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Mirrors > Home > ILE Home > Th. List > ax-mulass | GIF version |
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 6947. Proofs should normally use mulass 7012 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulass | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 6887 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1393 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1393 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 1393 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 885 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | cmul 6894 | . . . . 5 class · | |
10 | 1, 4, 9 | co 5512 | . . . 4 class (𝐴 · 𝐵) |
11 | 10, 6, 9 | co 5512 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
12 | 4, 6, 9 | co 5512 | . . . 4 class (𝐵 · 𝐶) |
13 | 1, 12, 9 | co 5512 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
14 | 11, 13 | wceq 1243 | . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
15 | 8, 14 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Colors of variables: wff set class |
This axiom is referenced by: mulass 7012 |
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