Metamath Proof Explorer


Theorem dalemply

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 dalemply
|- ( ph -> P .<_ Y )

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 dalempeb
 |-  ( ph -> P e. ( Base ` K ) )
9 1 dalemkehl
 |-  ( ph -> K e. HL )
10 1 dalemqea
 |-  ( ph -> Q e. A )
11 1 dalemrea
 |-  ( ph -> R e. A )
12 eqid
 |-  ( Base ` K ) = ( Base ` K )
13 12 3 4 hlatjcl
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
14 9 10 11 13 syl3anc
 |-  ( ph -> ( Q .\/ R ) e. ( Base ` K ) )
15 12 2 3 latlej1
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) -> P .<_ ( P .\/ ( Q .\/ R ) ) )
16 7 8 14 15 syl3anc
 |-  ( ph -> P .<_ ( P .\/ ( Q .\/ R ) ) )
17 1 dalempea
 |-  ( ph -> P e. A )
18 3 4 hlatjass
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )
19 9 17 10 11 18 syl13anc
 |-  ( ph -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )
20 16 19 breqtrrd
 |-  ( ph -> P .<_ ( ( P .\/ Q ) .\/ R ) )
21 20 6 breqtrrdi
 |-  ( ph -> P .<_ Y )