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Metamath Proof Explorer

Theorem hosubsub4

Description: Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion hosubsub4 ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 −op 𝑇 ) −op 𝑈 ) = ( 𝑆 −op ( 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 hosubsub ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ ( - 1 ·op 𝑈 ) : ℋ ⟶ ℋ ) → ( 𝑆 −op ( 𝑇 −op ( - 1 ·op 𝑈 ) ) ) = ( ( 𝑆 −op 𝑇 ) +op ( - 1 ·op 𝑈 ) ) )
5 3 4 syl3an3 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑆 −op ( 𝑇 −op ( - 1 ·op 𝑈 ) ) ) = ( ( 𝑆 −op 𝑇 ) +op ( - 1 ·op 𝑈 ) ) )
6 hosubneg ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 −op ( - 1 ·op 𝑈 ) ) = ( 𝑇 +op 𝑈 ) )
7 6 3adant1 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 −op ( - 1 ·op 𝑈 ) ) = ( 𝑇 +op 𝑈 ) )
8 7 oveq2d ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑆 −op ( 𝑇 −op ( - 1 ·op 𝑈 ) ) ) = ( 𝑆 −op ( 𝑇 +op 𝑈 ) ) )
9 hosubcl ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 −op 𝑇 ) : ℋ ⟶ ℋ )
10 honegsub ⊢ ( ( ( 𝑆 −op 𝑇 ) : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 −op 𝑇 ) +op ( - 1 ·op 𝑈 ) ) = ( ( 𝑆 −op 𝑇 ) −op 𝑈 ) )
11 9 10 stoic3 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 −op 𝑇 ) +op ( - 1 ·op 𝑈 ) ) = ( ( 𝑆 −op 𝑇 ) −op 𝑈 ) )
12 5 8 11 3eqtr3rd ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 −op 𝑇 ) −op 𝑈 ) = ( 𝑆 −op ( 𝑇 +op 𝑈 ) ) )