Theorem lnopaddmuli

Description: Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		lnopl.1	⊢ 𝑇 ∈ LinOp
2		hvmulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐶 ) ∈ ℋ )
3	1	lnopaddi	⊢ ( ( 𝐵 ∈ ℋ ∧ ( 𝐴 ·ℎ 𝐶 ) ∈ ℋ ) → ( 𝑇 ‘ ( 𝐵 +ℎ ( 𝐴 ·ℎ 𝐶 ) ) ) = ( ( 𝑇 ‘ 𝐵 ) +ℎ ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐶 ) ) ) )
4	2 3	sylan2	⊢ ( ( 𝐵 ∈ ℋ ∧ ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ‘ ( 𝐵 +ℎ ( 𝐴 ·ℎ 𝐶 ) ) ) = ( ( 𝑇 ‘ 𝐵 ) +ℎ ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐶 ) ) ) )
5	4	3impb	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐵 +ℎ ( 𝐴 ·ℎ 𝐶 ) ) ) = ( ( 𝑇 ‘ 𝐵 ) +ℎ ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐶 ) ) ) )
6	5	3com12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐵 +ℎ ( 𝐴 ·ℎ 𝐶 ) ) ) = ( ( 𝑇 ‘ 𝐵 ) +ℎ ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐶 ) ) ) )
7	1	lnopmuli	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐶 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐶 ) ) )
8	7	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐶 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐶 ) ) )
9	8	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐵 ) +ℎ ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐶 ) ) ) = ( ( 𝑇 ‘ 𝐵 ) +ℎ ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐶 ) ) ) )
10	6 9	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐵 +ℎ ( 𝐴 ·ℎ 𝐶 ) ) ) = ( ( 𝑇 ‘ 𝐵 ) +ℎ ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐶 ) ) ) )