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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 7648. Proofs should normally use addass 7718 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 7586 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1465 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1465 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 1465 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 947 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | caddc 7591 | . . . . 5 class + | |
10 | 1, 4, 9 | co 5742 | . . . 4 class (𝐴 + 𝐵) |
11 | 10, 6, 9 | co 5742 | . . 3 class ((𝐴 + 𝐵) + 𝐶) |
12 | 4, 6, 9 | co 5742 | . . . 4 class (𝐵 + 𝐶) |
13 | 1, 12, 9 | co 5742 | . . 3 class (𝐴 + (𝐵 + 𝐶)) |
14 | 11, 13 | wceq 1316 | . 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 7718 |
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