Theorem islln2a 34803

Description: The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)

Hypotheses

Ref	Expression
islln2a.j	|- .\/ = ( join ` K )
islln2a.a	|- A = ( Atoms ` K )
islln2a.n	|- N = ( LLines ` K )

Assertion

Ref	Expression
islln2a	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( P .\/ Q ) e. N <-> P =/= Q ) )

Proof of Theorem islln2a

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . . 6 |- ( P = Q -> ( P .\/ Q ) = ( Q .\/ Q ) )
2		islln2a.j	. . . . . . . 8 |- .\/ = ( join ` K )
3		islln2a.a	. . . . . . . 8 |- A = ( Atoms ` K )
4	2, 3	hlatjidm 34655	. . . . . . 7 |- ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q )
5	4	3adant2 1080	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( Q .\/ Q ) = Q )
6	1, 5	sylan9eqr 2678	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P = Q ) -> ( P .\/ Q ) = Q )
7		islln2a.n	. . . . . . . . . . 11 |- N = ( LLines ` K )
8	3, 7	llnneat 34800	. . . . . . . . . 10 |- ( ( K e. HL /\ Q e. N ) -> -. Q e. A )
9	8	adantlr 751	. . . . . . . . 9 |- ( ( ( K e. HL /\ P e. A ) /\ Q e. N ) -> -. Q e. A )
10	9	ex 450	. . . . . . . 8 |- ( ( K e. HL /\ P e. A ) -> ( Q e. N -> -. Q e. A ) )
11	10	con2d 129	. . . . . . 7 |- ( ( K e. HL /\ P e. A ) -> ( Q e. A -> -. Q e. N ) )
12	11	3impia 1261	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> -. Q e. N )
13	12	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P = Q ) -> -. Q e. N )
14	6, 13	eqneltrd 2720	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P = Q ) -> -. ( P .\/ Q ) e. N )
15	14	ex 450	. . 3 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P = Q -> -. ( P .\/ Q ) e. N ) )
16	15	necon2ad 2809	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( P .\/ Q ) e. N -> P =/= Q ) )
17	2, 3, 7	llni2 34798	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. N )
18	17	ex 450	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q -> ( P .\/ Q ) e. N ) )
19	16, 18	impbid 202	1 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( P .\/ Q ) e. N <-> P =/= Q ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   joincjn 16944   Atomscatm 34550   HLchlt 34637   LLinesclln 34777

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  df-llines 34784

This theorem is referenced by:  cdleme16d  35568