Metamath Proof Explorer


Theorem cdleme01N

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 5-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme0.l
|- .<_ = ( le ` K )
cdleme0.j
|- .\/ = ( join ` K )
cdleme0.m
|- ./\ = ( meet ` K )
cdleme0.a
|- A = ( Atoms ` K )
cdleme0.h
|- H = ( LHyp ` K )
cdleme0.u
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme01N
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) /\ U .<_ W ) )

Proof

Step Hyp Ref Expression
1 cdleme0.l
 |-  .<_ = ( le ` K )
2 cdleme0.j
 |-  .\/ = ( join ` K )
3 cdleme0.m
 |-  ./\ = ( meet ` K )
4 cdleme0.a
 |-  A = ( Atoms ` K )
5 cdleme0.h
 |-  H = ( LHyp ` K )
6 cdleme0.u
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp1l
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
8 7 hllatd
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. Lat )
9 simp2ll
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
10 simp2rl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
11 eqid
 |-  ( Base ` K ) = ( Base ` K )
12 11 2 4 hlatjcl
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
13 7 9 10 12 syl3anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) )
14 simp1r
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. H )
15 11 5 lhpbase
 |-  ( W e. H -> W e. ( Base ` K ) )
16 14 15 syl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. ( Base ` K ) )
17 11 1 3 latmle2
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
18 8 13 16 17 syl3anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
19 6 18 eqbrtrid
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U .<_ W )
20 simp2lr
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
21 nbrne2
 |-  ( ( U .<_ W /\ -. P .<_ W ) -> U =/= P )
22 19 20 21 syl2anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U =/= P )
23 simp2rr
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W )
24 nbrne2
 |-  ( ( U .<_ W /\ -. Q .<_ W ) -> U =/= Q )
25 19 23 24 syl2anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U =/= Q )
26 simp1
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
27 1 2 3 4 5 6 cdlemeulpq
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )
28 26 9 10 27 syl12anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U .<_ ( P .\/ Q ) )
29 22 25 28 3jca
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) )
30 29 19 jca
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) /\ U .<_ W ) )