Metamath Proof Explorer
Description: Reverse closure for addition: the second addend is real if the first
addend is real and the sum is real. (Contributed by SN, 25-Apr-2025)
|
|
Ref |
Expression |
|
Hypotheses |
readdrcl2d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
readdrcl2d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
readdrcl2d.c |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ ) |
|
Assertion |
readdrcl2d |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
readdrcl2d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
readdrcl2d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
readdrcl2d.c |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ ) |
4 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
5 |
4 2
|
pncan2d |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − 𝐴 ) = 𝐵 ) |
6 |
3 1
|
resubcld |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − 𝐴 ) ∈ ℝ ) |
7 |
5 6
|
eqeltrrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |