| Step | Hyp | Ref
| Expression |
| 1 | | phnv 30833 |
. . . . 5
⊢ (𝑈 ∈ CPreHilOLD
→ 𝑈 ∈
NrmCVec) |
| 2 | | ajval.1 |
. . . . . 6
⊢ 𝑋 = (BaseSet‘𝑈) |
| 3 | | ajval.2 |
. . . . . 6
⊢ 𝑌 = (BaseSet‘𝑊) |
| 4 | | ajval.3 |
. . . . . 6
⊢ 𝑃 =
(·𝑖OLD‘𝑈) |
| 5 | | ajval.4 |
. . . . . 6
⊢ 𝑄 =
(·𝑖OLD‘𝑊) |
| 6 | | ajval.5 |
. . . . . 6
⊢ 𝐴 = (𝑈adj𝑊) |
| 7 | 2, 3, 4, 5, 6 | ajfval 30828 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}) |
| 8 | 1, 7 | sylan 580 |
. . . 4
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝑊 ∈ NrmCVec)
→ 𝐴 = {〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}) |
| 9 | 8 | fveq1d 6908 |
. . 3
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝑊 ∈ NrmCVec)
→ (𝐴‘𝑇) = ({〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇)) |
| 10 | 9 | 3adant3 1133 |
. 2
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝑊 ∈ NrmCVec
∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = ({〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇)) |
| 11 | 2 | fvexi 6920 |
. . . . . 6
⊢ 𝑋 ∈ V |
| 12 | | fex 7246 |
. . . . . 6
⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑋 ∈ V) → 𝑇 ∈ V) |
| 13 | 11, 12 | mpan2 691 |
. . . . 5
⊢ (𝑇:𝑋⟶𝑌 → 𝑇 ∈ V) |
| 14 | | eqid 2737 |
. . . . . 6
⊢
{〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))} = {〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))} |
| 15 | | feq1 6716 |
. . . . . . 7
⊢ (𝑡 = 𝑇 → (𝑡:𝑋⟶𝑌 ↔ 𝑇:𝑋⟶𝑌)) |
| 16 | | fveq1 6905 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥)) |
| 17 | 16 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥)𝑄𝑦) = ((𝑇‘𝑥)𝑄𝑦)) |
| 18 | 17 | eqeq1d 2739 |
. . . . . . . 8
⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))) |
| 19 | 18 | 2ralbidv 3221 |
. . . . . . 7
⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))) |
| 20 | 15, 19 | 3anbi13d 1440 |
. . . . . 6
⊢ (𝑡 = 𝑇 → ((𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 21 | 14, 20 | fvopab5 7049 |
. . . . 5
⊢ (𝑇 ∈ V → ({〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 22 | 13, 21 | syl 17 |
. . . 4
⊢ (𝑇:𝑋⟶𝑌 → ({〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 23 | | 3anass 1095 |
. . . . . 6
⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑇:𝑋⟶𝑌 ∧ (𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 24 | 23 | baib 535 |
. . . . 5
⊢ (𝑇:𝑋⟶𝑌 → ((𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 25 | 24 | iotabidv 6545 |
. . . 4
⊢ (𝑇:𝑋⟶𝑌 → (℩𝑠(𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 26 | 22, 25 | eqtrd 2777 |
. . 3
⊢ (𝑇:𝑋⟶𝑌 → ({〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 27 | 26 | 3ad2ant3 1136 |
. 2
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝑊 ∈ NrmCVec
∧ 𝑇:𝑋⟶𝑌) → ({〈𝑡, 𝑠〉 ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |
| 28 | 10, 27 | eqtrd 2777 |
1
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝑊 ∈ NrmCVec
∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) |