Metamath Proof Explorer

Theorem 2llnma1b

Description: Generalization of 2llnma1 . (Contributed by NM, 26-Apr-2013)

Ref Expression
Hypotheses 2llnma1b.b 
|- B = ( Base ` K )
2llnma1b.l 
|- .<_ = ( le ` K )
2llnma1b.j 
|- .\/ = ( join ` K )
2llnma1b.m 
|- ./\ = ( meet ` K )
2llnma1b.a 
|- A = ( Atoms ` K )
Assertion 2llnma1b 
|- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( ( P .\/ X ) ./\ ( P .\/ Q ) ) = P )

Proof

Step Hyp Ref Expression
1 2llnma1b.b 
 |-  B = ( Base ` K )
2 2llnma1b.l 
 |-  .<_ = ( le ` K )
3 2llnma1b.j 
 |-  .\/ = ( join ` K )
4 2llnma1b.m 
 |-  ./\ = ( meet ` K )
5 2llnma1b.a 
 |-  A = ( Atoms ` K )
6 hllat 
 |-  ( K e. HL -> K e. Lat )
7 6 3ad2ant1 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> K e. Lat )
8 simp22 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P e. A )
9 1 5 atbase 
 |-  ( P e. A -> P e. B )
10 8 9 syl 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P e. B )
11 simp21 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> X e. B )
12 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ P e. B /\ X e. B ) -> P .<_ ( P .\/ X ) )
13 7 10 11 12 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P .<_ ( P .\/ X ) )
14 simp23 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> Q e. A )
15 1 5 atbase 
 |-  ( Q e. A -> Q e. B )
16 14 15 syl 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> Q e. B )
17 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ P e. B /\ Q e. B ) -> P .<_ ( P .\/ Q ) )
18 7 10 16 17 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P .<_ ( P .\/ Q ) )
19 1 3 latjcl 
 |-  ( ( K e. Lat /\ P e. B /\ X e. B ) -> ( P .\/ X ) e. B )
20 7 10 11 19 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( P .\/ X ) e. B )
21 simp1 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> K e. HL )
22 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. B )
23 21 8 14 22 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( P .\/ Q ) e. B )
24 1 2 4 latlem12 
 |-  ( ( K e. Lat /\ ( P e. B /\ ( P .\/ X ) e. B /\ ( P .\/ Q ) e. B ) ) -> ( ( P .<_ ( P .\/ X ) /\ P .<_ ( P .\/ Q ) ) <-> P .<_ ( ( P .\/ X ) ./\ ( P .\/ Q ) ) ) )
25 7 10 20 23 24 syl13anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( ( P .<_ ( P .\/ X ) /\ P .<_ ( P .\/ Q ) ) <-> P .<_ ( ( P .\/ X ) ./\ ( P .\/ Q ) ) ) )
26 13 18 25 mpbi2and 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P .<_ ( ( P .\/ X ) ./\ ( P .\/ Q ) ) )
27 hlatl 
 |-  ( K e. HL -> K e. AtLat )
28 27 3ad2ant1 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> K e. AtLat )
29 simp3 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> -. Q .<_ ( P .\/ X ) )
30 nbrne2 
 |-  ( ( P .<_ ( P .\/ X ) /\ -. Q .<_ ( P .\/ X ) ) -> P =/= Q )
31 13 29 30 syl2anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P =/= Q )
32 1 3 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ X ) e. B /\ Q e. B ) -> ( ( P .\/ X ) .\/ Q ) e. B )
33 7 20 16 32 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( ( P .\/ X ) .\/ Q ) e. B )
34 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ ( P .\/ X ) e. B /\ Q e. B ) -> ( P .\/ X ) .<_ ( ( P .\/ X ) .\/ Q ) )
35 7 20 16 34 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( P .\/ X ) .<_ ( ( P .\/ X ) .\/ Q ) )
36 1 2 7 10 20 33 13 35 lattrd 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P .<_ ( ( P .\/ X ) .\/ Q ) )
37 1 2 3 4 5 cvrat3 
 |-  ( ( K e. HL /\ ( ( P .\/ X ) e. B /\ P e. A /\ Q e. A ) ) -> ( ( P =/= Q /\ -. Q .<_ ( P .\/ X ) /\ P .<_ ( ( P .\/ X ) .\/ Q ) ) -> ( ( P .\/ X ) ./\ ( P .\/ Q ) ) e. A ) )
38 37 3impia 
 |-  ( ( K e. HL /\ ( ( P .\/ X ) e. B /\ P e. A /\ Q e. A ) /\ ( P =/= Q /\ -. Q .<_ ( P .\/ X ) /\ P .<_ ( ( P .\/ X ) .\/ Q ) ) ) -> ( ( P .\/ X ) ./\ ( P .\/ Q ) ) e. A )
39 21 20 8 14 31 29 36 38 syl133anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( ( P .\/ X ) ./\ ( P .\/ Q ) ) e. A )
40 2 5 atcmp 
 |-  ( ( K e. AtLat /\ P e. A /\ ( ( P .\/ X ) ./\ ( P .\/ Q ) ) e. A ) -> ( P .<_ ( ( P .\/ X ) ./\ ( P .\/ Q ) ) <-> P = ( ( P .\/ X ) ./\ ( P .\/ Q ) ) ) )
41 28 8 39 40 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( P .<_ ( ( P .\/ X ) ./\ ( P .\/ Q ) ) <-> P = ( ( P .\/ X ) ./\ ( P .\/ Q ) ) ) )
42 26 41 mpbid 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> P = ( ( P .\/ X ) ./\ ( P .\/ Q ) ) )
43 42 eqcomd 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ X ) ) -> ( ( P .\/ X ) ./\ ( P .\/ Q ) ) = P )