HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ax-hvdistr2 Structured version   Visualization version   GIF version

Axiom ax-hvdistr2 28790
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 10524 . . . 4 class
31, 2wcel 2114 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2114 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
7 chba 28700 . . . 4 class
86, 7wcel 2114 . . 3 wff 𝐶 ∈ ℋ
93, 5, 8w3a 1084 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)
10 caddc 10529 . . . . 5 class +
111, 4, 10co 7140 . . . 4 class (𝐴 + 𝐵)
12 csm 28702 . . . 4 class ·
1311, 6, 12co 7140 . . 3 class ((𝐴 + 𝐵) · 𝐶)
141, 6, 12co 7140 . . . 4 class (𝐴 · 𝐶)
154, 6, 12co 7140 . . . 4 class (𝐵 · 𝐶)
16 cva 28701 . . . 4 class +
1714, 15, 16co 7140 . . 3 class ((𝐴 · 𝐶) + (𝐵 · 𝐶))
1813, 17wceq 1538 . 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  28807  hvsubdistr2  28831  hv2times  28842  hilvc  28943  hhssnv  29045  hoadddir  29585  superpos  30135
  Copyright terms: Public domain W3C validator