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Metamath Proof Explorer

Theorem dvrval

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014) (Revised by Mario Carneiro, 2-Dec-2014)

Ref Expression
Hypotheses dvrval.b 
|- B = ( Base ` R )
dvrval.t 
|- .x. = ( .r ` R )
dvrval.u 
|- U = ( Unit ` R )
dvrval.i 
|- I = ( invr ` R )
dvrval.d 
|- ./ = ( /r ` R )
Assertion dvrval 
|- ( ( X e. B /\ Y e. U ) -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )

Proof

Step Hyp Ref Expression
1 dvrval.b 
 |-  B = ( Base ` R )
2 dvrval.t 
 |-  .x. = ( .r ` R )
3 dvrval.u 
 |-  U = ( Unit ` R )
4 dvrval.i 
 |-  I = ( invr ` R )
5 dvrval.d 
 |-  ./ = ( /r ` R )
6 oveq1 
 |-  ( x = X -> ( x .x. ( I ` y ) ) = ( X .x. ( I ` y ) ) )
7 fveq2 
 |-  ( y = Y -> ( I ` y ) = ( I ` Y ) )
8 7 oveq2d 
 |-  ( y = Y -> ( X .x. ( I ` y ) ) = ( X .x. ( I ` Y ) ) )
9 1 2 3 4 5 dvrfval 
 |-  ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) )
10 ovex 
 |-  ( X .x. ( I ` Y ) ) e. _V
11 6 8 9 10 ovmpo 
 |-  ( ( X e. B /\ Y e. U ) -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )