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Axiom ax-his2 27910
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 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))

Detailed syntax breakdown of Axiom ax-his2
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 chil 27746 . . . 4 class
31, 2wcel 1988 . . 3 wff 𝐴 ∈ ℋ
4 cB . . . 4 class 𝐵
54, 2wcel 1988 . . 3 wff 𝐵 ∈ ℋ
6 cC . . . 4 class 𝐶
76, 2wcel 1988 . . 3 wff 𝐶 ∈ ℋ
83, 5, 7w3a 1036 . 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)
9 cva 27747 . . . . 5 class +
101, 4, 9co 6635 . . . 4 class (𝐴 + 𝐵)
11 csp 27749 . . . 4 class ·ih
1210, 6, 11co 6635 . . 3 class ((𝐴 + 𝐵) ·ih 𝐶)
131, 6, 11co 6635 . . . 4 class (𝐴 ·ih 𝐶)
144, 6, 11co 6635 . . . 4 class (𝐵 ·ih 𝐶)
15 caddc 9924 . . . 4 class +
1613, 14, 15co 6635 . . 3 class ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))
1712, 16wceq 1481 . 2 wff ((𝐴 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))
188, 17wi 4 1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
Colors of variables: wff setvar class
This axiom is referenced by:  his7  27917  hiassdi  27918  his2sub  27919  normlem0  27936  normlem8  27944  ocsh  28112  pjspansn  28406  pjadjii  28503  braadd  28774  lnopunilem1  28839  hmops  28849  cnlnadjlem2  28897  adjadd  28922  leopadd  28961
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