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Metamath Proof Explorer

Theorem hlatexch4

Description: Exchange 2 atoms. (Contributed by NM, 13-May-2013)

Ref Expression
Hypotheses hlatexch4.j 
|- .\/ = ( join ` K )
hlatexch4.a 
|- A = ( Atoms ` K )
Assertion hlatexch4 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( P .\/ R ) = ( Q .\/ S ) )

Proof

Step Hyp Ref Expression
1 hlatexch4.j 
 |-  .\/ = ( join ` K )
2 hlatexch4.a 
 |-  A = ( Atoms ` K )
3 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> K e. HL )
4 simp2l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> R e. A )
5 simp2r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> S e. A )
6 eqid 
 |-  ( le ` K ) = ( le ` K )
7 6 1 2 hlatlej2 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> S ( le ` K ) ( R .\/ S ) )
8 3 4 5 7 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> S ( le ` K ) ( R .\/ S ) )
9 simp33 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( P .\/ Q ) = ( R .\/ S ) )
10 8 9 breqtrrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> S ( le ` K ) ( P .\/ Q ) )
11 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> P e. A )
12 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> Q e. A )
13 simp32 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> Q =/= S )
14 13 necomd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> S =/= Q )
15 6 1 2 hlatexch2 
 |-  ( ( K e. HL /\ ( S e. A /\ P e. A /\ Q e. A ) /\ S =/= Q ) -> ( S ( le ` K ) ( P .\/ Q ) -> P ( le ` K ) ( S .\/ Q ) ) )
16 3 5 11 12 14 15 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( S ( le ` K ) ( P .\/ Q ) -> P ( le ` K ) ( S .\/ Q ) ) )
17 10 16 mpd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> P ( le ` K ) ( S .\/ Q ) )
18 1 2 hlatjcom 
 |-  ( ( K e. HL /\ S e. A /\ Q e. A ) -> ( S .\/ Q ) = ( Q .\/ S ) )
19 3 5 12 18 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( S .\/ Q ) = ( Q .\/ S ) )
20 17 19 breqtrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> P ( le ` K ) ( Q .\/ S ) )
21 6 1 2 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q ( le ` K ) ( P .\/ Q ) )
22 3 11 12 21 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> Q ( le ` K ) ( P .\/ Q ) )
23 22 9 breqtrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> Q ( le ` K ) ( R .\/ S ) )
24 6 1 2 hlatexch2 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Q =/= S ) -> ( Q ( le ` K ) ( R .\/ S ) -> R ( le ` K ) ( Q .\/ S ) ) )
25 3 12 4 5 13 24 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( Q ( le ` K ) ( R .\/ S ) -> R ( le ` K ) ( Q .\/ S ) ) )
26 23 25 mpd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> R ( le ` K ) ( Q .\/ S ) )
27 3 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> K e. Lat )
28 eqid 
 |-  ( Base ` K ) = ( Base ` K )
29 28 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
30 11 29 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> P e. ( Base ` K ) )
31 28 2 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
32 4 31 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> R e. ( Base ` K ) )
33 28 1 2 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ S e. A ) -> ( Q .\/ S ) e. ( Base ` K ) )
34 3 12 5 33 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( Q .\/ S ) e. ( Base ` K ) )
35 28 6 1 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ R e. ( Base ` K ) /\ ( Q .\/ S ) e. ( Base ` K ) ) ) -> ( ( P ( le ` K ) ( Q .\/ S ) /\ R ( le ` K ) ( Q .\/ S ) ) <-> ( P .\/ R ) ( le ` K ) ( Q .\/ S ) ) )
36 27 30 32 34 35 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( ( P ( le ` K ) ( Q .\/ S ) /\ R ( le ` K ) ( Q .\/ S ) ) <-> ( P .\/ R ) ( le ` K ) ( Q .\/ S ) ) )
37 20 26 36 mpbi2and 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( P .\/ R ) ( le ` K ) ( Q .\/ S ) )
38 simp31 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> P =/= R )
39 6 1 2 ps-1 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ P =/= R ) /\ ( Q e. A /\ S e. A ) ) -> ( ( P .\/ R ) ( le ` K ) ( Q .\/ S ) <-> ( P .\/ R ) = ( Q .\/ S ) ) )
40 3 11 4 38 12 5 39 syl132anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( ( P .\/ R ) ( le ` K ) ( Q .\/ S ) <-> ( P .\/ R ) = ( Q .\/ S ) ) )
41 37 40 mpbid 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= R /\ Q =/= S /\ ( P .\/ Q ) = ( R .\/ S ) ) ) -> ( P .\/ R ) = ( Q .\/ S ) )