Theorem hosval

Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	hosval	⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hosmval	⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 +op 𝑇 ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ) )
2	1	fveq1d	⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) = ( ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ) ‘ 𝐴 ) )
3		fveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝐴 ) )
4		fveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑇 ‘ 𝑥 ) = ( 𝑇 ‘ 𝐴 ) )
5	3 4	oveq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) )
6		eqid	⊢ ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) )
7		ovex	⊢ ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) ∈ V
8	5 6 7	fvmpt	⊢ ( 𝐴 ∈ ℋ → ( ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) )
9	2 8	sylan9eq	⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) )
10	9	3impa	⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) )