Theorem hfsval

Description: Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	hfsval	⊢ ( ( 𝑆 : ℋ ⟶ ℂ ∧ 𝑇 : ℋ ⟶ ℂ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +fn 𝑇 ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑇 ‘ 𝐴 ) ) )

Proof

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