Theorem lnopmulsubi

Description: Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005) (New usage is discouraged.)

	Ref	Expression
Hypothesis	lnopl.1	⊢ 𝑇 ∈ LinOp
Assertion	lnopmulsubi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) −ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) −ℎ ( 𝑇 ‘ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lnopl.1	⊢ 𝑇 ∈ LinOp
2		hvmulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ )
3	1	lnopsubi	⊢ ( ( ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) −ℎ 𝐶 ) ) = ( ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) −ℎ ( 𝑇 ‘ 𝐶 ) ) )
4	2 3	stoic3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) −ℎ 𝐶 ) ) = ( ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) −ℎ ( 𝑇 ‘ 𝐶 ) ) )
5	1	lnopmuli	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )
6	5	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )
7	6	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) −ℎ ( 𝑇 ‘ 𝐶 ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) −ℎ ( 𝑇 ‘ 𝐶 ) ) )
8	4 7	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) −ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) −ℎ ( 𝑇 ‘ 𝐶 ) ) )