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Mirrors > Home > MPE Home > Th. List > ax-distr | Structured version Visualization version GIF version |
Description: Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by Theorem axdistr 10903. Proofs should normally use adddi 10949 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-distr | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 10858 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2106 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2106 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 2106 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 1086 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | caddc 10863 | . . . . 5 class + | |
10 | 4, 6, 9 | co 7269 | . . . 4 class (𝐵 + 𝐶) |
11 | cmul 10865 | . . . 4 class · | |
12 | 1, 10, 11 | co 7269 | . . 3 class (𝐴 · (𝐵 + 𝐶)) |
13 | 1, 4, 11 | co 7269 | . . . 4 class (𝐴 · 𝐵) |
14 | 1, 6, 11 | co 7269 | . . . 4 class (𝐴 · 𝐶) |
15 | 13, 14, 9 | co 7269 | . . 3 class ((𝐴 · 𝐵) + (𝐴 · 𝐶)) |
16 | 12, 15 | wceq 1539 | . 2 wff (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)) |
17 | 8, 16 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
Colors of variables: wff setvar class |
This axiom is referenced by: adddi 10949 |
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