Theorem lnfnli

Description: Basic property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	lnfnl.1	⊢ 𝑇 ∈ LinFn
Assertion	lnfnli	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lnfnl.1	⊢ 𝑇 ∈ LinFn
2		lnfnl	⊢ ( ( ( 𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ ) ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )
3	1 2	mpanl1	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )
4	3	3impb	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )