| Step | Hyp | Ref
| Expression |
| 1 | | simp1l 1198 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → 𝐾 ∈ HL) |
| 2 | | simp1r 1199 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → 𝑊 ∈ 𝐻) |
| 3 | | simp21 1207 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → 𝑈 ∈ 𝐸) |
| 4 | | simp22 1208 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → 𝑉 ∈ 𝐸) |
| 5 | | simp11 1204 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 6 | | simp12 1205 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹))) |
| 7 | | simp13l 1289 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → 𝐹 ∈ 𝑇) |
| 8 | | simp13r 1290 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → 𝐹 ≠ ( I ↾ 𝐵)) |
| 9 | | simp2 1138 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → ℎ ∈ 𝑇) |
| 10 | 7, 8, 9 | 3jca 1129 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ ℎ ∈ 𝑇)) |
| 11 | | simp3 1139 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → ℎ ≠ ( I ↾ 𝐵)) |
| 12 | | tendocan.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐾) |
| 13 | | tendocan.h |
. . . . . 6
⊢ 𝐻 = (LHyp‘𝐾) |
| 14 | | tendocan.t |
. . . . . 6
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| 15 | | eqid 2737 |
. . . . . 6
⊢
((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) |
| 16 | | tendocan.e |
. . . . . 6
⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
| 17 | 12, 13, 14, 15, 16 | cdlemj3 40825 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ ℎ ∈ 𝑇)) ∧ ℎ ≠ ( I ↾ 𝐵)) → (𝑈‘ℎ) = (𝑉‘ℎ)) |
| 18 | 5, 6, 10, 11, 17 | syl31anc 1375 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) → (𝑈‘ℎ) = (𝑉‘ℎ)) |
| 19 | 18 | 3exp 1120 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → (ℎ ∈ 𝑇 → (ℎ ≠ ( I ↾ 𝐵) → (𝑈‘ℎ) = (𝑉‘ℎ)))) |
| 20 | 19 | ralrimiv 3145 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ∀ℎ ∈ 𝑇 (ℎ ≠ ( I ↾ 𝐵) → (𝑈‘ℎ) = (𝑉‘ℎ))) |
| 21 | 12, 13, 14, 16 | tendoeq2 40776 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀ℎ ∈ 𝑇 (ℎ ≠ ( I ↾ 𝐵) → (𝑈‘ℎ) = (𝑉‘ℎ))) → 𝑈 = 𝑉) |
| 22 | 1, 2, 3, 4, 20, 21 | syl221anc 1383 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑉‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → 𝑈 = 𝑉) |