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Axiom ax-his4 21664
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  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )

Detailed syntax breakdown of Axiom ax-his4
StepHypRef Expression
1 cA . . . 4  class  A
2 chil 21499 . . . 4  class  ~H
31, 2wcel 1684 . . 3  wff  A  e. 
~H
4 c0v 21504 . . . 4  class  0h
51, 4wne 2446 . . 3  wff  A  =/= 
0h
63, 5wa 358 . 2  wff  ( A  e.  ~H  /\  A  =/=  0h )
7 cc0 8737 . . 3  class  0
8 csp 21502 . . . 4  class  .ih
91, 1, 8co 5858 . . 3  class  ( A 
.ih  A )
10 clt 8867 . . 3  class  <
117, 9, 10wbr 4023 . 2  wff  0  <  ( A  .ih  A
)
126, 11wi 4 1  wff  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
Colors of variables: wff set class
This axiom is referenced by:  hiidge0  21677  his6  21678  normgt0  21706  eigrei  22414  eigposi  22416
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