Axiom ax-his4 31113

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 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))

Detailed syntax breakdown of Axiom ax-his4

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		chba 30947	. . . 4 class ℋ
3	1, 2	wcel 2105	. . 3 wff 𝐴 ∈ ℋ
4		c0v 30952	. . . 4 class 0ℎ
5	1, 4	wne 2937	. . 3 wff 𝐴 ≠ 0ℎ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
7		cc0 11152	. . 3 class 0
8		csp 30950	. . . 4 class ·ih
9	1, 1, 8	co 7430	. . 3 class (𝐴 ·ih 𝐴)
10		clt 11292	. . 3 class <
11	7, 9, 10	wbr 5147	. 2 wff 0 < (𝐴 ·ih 𝐴)
12	6, 11	wi 4	1 wff ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))

This axiom is referenced by:  hiidge0  31126  his6  31127  normgt0  31155  eigrei  31862  eigposi  31864