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Metamath Proof Explorer

Theorem dihmeetlem3N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem3.b 
|- B = ( Base ` K )
dihmeetlem3.l 
|- .<_ = ( le ` K )
dihmeetlem3.j 
|- .\/ = ( join ` K )
dihmeetlem3.m 
|- ./\ = ( meet ` K )
dihmeetlem3.a 
|- A = ( Atoms ` K )
dihmeetlem3.h 
|- H = ( LHyp ` K )
Assertion dihmeetlem3N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) ) -> Q =/= R )

Proof

Step Hyp Ref Expression
1 dihmeetlem3.b 
 |-  B = ( Base ` K )
2 dihmeetlem3.l 
 |-  .<_ = ( le ` K )
3 dihmeetlem3.j 
 |-  .\/ = ( join ` K )
4 dihmeetlem3.m 
 |-  ./\ = ( meet ` K )
5 dihmeetlem3.a 
 |-  A = ( Atoms ` K )
6 dihmeetlem3.h 
 |-  H = ( LHyp ` K )
7 simp2lr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) ) -> -. Q .<_ W )
8 oveq1 
 |-  ( Q = R -> ( Q .\/ ( Y ./\ W ) ) = ( R .\/ ( Y ./\ W ) ) )
9 simpr 
 |-  ( ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) -> ( R .\/ ( Y ./\ W ) ) = Y )
10 8 9 sylan9eqr 
 |-  ( ( ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) /\ Q = R ) -> ( Q .\/ ( Y ./\ W ) ) = Y )
11 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> K e. HL )
12 11 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> K e. Lat )
13 simp2ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q e. A )
14 1 5 atbase 
 |-  ( Q e. A -> Q e. B )
15 13 14 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q e. B )
16 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> X e. B )
17 simp12r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Y e. B )
18 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
19 12 16 17 18 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> ( X ./\ Y ) e. B )
20 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> W e. H )
21 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
22 20 21 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> W e. B )
23 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
24 12 16 22 23 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> ( X ./\ W ) e. B )
25 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ Q e. B /\ ( X ./\ W ) e. B ) -> Q .<_ ( Q .\/ ( X ./\ W ) ) )
26 12 15 24 25 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q .<_ ( Q .\/ ( X ./\ W ) ) )
27 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> ( Q .\/ ( X ./\ W ) ) = X )
28 26 27 breqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q .<_ X )
29 1 4 latmcl 
 |-  ( ( K e. Lat /\ Y e. B /\ W e. B ) -> ( Y ./\ W ) e. B )
30 12 17 22 29 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> ( Y ./\ W ) e. B )
31 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ Q e. B /\ ( Y ./\ W ) e. B ) -> Q .<_ ( Q .\/ ( Y ./\ W ) ) )
32 12 15 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q .<_ ( Q .\/ ( Y ./\ W ) ) )
33 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> ( Q .\/ ( Y ./\ W ) ) = Y )
34 32 33 breqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q .<_ Y )
35 1 2 4 latlem12 
 |-  ( ( K e. Lat /\ ( Q e. B /\ X e. B /\ Y e. B ) ) -> ( ( Q .<_ X /\ Q .<_ Y ) <-> Q .<_ ( X ./\ Y ) ) )
36 12 15 16 17 35 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> ( ( Q .<_ X /\ Q .<_ Y ) <-> Q .<_ ( X ./\ Y ) ) )
37 28 34 36 mpbi2and 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q .<_ ( X ./\ Y ) )
38 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> ( X ./\ Y ) .<_ W )
39 1 2 12 15 19 22 37 38 lattrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( Q .\/ ( Y ./\ W ) ) = Y ) -> Q .<_ W )
40 39 3exp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) -> ( ( Q .\/ ( Y ./\ W ) ) = Y -> Q .<_ W ) ) )
41 10 40 syl7 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) -> ( ( ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) /\ Q = R ) -> Q .<_ W ) ) )
42 41 exp4a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) -> ( ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) -> ( Q = R -> Q .<_ W ) ) ) )
43 42 3imp 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) ) -> ( Q = R -> Q .<_ W ) )
44 43 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) ) -> ( -. Q .<_ W -> Q =/= R ) )
45 7 44 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( Y ./\ W ) ) = Y ) ) -> Q =/= R )