Metamath Proof Explorer


Theorem axhfvmul-zf

Description: Derive Axiom ax-hfvmul 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 axhfvmul-zf
|- .h : ( CC X. ~H ) --> ~H

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 hlmulf
 |-  ( U e. CHilOLD -> .h : ( CC X. ~H ) --> ~H )
9 2 8 ax-mp
 |-  .h : ( CC X. ~H ) --> ~H