Axiom ax-his2 27940

Description: Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-his2	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) .ih C ) = ( ( A .ih C ) + ( B .ih C ) ) )

Detailed syntax breakdown of Axiom ax-his2

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		chil 27776	. . . 4 class ~H
3	1, 2	wcel 1990	. . 3 wff A e. ~H
4		cB	. . . 4 class B
5	4, 2	wcel 1990	. . 3 wff B e. ~H
6		cC	. . . 4 class C
7	6, 2	wcel 1990	. . 3 wff C e. ~H
8	3, 5, 7	w3a 1037	. 2 wff ( A e. ~H /\ B e. ~H /\ C e. ~H )
9		cva 27777	. . . . 5 class +h
10	1, 4, 9	co 6650	. . . 4 class ( A +h B )
11		csp 27779	. . . 4 class .ih
12	10, 6, 11	co 6650	. . 3 class ( ( A +h B ) .ih C )
13	1, 6, 11	co 6650	. . . 4 class ( A .ih C )
14	4, 6, 11	co 6650	. . . 4 class ( B .ih C )
15		caddc 9939	. . . 4 class +
16	13, 14, 15	co 6650	. . 3 class ( ( A .ih C ) + ( B .ih C ) )
17	12, 16	wceq 1483	. 2 wff ( ( A +h B ) .ih C ) = ( ( A .ih C ) + ( B .ih C ) )
18	8, 17	wi 4	1 wff ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) .ih C ) = ( ( A .ih C ) + ( B .ih C ) ) )

This axiom is referenced by:  his7  27947  hiassdi  27948  his2sub  27949  normlem0  27966  normlem8  27974  ocsh  28142  pjspansn  28436  pjadjii  28533  braadd  28804  lnopunilem1  28869  hmops  28879  cnlnadjlem2  28927  adjadd  28952  leopadd  28991