| Step | Hyp | Ref
| Expression |
| 1 | | grpoinveu.1 |
. . . . 5
⊢ 𝑋 = ran 𝐺 |
| 2 | | grpoinveu.2 |
. . . . 5
⊢ 𝑈 = (GId‘𝐺) |
| 3 | 1, 2 | grpoidinv2 30534 |
. . . 4
⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))) |
| 4 | | simpl 482 |
. . . . . 6
⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) |
| 5 | 4 | reximi 3084 |
. . . . 5
⊢
(∃𝑦 ∈
𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) |
| 6 | 5 | adantl 481 |
. . . 4
⊢ ((((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) |
| 7 | 3, 6 | syl 17 |
. . 3
⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) |
| 8 | | eqtr3 2763 |
. . . . . . . . . . . 12
⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴)) |
| 9 | 1 | grporcan 30537 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧)) |
| 10 | 8, 9 | imbitrid 244 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)) |
| 11 | 10 | 3exp2 1355 |
. . . . . . . . . 10
⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝐴 ∈ 𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))))) |
| 12 | 11 | com24 95 |
. . . . . . . . 9
⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))))) |
| 13 | 12 | imp41 425 |
. . . . . . . 8
⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)) |
| 14 | 13 | an32s 652 |
. . . . . . 7
⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)) |
| 15 | 14 | expd 415 |
. . . . . 6
⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))) |
| 16 | 15 | ralrimdva 3154 |
. . . . 5
⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))) |
| 17 | 16 | ancld 550 |
. . . 4
⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))) |
| 18 | 17 | reximdva 3168 |
. . 3
⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈 → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))) |
| 19 | 7, 18 | mpd 15 |
. 2
⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))) |
| 20 | | oveq1 7438 |
. . . 4
⊢ (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴)) |
| 21 | 20 | eqeq1d 2739 |
. . 3
⊢ (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈)) |
| 22 | 21 | reu8 3739 |
. 2
⊢
(∃!𝑦 ∈
𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))) |
| 23 | 19, 22 | sylibr 234 |
1
⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) |