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Axiom ax-hvdistr2 27250
 Description: Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-hvdistr2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-hvdistr2
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 9813 . . . 4 class
31, 2wcel 1977 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1977 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
7 chil 27160 . . . 4 class
86, 7wcel 1977 . . 3 wff 𝐶 ∈ ℋ
93, 5, 8w3a 1031 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)
10 caddc 9818 . . . . 5 class +
111, 4, 10co 6549 . . . 4 class (𝐴 + 𝐵)
12 csm 27162 . . . 4 class ·
1311, 6, 12co 6549 . . 3 class ((𝐴 + 𝐵) · 𝐶)
141, 6, 12co 6549 . . . 4 class (𝐴 · 𝐶)
154, 6, 12co 6549 . . . 4 class (𝐵 · 𝐶)
16 cva 27161 . . . 4 class +
1714, 15, 16co 6549 . . 3 class ((𝐴 · 𝐶) + (𝐵 · 𝐶))
1813, 17wceq 1475 . 2 wff ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))
199, 18wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 Colors of variables: wff setvar class This axiom is referenced by:  hvsubid  27267  hvsubdistr2  27291  hv2times  27302  hilvc  27403  hhssnv  27505  hoadddir  28047  superpos  28597
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