Axiom ax-his4 9228

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

Assertion

Ref	Expression
ax-his4	|- ((A e. 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 9063	. . . 4 class H~
3	1, 2	wcel 994	. . 3 wff A e. H~
4		c0v 9066	. . . 4 class 0h
5	1, 4	wne 1628	. . 3 wff A =/= 0h
6	3, 5	wa 221	. 2 wff (A e. H~ /\ A =/= 0h)
7		cc0 5388	. . 3 class 0
8		csp 9068	. . . 4 class .ih
9	1, 1, 8	co 4021	. . 3 class (A .ih A)
10		clt 5640	. . 3 class <
11	7, 9, 10	wbr 2692	. 2 wff 0 < (A .ih A)
12	6, 11	wi 3	1 wff ((A e. H~ /\ A =/= 0h) -> 0 < (A .ih A))

This axiom is referenced by:  hiidge0 9240  his6 9241  normgt0OLD 9269  normgt0 9270  pjthlem2 9496  pjthlem3 9497  pjthlem7 9501  eigrei 10040  eigposi 10042

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