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Metamath Proof Explorer

Theorem subadd

Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997) (Revised by Mario Carneiro, 21-Dec-2013)

Ref Expression
Assertion subadd ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B = C ↔ B + C = A

Proof

Step Hyp Ref Expression
1 subval ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = ι x ∈ ℂ | B + x = A
2 1 eqeq1d ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = C ↔ ι x ∈ ℂ | B + x = A = C
3 2 3adant3 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B = C ↔ ι x ∈ ℂ | B + x = A = C
4 negeu ⊢ B ∈ ℂ ∧ A ∈ ℂ → ∃! x ∈ ℂ B + x = A
5 oveq2 ⊢ x = C → B + x = B + C
6 5 eqeq1d ⊢ x = C → B + x = A ↔ B + C = A
7 6 riota2 ⊢ C ∈ ℂ ∧ ∃! x ∈ ℂ B + x = A → B + C = A ↔ ι x ∈ ℂ | B + x = A = C
8 4 7 sylan2 ⊢ C ∈ ℂ ∧ B ∈ ℂ ∧ A ∈ ℂ → B + C = A ↔ ι x ∈ ℂ | B + x = A = C
9 8 3impb ⊢ C ∈ ℂ ∧ B ∈ ℂ ∧ A ∈ ℂ → B + C = A ↔ ι x ∈ ℂ | B + x = A = C
10 9 3com13 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → B + C = A ↔ ι x ∈ ℂ | B + x = A = C
11 3 10 bitr4d ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B = C ↔ B + C = A