Metamath Proof Explorer

Theorem dalempnes

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012)

Ref Expression
Hypotheses dalema.ph 
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalemc.l 
|- .<_ = ( le ` K )
dalemc.j 
|- .\/ = ( join ` K )
dalemc.a 
|- A = ( Atoms ` K )
dalempnes.o 
|- O = ( LPlanes ` K )
dalempnes.y 
|- Y = ( ( P .\/ Q ) .\/ R )
Assertion dalempnes 
|- ( ph -> P =/= S )

Proof

Step Hyp Ref Expression
1 dalema.ph 
 |-  ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
2 dalemc.l 
 |-  .<_ = ( le ` K )
3 dalemc.j 
 |-  .\/ = ( join ` K )
4 dalemc.a 
 |-  A = ( Atoms ` K )
5 dalempnes.o 
 |-  O = ( LPlanes ` K )
6 dalempnes.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 1 dalemkelat 
 |-  ( ph -> K e. Lat )
8 1 4 dalemceb 
 |-  ( ph -> C e. ( Base ` K ) )
9 1 4 dalemseb 
 |-  ( ph -> S e. ( Base ` K ) )
10 1 4 dalemteb 
 |-  ( ph -> T e. ( Base ` K ) )
11 simp321 
 |-  ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) -> -. C .<_ ( S .\/ T ) )
12 1 11 sylbi 
 |-  ( ph -> -. C .<_ ( S .\/ T ) )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 2 3 latnlej2l 
 |-  ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ S e. ( Base ` K ) /\ T e. ( Base ` K ) ) /\ -. C .<_ ( S .\/ T ) ) -> -. C .<_ S )
15 7 8 9 10 12 14 syl131anc 
 |-  ( ph -> -. C .<_ S )
16 1 dalemclpjs 
 |-  ( ph -> C .<_ ( P .\/ S ) )
17 oveq1 
 |-  ( P = S -> ( P .\/ S ) = ( S .\/ S ) )
18 17 breq2d 
 |-  ( P = S -> ( C .<_ ( P .\/ S ) <-> C .<_ ( S .\/ S ) ) )
19 16 18 syl5ibcom 
 |-  ( ph -> ( P = S -> C .<_ ( S .\/ S ) ) )
20 1 dalemkehl 
 |-  ( ph -> K e. HL )
21 1 dalemsea 
 |-  ( ph -> S e. A )
22 3 4 hlatjidm 
 |-  ( ( K e. HL /\ S e. A ) -> ( S .\/ S ) = S )
23 20 21 22 syl2anc 
 |-  ( ph -> ( S .\/ S ) = S )
24 23 breq2d 
 |-  ( ph -> ( C .<_ ( S .\/ S ) <-> C .<_ S ) )
25 19 24 sylibd 
 |-  ( ph -> ( P = S -> C .<_ S ) )
26 25 necon3bd 
 |-  ( ph -> ( -. C .<_ S -> P =/= S ) )
27 15 26 mpd 
 |-  ( ph -> P =/= S )