Theorem 2llnma1 35073

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)

Hypotheses

Ref	Expression
2llnm.l	|- .<_ = ( le ` K )
2llnm.j	|- .\/ = ( join ` K )
2llnm.m	|- ./\ = ( meet ` K )
2llnm.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
2llnma1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ R ) ) = Q )

Proof of Theorem 2llnma1

Step	Hyp	Ref	Expression
1		simp1 1061	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. HL )
2		simp21 1094	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. A )
3		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
4		2llnm.a	. . . 4 |- A = ( Atoms ` K )
5	3, 4	atbase 34576	. . 3 |- ( P e. A -> P e. ( Base ` K ) )
6	2, 5	syl 17	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. ( Base ` K ) )
7		simp22 1095	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. A )
8		simp23 1096	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A )
9		simp3 1063	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( P .\/ Q ) )
10		2llnm.j	. . . . . 6 |- .\/ = ( join ` K )
11	10, 4	hlatjcom 34654	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
12	1, 2, 7, 11	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
13	12	breq2d 4665	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( R .<_ ( P .\/ Q ) <-> R .<_ ( Q .\/ P ) ) )
14	9, 13	mtbid 314	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( Q .\/ P ) )
15		2llnm.l	. . 3 |- .<_ = ( le ` K )
16		2llnm.m	. . 3 |- ./\ = ( meet ` K )
17	3, 15, 10, 16, 4	2llnma1b 35072	. 2 |- ( ( K e. HL /\ ( P e. ( Base ` K ) /\ Q e. A /\ R e. A ) /\ -. R .<_ ( Q .\/ P ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ R ) ) = Q )
18	1, 6, 7, 8, 14, 17	syl131anc 1339	1 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ R ) ) = Q )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   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:  2llnma3r  35074  2llnma2  35075  cdleme17c  35575