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Metamath Proof Explorer

Theorem lhpat4N

Description: Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013) (New usage is discouraged.)

Ref Expression
Hypotheses lhpat.l 
|- .<_ = ( le ` K )
lhpat.j 
|- .\/ = ( join ` K )
lhpat.m 
|- ./\ = ( meet ` K )
lhpat.a 
|- A = ( Atoms ` K )
lhpat.h 
|- H = ( LHyp ` K )
Assertion lhpat4N 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( ( P .\/ U ) ./\ W ) = U )

Proof

Step Hyp Ref Expression
1 lhpat.l 
 |-  .<_ = ( le ` K )
2 lhpat.j 
 |-  .\/ = ( join ` K )
3 lhpat.m 
 |-  ./\ = ( meet ` K )
4 lhpat.a 
 |-  A = ( Atoms ` K )
5 lhpat.h 
 |-  H = ( LHyp ` K )
6 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( K e. HL /\ W e. H ) )
7 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( P e. A /\ -. P .<_ W ) )
8 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> U e. A )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 4 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
11 8 10 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> U e. ( Base ` K ) )
12 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> U .<_ W )
13 9 1 2 3 4 5 lhple 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. ( Base ` K ) /\ U .<_ W ) ) -> ( ( P .\/ U ) ./\ W ) = U )
14 6 7 11 12 13 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( ( P .\/ U ) ./\ W ) = U )