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Mirrors > Home > ILE Home > Th. List > ax-addrcl | GIF version |
Description: Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 6751. Proofs should normally use readdcl 6805 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-addrcl | ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A + B) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class A | |
2 | cr 6710 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 1390 | . . 3 wff A ∈ ℝ |
4 | cB | . . . 4 class B | |
5 | 4, 2 | wcel 1390 | . . 3 wff B ∈ ℝ |
6 | 3, 5 | wa 97 | . 2 wff (A ∈ ℝ ∧ B ∈ ℝ) |
7 | caddc 6714 | . . . 4 class + | |
8 | 1, 4, 7 | co 5455 | . . 3 class (A + B) |
9 | 8, 2 | wcel 1390 | . 2 wff (A + B) ∈ ℝ |
10 | 6, 9 | wi 4 | 1 wff ((A ∈ ℝ ∧ B ∈ ℝ) → (A + B) ∈ ℝ) |
Colors of variables: wff set class |
This axiom is referenced by: readdcl 6805 |
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