Metamath Proof Explorer


Theorem axhvdistr2-zf

Description: Derive Axiom ax-hvdistr2 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 axhvdistr2-zf A B C A + B C = A C + B C

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 h2hva + = + v U
8 1 6 h2hsm = 𝑠OLD U
9 5 7 8 hldir U CHil OLD A B C A + B C = A C + B C
10 2 9 mpan A B C A + B C = A C + B C