Description: Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | negidi.1 | ⊢ 𝐴 ∈ ℂ | |
pncan3i.2 | ⊢ 𝐵 ∈ ℂ | ||
Assertion | negsubdii | ⊢ - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | ⊢ 𝐴 ∈ ℂ | |
2 | pncan3i.2 | ⊢ 𝐵 ∈ ℂ | |
3 | 2 | negcli | ⊢ - 𝐵 ∈ ℂ |
4 | 1 3 | negdii | ⊢ - ( 𝐴 + - 𝐵 ) = ( - 𝐴 + - - 𝐵 ) |
5 | 1 2 | negsubi | ⊢ ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) |
6 | 5 | negeqi | ⊢ - ( 𝐴 + - 𝐵 ) = - ( 𝐴 − 𝐵 ) |
7 | 2 | negnegi | ⊢ - - 𝐵 = 𝐵 |
8 | 7 | oveq2i | ⊢ ( - 𝐴 + - - 𝐵 ) = ( - 𝐴 + 𝐵 ) |
9 | 4 6 8 | 3eqtr3i | ⊢ - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 ) |