Metamath Proof Explorer


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 = + norm
axhil.2 U CHil OLD
Assertion axhvmul0-zf A 0 A = 0

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 df-h0v 0 = 0 vec + norm
9 1 fveq2i 0 vec U = 0 vec + norm
10 8 9 eqtr4i 0 = 0 vec U
11 5 7 10 hlmul0 U CHil OLD A 0 A = 0
12 2 11 mpan A 0 A = 0