Theorem axhis1-zf

Description: Derive Axiom ax-his1 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	axhis1-zf	|- ( ( A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) )

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	6 3	hlipcj	|- ( ( U e. CHilOLD /\ A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) )
8	2 7	mp3an1	|- ( ( A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) )