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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 9857. Proofs should normally use mulass 9903 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 9813 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1977 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1977 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 1977 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 1031 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | cmul 9820 | . . . . 5 class · | |
10 | 1, 4, 9 | co 6549 | . . . 4 class (𝐴 · 𝐵) |
11 | 10, 6, 9 | co 6549 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
12 | 4, 6, 9 | co 6549 | . . . 4 class (𝐵 · 𝐶) |
13 | 1, 12, 9 | co 6549 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
14 | 11, 13 | wceq 1475 | . 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 9903 |
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