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Metamath Proof Explorer

Theorem 4atlem9

Description: Lemma for 4at . Substitute W for S . (Contributed by NM, 9-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4atlem9 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ ( R .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( R .\/ W ) ) ) )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> K e. HL )
5 simp22 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> S e. A )
6 simp23 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> W e. A )
7 4 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> K e. Lat )
8 simp1 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
11 8 10 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
12 simp21 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> R e. A )
13 9 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> R e. ( Base ` K ) )
15 9 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
16 7 11 14 15 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
17 simp3 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. S .<_ ( ( P .\/ Q ) .\/ R ) )
18 9 1 2 3 hlexchb2 
 |-  ( ( K e. HL /\ ( S e. A /\ W e. A /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .<_ ( W .\/ ( ( P .\/ Q ) .\/ R ) ) <-> ( S .\/ ( ( P .\/ Q ) .\/ R ) ) = ( W .\/ ( ( P .\/ Q ) .\/ R ) ) ) )
19 4 5 6 16 17 18 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .<_ ( W .\/ ( ( P .\/ Q ) .\/ R ) ) <-> ( S .\/ ( ( P .\/ Q ) .\/ R ) ) = ( W .\/ ( ( P .\/ Q ) .\/ R ) ) ) )
20 1 2 3 4atlem4d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ W e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( W .\/ ( ( P .\/ Q ) .\/ R ) ) )
21 8 12 6 20 syl12anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( W .\/ ( ( P .\/ Q ) .\/ R ) ) )
22 21 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ ( R .\/ W ) ) <-> S .<_ ( W .\/ ( ( P .\/ Q ) .\/ R ) ) ) )
23 1 2 3 4atlem4d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )
24 8 12 5 23 syl12anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )
25 24 21 eqeq12d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( R .\/ W ) ) <-> ( S .\/ ( ( P .\/ Q ) .\/ R ) ) = ( W .\/ ( ( P .\/ Q ) .\/ R ) ) ) )
26 19 22 25 3bitr4d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ ( R .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( R .\/ W ) ) ) )