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Metamath Proof Explorer

Theorem hosubneg

Description: Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion hosubneg ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 −op ( - 1 ·op 𝑈 ) ) = ( 𝑇 +op 𝑈 ) )

Proof

Step Hyp Ref Expression
1 neg1cn ⊢ - 1 ∈ ℂ
2 homulcl ⊢ ( ( - 1 ∈ ℂ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( - 1 ·op 𝑈 ) : ℋ ⟶ ℋ )
3 1 2 mpan ⊢ ( 𝑈 : ℋ ⟶ ℋ → ( - 1 ·op 𝑈 ) : ℋ ⟶ ℋ )
4 honegsub ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ ( - 1 ·op 𝑈 ) : ℋ ⟶ ℋ ) → ( 𝑇 +op ( - 1 ·op ( - 1 ·op 𝑈 ) ) ) = ( 𝑇 −op ( - 1 ·op 𝑈 ) ) )
5 3 4 sylan2 ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 +op ( - 1 ·op ( - 1 ·op 𝑈 ) ) ) = ( 𝑇 −op ( - 1 ·op 𝑈 ) ) )
6 honegneg ⊢ ( 𝑈 : ℋ ⟶ ℋ → ( - 1 ·op ( - 1 ·op 𝑈 ) ) = 𝑈 )
7 6 oveq2d ⊢ ( 𝑈 : ℋ ⟶ ℋ → ( 𝑇 +op ( - 1 ·op ( - 1 ·op 𝑈 ) ) ) = ( 𝑇 +op 𝑈 ) )
8 7 adantl ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 +op ( - 1 ·op ( - 1 ·op 𝑈 ) ) ) = ( 𝑇 +op 𝑈 ) )
9 5 8 eqtr3d ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 −op ( - 1 ·op 𝑈 ) ) = ( 𝑇 +op 𝑈 ) )