Theorem axhvmul0-zf

Description: Derive Axiom ax-hvmul0 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	axhvmul0-zf	|- ( A e. ~H -> ( 0 .h A ) = 0h )

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		df-h0v	|- 0h = ( 0vec ` <. <. +h , .h >. , normh >. )
9	1	fveq2i	|- ( 0vec ` U ) = ( 0vec ` <. <. +h , .h >. , normh >. )
10	8 9	eqtr4i	|- 0h = ( 0vec ` U )
11	5 7 10	hlmul0	|- ( ( U e. CHilOLD /\ A e. ~H ) -> ( 0 .h A ) = 0h )
12	2 11	mpan	|- ( A e. ~H -> ( 0 .h A ) = 0h )