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 Description: Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 adj2 29706 . . . . 5 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
2 dmadjrn 29667 . . . . . 6 (𝑇 ∈ dom adj → (adj𝑇) ∈ dom adj)
3 adj1 29705 . . . . . 6 (((adj𝑇) ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((adj𝑇)‘𝑦)) = (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦))
42, 3syl3an1 1160 . . . . 5 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((adj𝑇)‘𝑦)) = (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦))
51, 4eqtr2d 2860 . . . 4 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))
653expib 1119 . . 3 (𝑇 ∈ dom adj → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦)))
76ralrimivv 3184 . 2 (𝑇 ∈ dom adj → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))