Metamath Proof Explorer


Theorem axhis1-zf

Description: Derive Axiom ax-his1 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 UCHilOLD
axhfi.1 ih=𝑖OLDU
Assertion axhis1-zf ABAihB=BihA

Proof

Step Hyp Ref Expression
1 axhil.1 U=+norm
2 axhil.2 UCHilOLD
3 axhfi.1 ih=𝑖OLDU
4 df-hba =BaseSet+norm
5 1 fveq2i BaseSetU=BaseSet+norm
6 4 5 eqtr4i =BaseSetU
7 6 3 hlipcj UCHilOLDABAihB=BihA
8 2 7 mp3an1 ABAihB=BihA