Theorem imasmulval

Description: The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	imasaddf.f	|- ( ph -> F : V -onto-> B )
	imasaddf.e	|- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )
	imasaddf.u	|- ( ph -> U = ( F "s R ) )
	imasaddf.v	|- ( ph -> V = ( Base ` R ) )
	imasaddf.r	|- ( ph -> R e. Z )
	imasmulf.p	|- .x. = ( .r ` R )
	imasmulf.a	|- .xb = ( .r ` U )
Assertion	imasmulval	|- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		imasaddf.f	|- ( ph -> F : V -onto-> B )
2		imasaddf.e	|- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )
3		imasaddf.u	|- ( ph -> U = ( F "s R ) )
4		imasaddf.v	|- ( ph -> V = ( Base ` R ) )
5		imasaddf.r	|- ( ph -> R e. Z )
6		imasmulf.p	|- .x. = ( .r ` R )
7		imasmulf.a	|- .xb = ( .r ` U )
8	3 4 1 5 6 7	imasmulr	|- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
9	1 2 8	imasaddvallem	|- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )