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Metamath Proof Explorer

Theorem 2atjm

Description: The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012)

Ref Expression
Hypotheses 2atjm.b 
|- B = ( Base ` K )
2atjm.l 
|- .<_ = ( le ` K )
2atjm.j 
|- .\/ = ( join ` K )
2atjm.m 
|- ./\ = ( meet ` K )
2atjm.a 
|- A = ( Atoms ` K )
Assertion 2atjm 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) = P )

Proof

Step Hyp Ref Expression
1 2atjm.b 
 |-  B = ( Base ` K )
2 2atjm.l 
 |-  .<_ = ( le ` K )
3 2atjm.j 
 |-  .\/ = ( join ` K )
4 2atjm.m 
 |-  ./\ = ( meet ` K )
5 2atjm.a 
 |-  A = ( Atoms ` K )
6 hllat 
 |-  ( K e. HL -> K e. Lat )
7 6 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> K e. Lat )
8 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P e. A )
9 1 5 atbase 
 |-  ( P e. A -> P e. B )
10 8 9 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P e. B )
11 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> Q e. A )
12 1 5 atbase 
 |-  ( Q e. A -> Q e. B )
13 11 12 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> Q e. B )
14 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ P e. B /\ Q e. B ) -> P .<_ ( P .\/ Q ) )
15 7 10 13 14 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P .<_ ( P .\/ Q ) )
16 simp3l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P .<_ X )
17 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> K e. HL )
18 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. B )
19 17 8 11 18 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( P .\/ Q ) e. B )
20 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> X e. B )
21 1 2 4 latlem12 
 |-  ( ( K e. Lat /\ ( P e. B /\ ( P .\/ Q ) e. B /\ X e. B ) ) -> ( ( P .<_ ( P .\/ Q ) /\ P .<_ X ) <-> P .<_ ( ( P .\/ Q ) ./\ X ) ) )
22 7 10 19 20 21 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .<_ ( P .\/ Q ) /\ P .<_ X ) <-> P .<_ ( ( P .\/ Q ) ./\ X ) ) )
23 15 16 22 mpbi2and 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P .<_ ( ( P .\/ Q ) ./\ X ) )
24 hlatl 
 |-  ( K e. HL -> K e. AtLat )
25 24 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> K e. AtLat )
26 1 4 latmcom 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ X e. B ) -> ( ( P .\/ Q ) ./\ X ) = ( X ./\ ( P .\/ Q ) ) )
27 7 19 20 26 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) = ( X ./\ ( P .\/ Q ) ) )
28 20 8 11 3jca 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( X e. B /\ P e. A /\ Q e. A ) )
29 nbrne2 
 |-  ( ( P .<_ X /\ -. Q .<_ X ) -> P =/= Q )
30 29 3ad2ant3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P =/= Q )
31 simp3r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> -. Q .<_ X )
32 1 3 latjcl 
 |-  ( ( K e. Lat /\ X e. B /\ Q e. B ) -> ( X .\/ Q ) e. B )
33 7 20 13 32 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( X .\/ Q ) e. B )
34 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ X e. B /\ Q e. B ) -> X .<_ ( X .\/ Q ) )
35 7 20 13 34 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> X .<_ ( X .\/ Q ) )
36 1 2 7 10 20 33 16 35 lattrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P .<_ ( X .\/ Q ) )
37 1 2 3 4 5 cvrat3 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( P =/= Q /\ -. Q .<_ X /\ P .<_ ( X .\/ Q ) ) -> ( X ./\ ( P .\/ Q ) ) e. A ) )
38 37 imp 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( P =/= Q /\ -. Q .<_ X /\ P .<_ ( X .\/ Q ) ) ) -> ( X ./\ ( P .\/ Q ) ) e. A )
39 17 28 30 31 36 38 syl23anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( X ./\ ( P .\/ Q ) ) e. A )
40 27 39 eqeltrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) e. A )
41 2 5 atcmp 
 |-  ( ( K e. AtLat /\ P e. A /\ ( ( P .\/ Q ) ./\ X ) e. A ) -> ( P .<_ ( ( P .\/ Q ) ./\ X ) <-> P = ( ( P .\/ Q ) ./\ X ) ) )
42 25 8 40 41 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( P .<_ ( ( P .\/ Q ) ./\ X ) <-> P = ( ( P .\/ Q ) ./\ X ) ) )
43 23 42 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> P = ( ( P .\/ Q ) ./\ X ) )
44 43 eqcomd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) = P )