Axiom ax-his2 31010

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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ 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		chba 30846	. . . 4 class ℋ
3	1, 2	wcel 2108	. . 3 wff 𝐴 ∈ ℋ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2108	. . 3 wff 𝐵 ∈ ℋ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2108	. . 3 wff 𝐶 ∈ ℋ
8	3, 5, 7	w3a 1086	. 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)
9		cva 30847	. . . . 5 class +ℎ
10	1, 4, 9	co 7403	. . . 4 class (𝐴 +ℎ 𝐵)
11		csp 30849	. . . 4 class ·ih
12	10, 6, 11	co 7403	. . 3 class ((𝐴 +ℎ 𝐵) ·ih 𝐶)
13	1, 6, 11	co 7403	. . . 4 class (𝐴 ·ih 𝐶)
14	4, 6, 11	co 7403	. . . 4 class (𝐵 ·ih 𝐶)
15		caddc 11130	. . . 4 class +
16	13, 14, 15	co 7403	. . 3 class ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))
17	12, 16	wceq 1540	. 2 wff ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))
18	8, 17	wi 4	1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))

This axiom is referenced by:  his7  31017  hiassdi  31018  his2sub  31019  normlem0  31036  normlem8  31044  ocsh  31210  pjspansn  31504  pjadjii  31601  braadd  31872  lnopunilem1  31937  hmops  31947  cnlnadjlem2  31995  adjadd  32020  leopadd  32059