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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 10568. Proofs should normally use mulass 10614 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 10524 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2105 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2105 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 2105 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 1079 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | cmul 10531 | . . . . 5 class · | |
10 | 1, 4, 9 | co 7145 | . . . 4 class (𝐴 · 𝐵) |
11 | 10, 6, 9 | co 7145 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
12 | 4, 6, 9 | co 7145 | . . . 4 class (𝐵 · 𝐶) |
13 | 1, 12, 9 | co 7145 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
14 | 11, 13 | wceq 1528 | . 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 10614 |
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