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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 11197. Proofs should normally use mulass 11243 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 11153 | . . . 4 class ℂ | |
| 3 | 1, 2 | wcel 2108 | . . 3 wff 𝐴 ∈ ℂ |
| 4 | cB | . . . 4 class 𝐵 | |
| 5 | 4, 2 | wcel 2108 | . . 3 wff 𝐵 ∈ ℂ |
| 6 | cC | . . . 4 class 𝐶 | |
| 7 | 6, 2 | wcel 2108 | . . 3 wff 𝐶 ∈ ℂ |
| 8 | 3, 5, 7 | w3a 1087 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
| 9 | cmul 11160 | . . . . 5 class · | |
| 10 | 1, 4, 9 | co 7431 | . . . 4 class (𝐴 · 𝐵) |
| 11 | 10, 6, 9 | co 7431 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
| 12 | 4, 6, 9 | co 7431 | . . . 4 class (𝐵 · 𝐶) |
| 13 | 1, 12, 9 | co 7431 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
| 14 | 11, 13 | wceq 1540 | . 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 11243 |
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