Metamath Proof Explorer
Description: Relationship between addition and subtraction for surreals.
(Contributed by Scott Fenton, 5-Feb-2025)
|
|
Ref |
Expression |
|
Hypotheses |
subaddsd.1 |
⊢ ( 𝜑 → 𝐴 ∈ No ) |
|
|
subaddsd.2 |
⊢ ( 𝜑 → 𝐵 ∈ No ) |
|
|
subaddsd.3 |
⊢ ( 𝜑 → 𝐶 ∈ No ) |
|
Assertion |
subaddsd |
⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) = 𝐶 ↔ ( 𝐵 +s 𝐶 ) = 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
subaddsd.1 |
⊢ ( 𝜑 → 𝐴 ∈ No ) |
2 |
|
subaddsd.2 |
⊢ ( 𝜑 → 𝐵 ∈ No ) |
3 |
|
subaddsd.3 |
⊢ ( 𝜑 → 𝐶 ∈ No ) |
4 |
|
subadds |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 -s 𝐵 ) = 𝐶 ↔ ( 𝐵 +s 𝐶 ) = 𝐴 ) ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) = 𝐶 ↔ ( 𝐵 +s 𝐶 ) = 𝐴 ) ) |