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Metamath Proof Explorer

Theorem dvdsrmul

Description: A left-multiple of X is divisible by X . (Contributed by Mario Carneiro, 1-Dec-2014)

Ref Expression
Hypotheses dvdsr.1 
|- B = ( Base ` R )
dvdsr.2 
|- .|| = ( ||r ` R )
dvdsr.3 
|- .x. = ( .r ` R )
Assertion dvdsrmul 
|- ( ( X e. B /\ Y e. B ) -> X .|| ( Y .x. X ) )

Proof

Step Hyp Ref Expression
1 dvdsr.1 
 |-  B = ( Base ` R )
2 dvdsr.2 
 |-  .|| = ( ||r ` R )
3 dvdsr.3 
 |-  .x. = ( .r ` R )
4 simpl 
 |-  ( ( X e. B /\ Y e. B ) -> X e. B )
5 simpr 
 |-  ( ( X e. B /\ Y e. B ) -> Y e. B )
6 eqid 
 |-  ( Y .x. X ) = ( Y .x. X )
7 oveq1 
 |-  ( z = Y -> ( z .x. X ) = ( Y .x. X ) )
8 7 eqeq1d 
 |-  ( z = Y -> ( ( z .x. X ) = ( Y .x. X ) <-> ( Y .x. X ) = ( Y .x. X ) ) )
9 8 rspcev 
 |-  ( ( Y e. B /\ ( Y .x. X ) = ( Y .x. X ) ) -> E. z e. B ( z .x. X ) = ( Y .x. X ) )
10 5 6 9 sylancl 
 |-  ( ( X e. B /\ Y e. B ) -> E. z e. B ( z .x. X ) = ( Y .x. X ) )
11 1 2 3 dvdsr 
 |-  ( X .|| ( Y .x. X ) <-> ( X e. B /\ E. z e. B ( z .x. X ) = ( Y .x. X ) ) )
12 4 10 11 sylanbrc 
 |-  ( ( X e. B /\ Y e. B ) -> X .|| ( Y .x. X ) )