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 Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adj1 ((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵))

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funadj 28591 . . . . . . 7 Fun adj
31, 2mpan 705 . . . . . 6 (𝑇 ∈ dom adj → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
4 dfadj2 28590 . . . . . 6 adj = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))}
53, 4syl6eleq 2708 . . . . 5 (𝑇 ∈ dom adj → ⟨𝑇, (adj𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))})
6 fvex 6158 . . . . . 6 (adj𝑇) ∈ V
7 feq1 5983 . . . . . . . 8 (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8 fveq1 6147 . . . . . . . . . . 11 (𝑧 = 𝑇 → (𝑧𝑦) = (𝑇𝑦))
98oveq2d 6620 . . . . . . . . . 10 (𝑧 = 𝑇 → (𝑥 ·ih (𝑧𝑦)) = (𝑥 ·ih (𝑇𝑦)))
109eqeq1d 2623 . . . . . . . . 9 (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)))
11102ralbidv 2983 . . . . . . . 8 (𝑧 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)))
127, 113anbi13d 1398 . . . . . . 7 (𝑧 = 𝑇 → ((𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦))))
13 feq1 5983 . . . . . . . 8 (𝑤 = (adj𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj𝑇): ℋ⟶ ℋ))
14 fveq1 6147 . . . . . . . . . . 11 (𝑤 = (adj𝑇) → (𝑤𝑥) = ((adj𝑇)‘𝑥))
1514oveq1d 6619 . . . . . . . . . 10 (𝑤 = (adj𝑇) → ((𝑤𝑥) ·ih 𝑦) = (((adj𝑇)‘𝑥) ·ih 𝑦))
1615eqeq2d 2631 . . . . . . . . 9 (𝑤 = (adj𝑇) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
17162ralbidv 2983 . . . . . . . 8 (𝑤 = (adj𝑇) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
1813, 173anbi23d 1399 . . . . . . 7 (𝑤 = (adj𝑇) → ((𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))))
1912, 18opelopabg 4953 . . . . . 6 ((𝑇 ∈ dom adj ∧ (adj𝑇) ∈ V) → (⟨𝑇, (adj𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))))
206, 19mpan2 706 . . . . 5 (𝑇 ∈ dom adj → (⟨𝑇, (adj𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))))
215, 20mpbid 222 . . . 4 (𝑇 ∈ dom adj → (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
2221simp3d 1073 . . 3 (𝑇 ∈ dom adj → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))
23 oveq1 6611 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
24 fveq2 6148 . . . . . 6 (𝑥 = 𝐴 → ((adj𝑇)‘𝑥) = ((adj𝑇)‘𝐴))
2524oveq1d 6619 . . . . 5 (𝑥 = 𝐴 → (((adj𝑇)‘𝑥) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝑦))
2623, 25eqeq12d 2636 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦)))
27 fveq2 6148 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
2827oveq2d 6620 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
29 oveq2 6612 . . . . 5 (𝑦 = 𝐵 → (((adj𝑇)‘𝐴) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝐵))
3028, 29eqeq12d 2636 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
3126, 30rspc2v 3306 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
3222, 31syl5com 31 . 2 (𝑇 ∈ dom adj → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
33323impib 1259 1 ((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵))