat

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> K e. HL )
2		simpl2 1065	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> P e. A )
3		eqid 2622	. . . . . . 7 |- ( 0. ` K ) = ( 0. ` K )
4		eqid 2622	. . . . . . 7 |- ( <o ` K ) = ( <o ` K )
5		lnnat.a	. . . . . . 7 |- A = ( Atoms ` K )
6	3, 4, 5	atcvr0 34575	. . . . . 6 |- ( ( K e. HL /\ P e. A ) -> ( 0. ` K ) ( <o ` K ) P )
7	1, 2, 6	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( 0. ` K ) ( <o ` K ) P )
8		lnnat.j	. . . . . . 7 |- .\/ = ( join ` K )
9	8, 4, 5	atcvr1 34703	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P ( <o ` K ) ( P .\/ Q ) ) )
10	9	biimpa 501	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> P ( <o ` K ) ( P .\/ Q ) )
11		hlop 34649	. . . . . . 7 |- ( K e. HL -> K e. OP )
12		eqid 2622	. . . . . . . 8 |- ( Base ` K ) = ( Base ` K )
13	12, 3	op0cl 34471	. . . . . . 7 |- ( K e. OP -> ( 0. ` K ) e. ( Base ` K ) )
14	1, 11, 13	3syl 18	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( 0. ` K ) e. ( Base ` K ) )
15	12, 5	atbase 34576	. . . . . . 7 |- ( P e. A -> P e. ( Base ` K ) )
16	2, 15	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> P e. ( Base ` K ) )
17		hllat 34650	. . . . . . . 8 |- ( K e. HL -> K e. Lat )
18	1, 17	syl 17	. . . . . . 7 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> K e. Lat )
19		simpl3 1066	. . . . . . . 8 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> Q e. A )
20	12, 5	atbase 34576	. . . . . . . 8 |- ( Q e. A -> Q e. ( Base ` K ) )
21	19, 20	syl 17	. . . . . . 7 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> Q e. ( Base ` K ) )
22	12, 8	latjcl 17051	. . . . . . 7 |- ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
23	18, 16, 21, 22	syl3anc 1326	. . . . . 6 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) )
24	12, 4	cvrntr 34711	. . . . . 6 |- ( ( K e. HL /\ ( ( 0. ` K ) e. ( Base ` K ) /\ P e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( ( 0. ` K ) ( <o ` K ) P /\ P ( <o ` K ) ( P .\/ Q ) ) -> -. ( 0. ` K ) ( <o ` K ) ( P .\/ Q ) ) )
25	1, 14, 16, 23, 24	syl13anc 1328	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( ( ( 0. ` K ) ( <o ` K ) P /\ P ( <o ` K ) ( P .\/ Q ) ) -> -. ( 0. ` K ) ( <o ` K ) ( P .\/ Q ) ) )
26	7, 10, 25	mp2and 715	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> -. ( 0. ` K ) ( <o ` K ) ( P .\/ Q ) )
27		simpll1 1100	. . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) /\ ( P .\/ Q ) e. A ) -> K e. HL )
28	3, 4, 5	atcvr0 34575	. . . . 5 |- ( ( K e. HL /\ ( P .\/ Q ) e. A ) -> ( 0. ` K ) ( <o ` K ) ( P .\/ Q ) )
29	27, 28	sylancom 701	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) /\ ( P .\/ Q ) e. A ) -> ( 0. ` K ) ( <o ` K ) ( P .\/ Q ) )
30	26, 29	mtand 691	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> -. ( P .\/ Q ) e. A )
31	30	ex 450	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q -> -. ( P .\/ Q ) e. A ) )
32	8, 5	hlatjidm 34655	. . . . . 6 |- ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
33	32	3adant3 1081	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ P ) = P )
34		simp2 1062	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> P e. A )
35	33, 34	eqeltrd 2701	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ P ) e. A )
36		oveq2 6658	. . . . 5 |- ( P = Q -> ( P .\/ P ) = ( P .\/ Q ) )
37	36	eleq1d 2686	. . . 4 |- ( P = Q -> ( ( P .\/ P ) e. A <-> ( P .\/ Q ) e. A ) )
38	35, 37	syl5ibcom 235	. . 3 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P = Q -> ( P .\/ Q ) e. A ) )
39	38	necon3bd 2808	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( -. ( P .\/ Q ) e. A -> P =/= Q ) )
40	31, 39	impbid 202	1 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> -. ( P .\/ Q ) e. A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   0.cp0 17037   Latclat 17045   OPcops 34459   <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:  2atjlej  34765  cdleme11h  35553