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Axiom ax-hvdistr1 27993
Description: Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-hvdistr1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Detailed syntax breakdown of Axiom ax-hvdistr1
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 9972 . . . 4 class
31, 2wcel 2030 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
5 chil 27904 . . . 4 class
64, 5wcel 2030 . . 3 wff 𝐵 ∈ ℋ
7 cC . . . 4 class 𝐶
87, 5wcel 2030 . . 3 wff 𝐶 ∈ ℋ
93, 6, 8w3a 1054 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)
10 cva 27905 . . . . 5 class +
114, 7, 10co 6690 . . . 4 class (𝐵 + 𝐶)
12 csm 27906 . . . 4 class ·
131, 11, 12co 6690 . . 3 class (𝐴 · (𝐵 + 𝐶))
141, 4, 12co 6690 . . . 4 class (𝐴 · 𝐵)
151, 7, 12co 6690 . . . 4 class (𝐴 · 𝐶)
1614, 15, 10co 6690 . . 3 class ((𝐴 · 𝐵) + (𝐴 · 𝐶))
1713, 16wceq 1523 . 2 wff (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
189, 17wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Colors of variables: wff setvar class
This axiom is referenced by:  hvsub4  28022  hvsubass  28029  hvsubdistr1  28034  hvdistr1i  28036  hv2times  28046  hilvc  28147  hhssnv  28249  shscli  28304  spanunsni  28566  hoadddi  28790  lnopmi  28987  lnophsi  28988
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