Metamath Proof Explorer


Theorem axhis4-zf

Description: Derive Axiom ax-his4 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 axhis4-zf A A 0 0 < A ih A

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 df-h0v 0 = 0 vec + norm
8 1 fveq2i 0 vec U = 0 vec + norm
9 7 8 eqtr4i 0 = 0 vec U
10 6 9 3 hlipgt0 U CHil OLD A A 0 0 < A ih A
11 2 10 mp3an1 A A 0 0 < A ih A