Theorem axhis3-zf

Description: Derive Axiom ax-his3 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
	axhfi.1	|- .ih = ( .iOLD ` U )
Assertion	axhis3-zf	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( A .h B ) .ih C ) = ( A x. ( B .ih C ) ) )

Proof

Step	Hyp	Ref	Expression
1		axhil.1	|- U = <. <. +h , .h >. , normh >.
2		axhil.2	|- U e. CHilOLD
3		axhfi.1	|- .ih = ( .iOLD ` U )
4		df-hba	|- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
5	1	fveq2i	|- ( BaseSet ` U ) = ( BaseSet ` <. <. +h , .h >. , normh >. )
6	4 5	eqtr4i	|- ~H = ( BaseSet ` U )
7	2	hlnvi	|- U e. NrmCVec
8	1 7	h2hsm	|- .h = ( .sOLD ` U )
9	6 8 3	hlipass	|- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. ~H /\ C e. ~H ) ) -> ( ( A .h B ) .ih C ) = ( A x. ( B .ih C ) ) )
10	2 9	mpan	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( A .h B ) .ih C ) = ( A x. ( B .ih C ) ) )