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Metamath Proof Explorer

Theorem subneg

Description: Relationship between subtraction and negative. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion subneg ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − - 𝐵 ) = ( 𝐴 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 df-neg ⊢ - 𝐵 = ( 0 − 𝐵 )
2 1 oveq2i ⊢ ( 𝐴 − - 𝐵 ) = ( 𝐴 − ( 0 − 𝐵 ) )
3 0cn ⊢ 0 ∈ ℂ
4 subsub ⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − ( 0 − 𝐵 ) ) = ( ( 𝐴 − 0 ) + 𝐵 ) )
5 3 4 mp3an2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − ( 0 − 𝐵 ) ) = ( ( 𝐴 − 0 ) + 𝐵 ) )
6 2 5 eqtrid ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − - 𝐵 ) = ( ( 𝐴 − 0 ) + 𝐵 ) )
7 subid1 ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 0 ) = 𝐴 )
8 7 adantr ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 0 ) = 𝐴 )
9 8 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 0 ) + 𝐵 ) = ( 𝐴 + 𝐵 ) )
10 6 9 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − - 𝐵 ) = ( 𝐴 + 𝐵 ) )