Axiom ax-his4 29022

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 28856	. . . 4 class ℋ
3	1, 2	wcel 2114	. . 3 wff 𝐴 ∈ ℋ
4		c0v 28861	. . . 4 class 0ℎ
5	1, 4	wne 2934	. . 3 wff 𝐴 ≠ 0ℎ
6	3, 5	wa 399	. 2 wff (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
7		cc0 10617	. . 3 class 0
8		csp 28859	. . . 4 class ·ih
9	1, 1, 8	co 7172	. . 3 class (𝐴 ·ih 𝐴)
10		clt 10755	. . 3 class <
11	7, 9, 10	wbr 5030	. 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  29035  his6  29036  normgt0  29064  eigrei  29771  eigposi  29773