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Metamath Proof Explorer

Theorem cvrcmp

Description: If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012)

Ref Expression
Hypotheses cvrcmp.b 
|- B = ( Base ` K )
cvrcmp.l 
|- .<_ = ( le ` K )
cvrcmp.c 
|- C = (
Assertion cvrcmp 
|- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( X .<_ Y <-> X = Y ) )

Proof

Step Hyp Ref Expression
1 cvrcmp.b 
 |-  B = ( Base ` K )
2 cvrcmp.l 
 |-  .<_ = ( le ` K )
3 cvrcmp.c 
 |-  C = (
4 simpl1 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> K e. Poset )
5 simpl23 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z e. B )
6 simpl21 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> X e. B )
7 simpl3l 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z C X )
8 1 3 cvrne 
 |-  ( ( ( K e. Poset /\ Z e. B /\ X e. B ) /\ Z C X ) -> Z =/= X )
9 4 5 6 7 8 syl31anc 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z =/= X )
10 1 2 3 cvrle 
 |-  ( ( ( K e. Poset /\ Z e. B /\ X e. B ) /\ Z C X ) -> Z .<_ X )
11 4 5 6 7 10 syl31anc 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z .<_ X )
12 simpr 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> X .<_ Y )
13 simpl22 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Y e. B )
14 simpl3r 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z C Y )
15 1 2 3 cvrnbtwn4 
 |-  ( ( K e. Poset /\ ( Z e. B /\ Y e. B /\ X e. B ) /\ Z C Y ) -> ( ( Z .<_ X /\ X .<_ Y ) <-> ( Z = X \/ X = Y ) ) )
16 4 5 13 6 14 15 syl131anc 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> ( ( Z .<_ X /\ X .<_ Y ) <-> ( Z = X \/ X = Y ) ) )
17 11 12 16 mpbi2and 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> ( Z = X \/ X = Y ) )
18 neor 
 |-  ( ( Z = X \/ X = Y ) <-> ( Z =/= X -> X = Y ) )
19 17 18 sylib 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> ( Z =/= X -> X = Y ) )
20 9 19 mpd 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> X = Y )
21 20 ex 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( X .<_ Y -> X = Y ) )
22 simp1 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> K e. Poset )
23 simp21 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> X e. B )
24 1 2 posref 
 |-  ( ( K e. Poset /\ X e. B ) -> X .<_ X )
25 22 23 24 syl2anc 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> X .<_ X )
26 breq2 
 |-  ( X = Y -> ( X .<_ X <-> X .<_ Y ) )
27 25 26 syl5ibcom 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( X = Y -> X .<_ Y ) )
28 21 27 impbid 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( X .<_ Y <-> X = Y ) )