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Metamath Proof Explorer

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 ) ) )