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Metamath Proof Explorer

Theorem hosubdi

Description: Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion hosubdi ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑇 −op 𝑈 ) ) = ( ( 𝐴 ·op 𝑇 ) −op ( 𝐴 ·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 hoadddi ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ ( - 1 ·op 𝑈 ) : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑇 +op ( - 1 ·op 𝑈 ) ) ) = ( ( 𝐴 ·op 𝑇 ) +op ( 𝐴 ·op ( - 1 ·op 𝑈 ) ) ) )
5 3 4 syl3an3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑇 +op ( - 1 ·op 𝑈 ) ) ) = ( ( 𝐴 ·op 𝑇 ) +op ( 𝐴 ·op ( - 1 ·op 𝑈 ) ) ) )
6 homul12 ⊢ ( ( 𝐴 ∈ ℂ ∧ - 1 ∈ ℂ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( - 1 ·op 𝑈 ) ) = ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) )
7 1 6 mp3an2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( - 1 ·op 𝑈 ) ) = ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) )
8 7 3adant2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( - 1 ·op 𝑈 ) ) = ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) )
9 8 oveq2d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝐴 ·op 𝑇 ) +op ( 𝐴 ·op ( - 1 ·op 𝑈 ) ) ) = ( ( 𝐴 ·op 𝑇 ) +op ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) ) )
10 5 9 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑇 +op ( - 1 ·op 𝑈 ) ) ) = ( ( 𝐴 ·op 𝑇 ) +op ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) ) )
11 honegsub ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 +op ( - 1 ·op 𝑈 ) ) = ( 𝑇 −op 𝑈 ) )
12 11 oveq2d ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑇 +op ( - 1 ·op 𝑈 ) ) ) = ( 𝐴 ·op ( 𝑇 −op 𝑈 ) ) )
13 12 3adant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑇 +op ( - 1 ·op 𝑈 ) ) ) = ( 𝐴 ·op ( 𝑇 −op 𝑈 ) ) )
14 homulcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝐴 ·op 𝑇 ) : ℋ ⟶ ℋ )
15 homulcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op 𝑈 ) : ℋ ⟶ ℋ )
16 honegsub ⊢ ( ( ( 𝐴 ·op 𝑇 ) : ℋ ⟶ ℋ ∧ ( 𝐴 ·op 𝑈 ) : ℋ ⟶ ℋ ) → ( ( 𝐴 ·op 𝑇 ) +op ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) ) = ( ( 𝐴 ·op 𝑇 ) −op ( 𝐴 ·op 𝑈 ) ) )
17 14 15 16 syl2an ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ ( 𝐴 ∈ ℂ ∧ 𝑈 : ℋ ⟶ ℋ ) ) → ( ( 𝐴 ·op 𝑇 ) +op ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) ) = ( ( 𝐴 ·op 𝑇 ) −op ( 𝐴 ·op 𝑈 ) ) )
18 17 3impdi ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝐴 ·op 𝑇 ) +op ( - 1 ·op ( 𝐴 ·op 𝑈 ) ) ) = ( ( 𝐴 ·op 𝑇 ) −op ( 𝐴 ·op 𝑈 ) ) )
19 10 13 18 3eqtr3d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑇 −op 𝑈 ) ) = ( ( 𝐴 ·op 𝑇 ) −op ( 𝐴 ·op 𝑈 ) ) )