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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 10174. Proofs should normally use addcl 10220 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 10136 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2145 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2145 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 382 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | caddc 10141 | . . . 4 class + | |
8 | 1, 4, 7 | co 6793 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 2145 | . 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 10220 |
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