Metamath Proof Explorer

Theorem lhp2at0ne

Description: Inequality for joins with 2 different atoms (or an atom and zero) under co-atom W . (Contributed by NM, 28-Jul-2013)

Ref Expression
Hypotheses lhp2at0nle.l 
|- .<_ = ( le ` K )
lhp2at0nle.j 
|- .\/ = ( join ` K )
lhp2at0nle.z 
|- .0. = ( 0. ` K )
lhp2at0nle.a 
|- A = ( Atoms ` K )
lhp2at0nle.h 
|- H = ( LHyp ` K )
Assertion lhp2at0ne 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( P .\/ U ) =/= ( Q .\/ V ) )

Proof

Step Hyp Ref Expression
1 lhp2at0nle.l 
 |-  .<_ = ( le ` K )
2 lhp2at0nle.j 
 |-  .\/ = ( join ` K )
3 lhp2at0nle.z 
 |-  .0. = ( 0. ` K )
4 lhp2at0nle.a 
 |-  A = ( Atoms ` K )
5 lhp2at0nle.h 
 |-  H = ( LHyp ` K )
6 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( K e. HL /\ W e. H ) )
7 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( P e. A /\ -. P .<_ W ) )
8 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> U =/= V )
9 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( ( U e. A \/ U = .0. ) /\ U .<_ W ) )
10 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( V e. A /\ V .<_ W ) )
11 1 2 3 4 5 lhp2at0nle 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) )
12 6 7 8 9 10 11 syl311anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> -. V .<_ ( P .\/ U ) )
13 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> K e. HL )
14 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> Q e. A )
15 simp2rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> V e. A )
16 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ Q e. A /\ V e. A ) -> V .<_ ( Q .\/ V ) )
17 13 14 15 16 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> V .<_ ( Q .\/ V ) )
18 17 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) /\ ( P .\/ U ) = ( Q .\/ V ) ) -> V .<_ ( Q .\/ V ) )
19 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) /\ ( P .\/ U ) = ( Q .\/ V ) ) -> ( P .\/ U ) = ( Q .\/ V ) )
20 18 19 breqtrrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) /\ ( P .\/ U ) = ( Q .\/ V ) ) -> V .<_ ( P .\/ U ) )
21 20 ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( ( P .\/ U ) = ( Q .\/ V ) -> V .<_ ( P .\/ U ) ) )
22 21 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( -. V .<_ ( P .\/ U ) -> ( P .\/ U ) =/= ( Q .\/ V ) ) )
23 12 22 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( P .\/ U ) =/= ( Q .\/ V ) )