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 Description: A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjeq ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))) → (adj𝑇) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝑇,𝑦

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
2 df-adjh 29039 . . . . . 6 adj = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑧𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)))}
32eleq2i 2832 . . . . 5 (⟨𝑇, 𝑆⟩ ∈ adj ↔ ⟨𝑇, 𝑆⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑧𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)))})
4 ax-hilex 28187 . . . . . . 7 ℋ ∈ V
5 fex 6655 . . . . . . 7 ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V)
64, 5mpan2 709 . . . . . 6 (𝑇: ℋ⟶ ℋ → 𝑇 ∈ V)
7 fex 6655 . . . . . . 7 ((𝑆: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑆 ∈ V)
84, 7mpan2 709 . . . . . 6 (𝑆: ℋ⟶ ℋ → 𝑆 ∈ V)
9 feq1 6188 . . . . . . . 8 (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
10 fveq1 6353 . . . . . . . . . . 11 (𝑧 = 𝑇 → (𝑧𝑥) = (𝑇𝑥))
1110oveq1d 6830 . . . . . . . . . 10 (𝑧 = 𝑇 → ((𝑧𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))
1211eqeq1d 2763 . . . . . . . . 9 (𝑧 = 𝑇 → (((𝑧𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)) ↔ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦))))
13122ralbidv 3128 . . . . . . . 8 (𝑧 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑧𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦))))
149, 133anbi13d 1550 . . . . . . 7 (𝑧 = 𝑇 → ((𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑧𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦))) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)))))
15 feq1 6188 . . . . . . . 8 (𝑤 = 𝑆 → (𝑤: ℋ⟶ ℋ ↔ 𝑆: ℋ⟶ ℋ))
16 fveq1 6353 . . . . . . . . . . 11 (𝑤 = 𝑆 → (𝑤𝑦) = (𝑆𝑦))
1716oveq2d 6831 . . . . . . . . . 10 (𝑤 = 𝑆 → (𝑥 ·ih (𝑤𝑦)) = (𝑥 ·ih (𝑆𝑦)))
1817eqeq2d 2771 . . . . . . . . 9 (𝑤 = 𝑆 → (((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)) ↔ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))))
19182ralbidv 3128 . . . . . . . 8 (𝑤 = 𝑆 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))))
2015, 193anbi23d 1551 . . . . . . 7 (𝑤 = 𝑆 → ((𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦))) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦)))))
2114, 20opelopabg 5144 . . . . . 6 ((𝑇 ∈ V ∧ 𝑆 ∈ V) → (⟨𝑇, 𝑆⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑧𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)))} ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦)))))
226, 8, 21syl2an 495 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) → (⟨𝑇, 𝑆⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑧𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑤𝑦)))} ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦)))))
233, 22syl5bb 272 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) → (⟨𝑇, 𝑆⟩ ∈ adj ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦)))))
24 df-3an 1074 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))) ↔ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))))
2524baibr 983 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦)))))
2623, 25bitr4d 271 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) → (⟨𝑇, 𝑆⟩ ∈ adj ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))))
2726biimp3ar 1582 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))) → ⟨𝑇, 𝑆⟩ ∈ adj)