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Metamath Proof Explorer

Theorem rnghmmul

Description: A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020)

Ref Expression
Hypotheses rnghmmul.x 
|- X = ( Base ` R )
rnghmmul.m 
|- .x. = ( .r ` R )
rnghmmul.n 
|- .X. = ( .r ` S )
Assertion rnghmmul 
|- ( ( F e. ( R RngHom S ) /\ A e. X /\ B e. X ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) )

Proof

Step Hyp Ref Expression
1 rnghmmul.x 
 |-  X = ( Base ` R )
2 rnghmmul.m 
 |-  .x. = ( .r ` R )
3 rnghmmul.n 
 |-  .X. = ( .r ` S )
4 1 2 3 isrnghm 
 |-  ( F e. ( R RngHom S ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ A. x e. X A. y e. X ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) ) ) )
5 fvoveq1 
 |-  ( x = A -> ( F ` ( x .x. y ) ) = ( F ` ( A .x. y ) ) )
6 fveq2 
 |-  ( x = A -> ( F ` x ) = ( F ` A ) )
7 6 oveq1d 
 |-  ( x = A -> ( ( F ` x ) .X. ( F ` y ) ) = ( ( F ` A ) .X. ( F ` y ) ) )
8 5 7 eqeq12d 
 |-  ( x = A -> ( ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) <-> ( F ` ( A .x. y ) ) = ( ( F ` A ) .X. ( F ` y ) ) ) )
9 oveq2 
 |-  ( y = B -> ( A .x. y ) = ( A .x. B ) )
10 9 fveq2d 
 |-  ( y = B -> ( F ` ( A .x. y ) ) = ( F ` ( A .x. B ) ) )
11 fveq2 
 |-  ( y = B -> ( F ` y ) = ( F ` B ) )
12 11 oveq2d 
 |-  ( y = B -> ( ( F ` A ) .X. ( F ` y ) ) = ( ( F ` A ) .X. ( F ` B ) ) )
13 10 12 eqeq12d 
 |-  ( y = B -> ( ( F ` ( A .x. y ) ) = ( ( F ` A ) .X. ( F ` y ) ) <-> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) ) )
14 8 13 rspc2va 
 |-  ( ( ( A e. X /\ B e. X ) /\ A. x e. X A. y e. X ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) )
15 14 expcom 
 |-  ( A. x e. X A. y e. X ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) -> ( ( A e. X /\ B e. X ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) ) )
16 15 ad2antll 
 |-  ( ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ A. x e. X A. y e. X ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) ) ) -> ( ( A e. X /\ B e. X ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) ) )
17 4 16 sylbi 
 |-  ( F e. ( R RngHom S ) -> ( ( A e. X /\ B e. X ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) ) )
18 17 3impib 
 |-  ( ( F e. ( R RngHom S ) /\ A e. X /\ B e. X ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) )