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Metamath Proof Explorer

Theorem dihmeetlem5

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014)

Ref Expression
Hypotheses dihmeetlem5.b 
|- B = ( Base ` K )
dihmeetlem5.l 
|- .<_ = ( le ` K )
dihmeetlem5.j 
|- .\/ = ( join ` K )
dihmeetlem5.m 
|- ./\ = ( meet ` K )
dihmeetlem5.a 
|- A = ( Atoms ` K )
Assertion dihmeetlem5 
|- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> ( X ./\ ( Y .\/ Q ) ) = ( ( X ./\ Y ) .\/ Q ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem5.b 
 |-  B = ( Base ` K )
2 dihmeetlem5.l 
 |-  .<_ = ( le ` K )
3 dihmeetlem5.j 
 |-  .\/ = ( join ` K )
4 dihmeetlem5.m 
 |-  ./\ = ( meet ` K )
5 dihmeetlem5.a 
 |-  A = ( Atoms ` K )
6 simpl1 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> K e. HL )
7 simprl 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> Q e. A )
8 simpl2 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> X e. B )
9 simpl3 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> Y e. B )
10 simprr 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> Q .<_ X )
11 1 2 3 4 5 atmod2i1 
 |-  ( ( K e. HL /\ ( Q e. A /\ X e. B /\ Y e. B ) /\ Q .<_ X ) -> ( ( X ./\ Y ) .\/ Q ) = ( X ./\ ( Y .\/ Q ) ) )
12 6 7 8 9 10 11 syl131anc 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> ( ( X ./\ Y ) .\/ Q ) = ( X ./\ ( Y .\/ Q ) ) )
13 12 eqcomd 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ Q .<_ X ) ) -> ( X ./\ ( Y .\/ Q ) ) = ( ( X ./\ Y ) .\/ Q ) )