Metamath Proof Explorer


Theorem axhis3-zf

Description: Derive Axiom ax-his3 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
axhfi.1 ih = 𝑖OLD U
Assertion axhis3-zf A B C A B ih C = A B ih C

Proof

Step Hyp Ref Expression
1 axhil.1 U = + norm
2 axhil.2 U CHil OLD
3 axhfi.1 ih = 𝑖OLD U
4 df-hba = BaseSet + norm
5 1 fveq2i BaseSet U = BaseSet + norm
6 4 5 eqtr4i = BaseSet U
7 2 hlnvi U NrmCVec
8 1 7 h2hsm = 𝑠OLD U
9 6 8 3 hlipass U CHil OLD A B C A B ih C = A B ih C
10 2 9 mpan A B C A B ih C = A B ih C