Step | Hyp | Ref
| Expression |
1 | | grplmulf1o.n |
. 2
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑋 + 𝑥)) |
2 | | grplmulf1o.b |
. . . 4
⊢ 𝐵 = (Base‘𝐺) |
3 | | grplmulf1o.p |
. . . 4
⊢ + =
(+g‘𝐺) |
4 | 2, 3 | grpcl 18373 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵) |
5 | 4 | 3expa 1120 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵) |
6 | | eqid 2737 |
. . . 4
⊢
(invg‘𝐺) = (invg‘𝐺) |
7 | 2, 6 | grpinvcl 18415 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵) |
8 | 2, 3 | grpcl 18373 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧
((invg‘𝐺)‘𝑋) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵) |
9 | 8 | 3expa 1120 |
. . 3
⊢ (((𝐺 ∈ Grp ∧
((invg‘𝐺)‘𝑋) ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵) |
10 | 7, 9 | syldanl 605 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵) |
11 | | eqcom 2744 |
. . 3
⊢ (𝑥 =
(((invg‘𝐺)‘𝑋) + 𝑦) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥) |
12 | | simpll 767 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp) |
13 | 10 | adantrl 716 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵) |
14 | | simprl 771 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵) |
15 | | simplr 769 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵) |
16 | 2, 3 | grplcan 18425 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧
((((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 +
(((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥)) |
17 | 12, 13, 14, 15, 16 | syl13anc 1374 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 +
(((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥)) |
18 | | eqid 2737 |
. . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) |
19 | 2, 3, 18, 6 | grprinv 18417 |
. . . . . . . 8
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 +
((invg‘𝐺)‘𝑋)) = (0g‘𝐺)) |
20 | 19 | adantr 484 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 +
((invg‘𝐺)‘𝑋)) = (0g‘𝐺)) |
21 | 20 | oveq1d 7228 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 +
((invg‘𝐺)‘𝑋)) + 𝑦) = ((0g‘𝐺) + 𝑦)) |
22 | 7 | adantr 484 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑋) ∈ 𝐵) |
23 | | simprr 773 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) |
24 | 2, 3 | grpass 18374 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 +
((invg‘𝐺)‘𝑋)) + 𝑦) = (𝑋 +
(((invg‘𝐺)‘𝑋) + 𝑦))) |
25 | 12, 15, 22, 23, 24 | syl13anc 1374 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 +
((invg‘𝐺)‘𝑋)) + 𝑦) = (𝑋 +
(((invg‘𝐺)‘𝑋) + 𝑦))) |
26 | 2, 3, 18 | grplid 18397 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦) |
27 | 26 | ad2ant2rl 749 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦) |
28 | 21, 25, 27 | 3eqtr3d 2785 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 +
(((invg‘𝐺)‘𝑋) + 𝑦)) = 𝑦) |
29 | 28 | eqeq1d 2739 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 +
(((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ 𝑦 = (𝑋 + 𝑥))) |
30 | 17, 29 | bitr3d 284 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥 ↔ 𝑦 = (𝑋 + 𝑥))) |
31 | 11, 30 | syl5bb 286 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (((invg‘𝐺)‘𝑋) + 𝑦) ↔ 𝑦 = (𝑋 + 𝑥))) |
32 | 1, 5, 10, 31 | f1o2d 7459 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵) |