Axiom ax-his4 9101

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 8968	. . . 4 class H~
3	1, 2	wcel 1105	. . 3 wff A e. H~
4		c0v 8971	. . . 4 class 0h
5	1, 4	wne 1561	. . 3 wff A =/= 0h
6	3, 5	wa 223	. 2 wff (A e. H~ /\ A =/= 0h)
7		cc0 5157	. . 3 class 0
8		csp 8973	. . . 4 class .ih
9	1, 1, 8	co 3902	. . 3 class (A .ih A)
10		clt 5409	. . 3 class <
11	7, 9, 10	wbr 2587	. 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:  hiidge0t 9113  his6t 9114  normgt0tOLD 9142  normgt0t 9143  pjthlem2 9349  pjthlem3 9350  pjthlem7 9354  eigre 9891  eigpos 9893

Copyright terms: Public domain