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Metamath Proof Explorer

Theorem joincomALT

Description: The join of a poset is commutative. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses joincom.b 
|- B = ( Base ` K )
joincom.j 
|- .\/ = ( join ` K )
Assertion joincomALT 
|- ( ( K e. V /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) = ( Y .\/ X ) )

Proof

Step Hyp Ref Expression
1 joincom.b 
 |-  B = ( Base ` K )
2 joincom.j 
 |-  .\/ = ( join ` K )
3 prcom 
 |-  { Y , X } = { X , Y }
4 3 fveq2i 
 |-  ( ( lub ` K ) ` { Y , X } ) = ( ( lub ` K ) ` { X , Y } )
5 4 a1i 
 |-  ( ( K e. V /\ X e. B /\ Y e. B ) -> ( ( lub ` K ) ` { Y , X } ) = ( ( lub ` K ) ` { X , Y } ) )
6 eqid 
 |-  ( lub ` K ) = ( lub ` K )
7 simp1 
 |-  ( ( K e. V /\ X e. B /\ Y e. B ) -> K e. V )
8 simp3 
 |-  ( ( K e. V /\ X e. B /\ Y e. B ) -> Y e. B )
9 simp2 
 |-  ( ( K e. V /\ X e. B /\ Y e. B ) -> X e. B )
10 6 2 7 8 9 joinval 
 |-  ( ( K e. V /\ X e. B /\ Y e. B ) -> ( Y .\/ X ) = ( ( lub ` K ) ` { Y , X } ) )
11 6 2 7 9 8 joinval 
 |-  ( ( K e. V /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) = ( ( lub ` K ) ` { X , Y } ) )
12 5 10 11 3eqtr4rd 
 |-  ( ( K e. V /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) = ( Y .\/ X ) )