Theorem axhvmulass-zf

Description: Derive Axiom ax-hvmulass from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	axhil.1	|- U = <. <. +h , .h >. , normh >.
	axhil.2	|- U e. CHilOLD
Assertion	axhvmulass-zf	|- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )

Proof

Step	Hyp	Ref	Expression
1		axhil.1	|- U = <. <. +h , .h >. , normh >.
2		axhil.2	|- U e. CHilOLD
3		df-hba	|- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
4	1	fveq2i	|- ( BaseSet ` U ) = ( BaseSet ` <. <. +h , .h >. , normh >. )
5	3 4	eqtr4i	|- ~H = ( BaseSet ` U )
6	2	hlnvi	|- U e. NrmCVec
7	1 6	h2hsm	|- .h = ( .sOLD ` U )
8	5 7	hlmulass	|- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. CC /\ C e. ~H ) ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )
9	2 8	mpan	|- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )