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Mirrors > Home > ILE Home > Th. List > addcli | GIF version |
Description: Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
addcli | ⊢ (𝐴 + 𝐵) ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | axi.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | addcl 7713 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ (𝐴 + 𝐵) ∈ ℂ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 + caddc 7591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia3 107 ax-addcl 7684 |
This theorem is referenced by: negdii 8014 eqneg 8460 nummac 9194 binom2i 10369 3dvds2dec 11490 |
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