Theorem 2atmat0 34812

Description: The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)

Hypotheses

Ref	Expression
2atmatz.j	|- .\/ = ( join ` K )
2atmatz.m	|- ./\ = ( meet ` K )
2atmatz.z	|- .0. = ( 0. ` K )
2atmatz.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
2atmat0	|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A \/ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = .0. ) )

Proof of Theorem 2atmat0

Step	Hyp	Ref	Expression
1		simpl 473	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) )
2		simpr1 1067	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> R e. A )
3		simpr2 1068	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> S e. A )
4	3	orcd 407	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( S e. A \/ S = .0. ) )
5		simpr3 1069	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( P .\/ Q ) =/= ( R .\/ S ) )
6		2atmatz.j	. . 3 |- .\/ = ( join ` K )
7		2atmatz.m	. . 3 |- ./\ = ( meet ` K )
8		2atmatz.z	. . 3 |- .0. = ( 0. ` K )
9		2atmatz.a	. . 3 |- A = ( Atoms ` K )
10	6, 7, 8, 9	2at0mat0 34811	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ ( S e. A \/ S = .0. ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A \/ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = .0. ) )
11	1, 2, 4, 5, 10	syl13anc 1328	1 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A \/ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = .0. ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   joincjn 16944   meetcmee 16945   0.cp0 17037   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  df-llines 34784

This theorem is referenced by:  2atm  34813  trlval3  35474  cdleme22b  35629  cdlemg31b0N  35982  cdlemh  36105