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Mirrors > Home > ILE Home > Th. List > ax-addcl | GIF version |
Description: Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 7893. Proofs should normally use addcl 7966 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-addcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 7839 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2160 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2160 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 104 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | caddc 7844 | . . . 4 class + | |
8 | 1, 4, 7 | co 5896 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 2160 | . 2 wff (𝐴 + 𝐵) ∈ ℂ |
10 | 6, 9 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
Colors of variables: wff set class |
This axiom is referenced by: addcl 7966 |
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