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Mirrors > Home > ILE Home > Th. List > ax-mulcom | GIF version |
Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7679. Proofs should normally use mulcom 7749 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulcom | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 7618 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1480 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1480 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | cmul 7625 | . . . 4 class · | |
8 | 1, 4, 7 | co 5774 | . . 3 class (𝐴 · 𝐵) |
9 | 4, 1, 7 | co 5774 | . . 3 class (𝐵 · 𝐴) |
10 | 8, 9 | wceq 1331 | . 2 wff (𝐴 · 𝐵) = (𝐵 · 𝐴) |
11 | 6, 10 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
Colors of variables: wff set class |
This axiom is referenced by: mulcom 7749 |
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