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Metamath Proof Explorer

Theorem subsub4

Description: Law for double subtraction. (Contributed by NM, 19-Aug-2005) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion subsub4 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - B - C = A − B + C

Proof

Step Hyp Ref Expression
1 nppcan2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - B + C + C = A − B
2 simp1 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A ∈ ℂ
3 simp2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → B ∈ ℂ
4 subcl ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B ∈ ℂ
5 2 3 4 syl2anc ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B ∈ ℂ
6 simp3 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → C ∈ ℂ
7 3 6 addcld ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → B + C ∈ ℂ
8 subcl ⊢ A ∈ ℂ ∧ B + C ∈ ℂ → A − B + C ∈ ℂ
9 2 7 8 syl2anc ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B + C ∈ ℂ
10 subadd2 ⊢ A − B ∈ ℂ ∧ C ∈ ℂ ∧ A − B + C ∈ ℂ → A - B - C = A − B + C ↔ A - B + C + C = A − B
11 5 6 9 10 syl3anc ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - B - C = A − B + C ↔ A - B + C + C = A − B
12 1 11 mpbird ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - B - C = A − B + C