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Axiom ax-his4 31104
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 30938 . . . 4 class
31, 2wcel 2108 . . 3 wff 𝐴 ∈ ℋ
4 c0v 30943 . . . 4 class 0
51, 4wne 2940 . . 3 wff 𝐴 ≠ 0
63, 5wa 395 . 2 wff (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0)
7 cc0 11155 . . 3 class 0
8 csp 30941 . . . 4 class ·ih
91, 1, 8co 7431 . . 3 class (𝐴 ·ih 𝐴)
10 clt 11295 . . 3 class <
117, 9, 10wbr 5143 . 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  31117  his6  31118  normgt0  31146  eigrei  31853  eigposi  31855
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