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Metamath Proof Explorer

Theorem adjval

Description: Value of the adjoint function for T in the domain of adjh . (Contributed by NM, 19-Feb-2006) (Revised by Mario Carneiro, 24-Dec-2016) (New usage is discouraged.)

Ref Expression
Assertion adjval ( 𝑇 ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) = ( ℩ 𝑢 ∈ ( ℋ ↑m ℋ ) ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 dmadjop ⊢ ( 𝑇 ∈ dom adjℎ → 𝑇 : ℋ ⟶ ℋ )
2 1 biantrurd ⊢ ( 𝑇 ∈ dom adjℎ → ( ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) )
3 ax-hilex ⊢ ℋ ∈ V
4 3 3 elmap ⊢ ( 𝑢 ∈ ( ℋ ↑m ℋ ) ↔ 𝑢 : ℋ ⟶ ℋ )
5 4 anbi1i ⊢ ( ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) )
6 3anass ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) )
7 2 5 6 3bitr4g ⊢ ( 𝑇 ∈ dom adjℎ → ( ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) )
8 7 iotabidv ⊢ ( 𝑇 ∈ dom adjℎ → ( ℩ 𝑢 ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) = ( ℩ 𝑢 ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) )
9 df-riota ⊢ ( ℩ 𝑢 ∈ ( ℋ ↑m ℋ ) ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) = ( ℩ 𝑢 ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) )
10 9 a1i ⊢ ( 𝑇 ∈ dom adjℎ → ( ℩ 𝑢 ∈ ( ℋ ↑m ℋ ) ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) = ( ℩ 𝑢 ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) )
11 dfadj2 ⊢ adjℎ = { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) }
12 feq1 ⊢ ( 𝑡 = 𝑇 → ( 𝑡 : ℋ ⟶ ℋ ↔ 𝑇 : ℋ ⟶ ℋ ) )
13 fveq1 ⊢ ( 𝑡 = 𝑇 → ( 𝑡 ‘ 𝑦 ) = ( 𝑇 ‘ 𝑦 ) )
14 13 oveq2d ⊢ ( 𝑡 = 𝑇 → ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) )
15 14 eqeq1d ⊢ ( 𝑡 = 𝑇 → ( ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) )
16 15 2ralbidv ⊢ ( 𝑡 = 𝑇 → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) )
17 12 16 3anbi13d ⊢ ( 𝑡 = 𝑇 → ( ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) )
18 11 17 fvopab5 ⊢ ( 𝑇 ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) = ( ℩ 𝑢 ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) )
19 8 10 18 3eqtr4rd ⊢ ( 𝑇 ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) = ( ℩ 𝑢 ∈ ( ℋ ↑m ℋ ) ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) )