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Mirrors > Home > ILE Home > Th. List > ax-addass | GIF version |
Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7804. Proofs should normally use addass 7874 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-addass | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 7742 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2135 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2135 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 2135 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 967 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | caddc 7747 | . . . . 5 class + | |
10 | 1, 4, 9 | co 5836 | . . . 4 class (𝐴 + 𝐵) |
11 | 10, 6, 9 | co 5836 | . . 3 class ((𝐴 + 𝐵) + 𝐶) |
12 | 4, 6, 9 | co 5836 | . . . 4 class (𝐵 + 𝐶) |
13 | 1, 12, 9 | co 5836 | . . 3 class (𝐴 + (𝐵 + 𝐶)) |
14 | 11, 13 | wceq 1342 | . 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 set class |
This axiom is referenced by: addass 7874 |
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