Description: Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013) (Proof shortened by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | subadd2 | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subadd | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) ) | |
2 | simp2 | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ ) | |
3 | simp3 | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ ) | |
4 | 2 3 | addcomd | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) ) |
5 | 4 | eqeq1d | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( 𝐶 + 𝐵 ) = 𝐴 ) ) |
6 | 1 5 | bitrd | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 ) ) |