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Metamath Proof Explorer

Theorem lhp2atne

Description: Inequality for joins with 2 different atoms under co-atom W . (Contributed by NM, 22-Jul-2013)

Ref Expression
Hypotheses lhp2atnle.l 
|- .<_ = ( le ` K )
lhp2atnle.j 
|- .\/ = ( join ` K )
lhp2atnle.a 
|- A = ( Atoms ` K )
lhp2atnle.h 
|- H = ( LHyp ` K )
Assertion lhp2atne 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( P .\/ U ) =/= ( Q .\/ V ) )

Proof

Step Hyp Ref Expression
1 lhp2atnle.l 
 |-  .<_ = ( le ` K )
2 lhp2atnle.j 
 |-  .\/ = ( join ` K )
3 lhp2atnle.a 
 |-  A = ( Atoms ` K )
4 lhp2atnle.h 
 |-  H = ( LHyp ` K )
5 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( K e. HL /\ W e. H ) )
6 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( P e. A /\ -. P .<_ W ) )
7 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> U =/= V )
8 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( U e. A /\ U .<_ W ) )
9 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( V e. A /\ V .<_ W ) )
10 1 2 3 4 lhp2atnle 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) )
11 5 6 7 8 9 10 syl311anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> -. V .<_ ( P .\/ U ) )
12 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> K e. HL )
13 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> Q e. A )
14 simp2rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> V e. A )
15 1 2 3 hlatlej2 
 |-  ( ( K e. HL /\ Q e. A /\ V e. A ) -> V .<_ ( Q .\/ V ) )
16 12 13 14 15 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> V .<_ ( Q .\/ V ) )
17 16 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) /\ ( P .\/ U ) = ( Q .\/ V ) ) -> V .<_ ( Q .\/ V ) )
18 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) /\ ( P .\/ U ) = ( Q .\/ V ) ) -> ( P .\/ U ) = ( Q .\/ V ) )
19 17 18 breqtrrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) /\ ( P .\/ U ) = ( Q .\/ V ) ) -> V .<_ ( P .\/ U ) )
20 19 ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( ( P .\/ U ) = ( Q .\/ V ) -> V .<_ ( P .\/ U ) ) )
21 20 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( -. V .<_ ( P .\/ U ) -> ( P .\/ U ) =/= ( Q .\/ V ) ) )
22 11 21 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U =/= V ) -> ( P .\/ U ) =/= ( Q .\/ V ) )