Metamath Proof Explorer


Theorem axhvass-zf

Description: Derive Axiom ax-hvass 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 axhvass-zf A B C A + B + C = A + 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 5 7 hlass U CHil OLD A B C A + B + C = A + B + C
9 2 8 mpan A B C A + B + C = A + B + C