Metamath Proof Explorer


Theorem axhvmulid-zf

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

Ref Expression
Hypotheses axhil.1 U = + norm
axhil.2 U CHil OLD
Assertion axhvmulid-zf A 1 A = A

Proof

Step Hyp Ref Expression
1 axhil.1 U = + norm
2 axhil.2 U CHil OLD
3 df-hba = BaseSet + norm
4 1 fveq2i BaseSet U = BaseSet + norm
5 3 4 eqtr4i = BaseSet U
6 2 hlnvi U NrmCVec
7 1 6 h2hsm = 𝑠OLD U
8 5 7 hlmulid U CHil OLD A 1 A = A
9 2 8 mpan A 1 A = A