Axiom ax-his4 8873

Description: Identity law for inner product. Postulate (S4) of [Beran] p. 95.

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		chil 8727	. . . 4 class ℋ
3	1, 2	wcel 955	. . 3 wff A ∈ ℋ
4		c0v 8730	. . . 4 class 0h
5	1, 4	wne 1577	. . 3 wff A ≠ 0h
6	3, 5	wa 223	. 2 wff (A ∈ ℋ ⋀ A ≠ 0h)
7		cc0 5206	. . 3 class 0
8		csp 8732	. . . 4 class ·ih
9	1, 1, 8	co 3948	. . 3 class (A ·ih A)
10		clt 5458	. . 3 class <
11	7, 9, 10	wbr 2609	. 2 wff 0 < (A ·ih A)
12	6, 11	wi 3	1 wff ((A ∈ ℋ ⋀ A ≠ 0h) → 0 < (A ·ih A))

This axiom is referenced by:  hiidge0t 8885  his6t 8886  normgt0tOLD 8914  normgt0t 8915  pjthlem2 9135  pjthlem3 9136  pjthlem7 9140  eigre 9677  eigpos 9679

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