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Axiom ax-his4 27912
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 chil 27746 . . . 4 class
31, 2wcel 1988 . . 3 wff 𝐴 ∈ ℋ
4 c0v 27751 . . . 4 class 0
51, 4wne 2791 . . 3 wff 𝐴 ≠ 0
63, 5wa 384 . 2 wff (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0)
7 cc0 9921 . . 3 class 0
8 csp 27749 . . . 4 class ·ih
91, 1, 8co 6635 . . 3 class (𝐴 ·ih 𝐴)
10 clt 10059 . . 3 class <
117, 9, 10wbr 4644 . 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  27925  his6  27926  normgt0  27954  eigrei  28663  eigposi  28665
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