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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 6945. Proofs should normally use mulcom 7010 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 6887 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1393 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1393 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 97 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | cmul 6894 | . . . 4 class · | |
8 | 1, 4, 7 | co 5512 | . . 3 class (𝐴 · 𝐵) |
9 | 4, 1, 7 | co 5512 | . . 3 class (𝐵 · 𝐴) |
10 | 8, 9 | wceq 1243 | . 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 7010 |
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