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Mirrors > Home > MPE Home > Th. List > ax-addcl | Structured version Visualization version GIF version |
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10562. Proofs should normally use addcl 10608 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 10524 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2105 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2105 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 396 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | caddc 10529 | . . . 4 class + | |
8 | 1, 4, 7 | co 7145 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 2105 | . 2 wff (𝐴 + 𝐵) ∈ ℂ |
10 | 6, 9 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
This axiom is referenced by: addcl 10608 |
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