Theorem lnopli

Description: Basic scalar product property of a linear Hilbert space operator. (Contributed by NM, 23-Jan-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	lnopl.1	⊢ 𝑇 ∈ LinOp
Assertion	lnopli	⊢ ( ( 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		lnopl	⊢ ( ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ) ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 𝐶 ) ) )
3	1 2	mpanl1	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 𝐶 ) ) )
4	3	3impb	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 𝐶 ) ) )