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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 7551. Proofs should normally use addcl 7617 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 7498 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1448 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1448 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | caddc 7503 | . . . 4 class + | |
8 | 1, 4, 7 | co 5706 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 1448 | . 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 7617 |
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