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Mirrors > Home > MPE Home > Th. List > ax-mulass | Structured version Visualization version GIF version |
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11182. Proofs should normally use mulass 11228 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 11138 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2098 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2098 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 2098 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 1084 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | cmul 11145 | . . . . 5 class · | |
10 | 1, 4, 9 | co 7419 | . . . 4 class (𝐴 · 𝐵) |
11 | 10, 6, 9 | co 7419 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
12 | 4, 6, 9 | co 7419 | . . . 4 class (𝐵 · 𝐶) |
13 | 1, 12, 9 | co 7419 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
14 | 11, 13 | wceq 1533 | . 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 setvar class |
This axiom is referenced by: mulass 11228 |
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