HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ax-his4 Structured version   Visualization version   GIF version

Axiom ax-his4 29447
Description: Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-his4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))

Detailed syntax breakdown of Axiom ax-his4
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 chba 29281 . . . 4 class
31, 2wcel 2106 . . 3 wff 𝐴 ∈ ℋ
4 c0v 29286 . . . 4 class 0
51, 4wne 2943 . . 3 wff 𝐴 ≠ 0
63, 5wa 396 . 2 wff (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0)
7 cc0 10871 . . 3 class 0
8 csp 29284 . . . 4 class ·ih
91, 1, 8co 7275 . . 3 class (𝐴 ·ih 𝐴)
10 clt 11009 . . 3 class <
117, 9, 10wbr 5074 . 2 wff 0 < (𝐴 ·ih 𝐴)
126, 11wi 4 1 wff ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))
Colors of variables: wff setvar class
This axiom is referenced by:  hiidge0  29460  his6  29461  normgt0  29489  eigrei  30196  eigposi  30198
  Copyright terms: Public domain W3C validator