Step | Hyp | Ref
| Expression |
1 | | gexex.1 |
. . 3
⊢ 𝑋 = (Base‘𝐺) |
2 | 1 | a1i 11 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝑋 = (Base‘𝐺)) |
3 | | eqidd 2739 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) →
(+g‘𝐺) =
(+g‘𝐺)) |
4 | | simpl 486 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝐺 ∈ Grp) |
5 | | simp1l 1198 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ Grp) |
6 | | simp2 1138 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
7 | | simp3 1139 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋) |
8 | | eqid 2738 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
9 | 1, 8 | grpass 18228 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑦))) |
10 | 5, 6, 7, 7, 9 | syl13anc 1373 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑦))) |
11 | | eqid 2738 |
. . . . . . . . . . . 12
⊢
(.g‘𝐺) = (.g‘𝐺) |
12 | 1, 11, 8 | mulg2 18355 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ 𝑋 → (2(.g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑦)) |
13 | 7, 12 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑦)) |
14 | | simp1r 1199 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝐸 ∥ 2) |
15 | | gexex.2 |
. . . . . . . . . . . 12
⊢ 𝐸 = (gEx‘𝐺) |
16 | | eqid 2738 |
. . . . . . . . . . . 12
⊢
(0g‘𝐺) = (0g‘𝐺) |
17 | 1, 15, 11, 16 | gexdvdsi 18826 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝐸 ∥ 2) →
(2(.g‘𝐺)𝑦) = (0g‘𝐺)) |
18 | 5, 7, 14, 17 | syl3anc 1372 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)𝑦) = (0g‘𝐺)) |
19 | 13, 18 | eqtr3d 2775 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑦) = (0g‘𝐺)) |
20 | 19 | oveq2d 7186 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑦)) = (𝑥(+g‘𝐺)(0g‘𝐺))) |
21 | 1, 8, 16 | grprid 18252 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) |
22 | 5, 6, 21 | syl2anc 587 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) |
23 | 10, 20, 22 | 3eqtrd 2777 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦) = 𝑥) |
24 | 23 | oveq1d 7185 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝑥)) |
25 | 1, 11, 8 | mulg2 18355 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑋 → (2(.g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝑥)) |
26 | 6, 25 | syl 17 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝑥)) |
27 | 1, 15, 11, 16 | gexdvdsi 18826 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ 2) →
(2(.g‘𝐺)𝑥) = (0g‘𝐺)) |
28 | 5, 6, 14, 27 | syl3anc 1372 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)𝑥) = (0g‘𝐺)) |
29 | 24, 26, 28 | 3eqtr2d 2779 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦)(+g‘𝐺)𝑥) = (0g‘𝐺)) |
30 | 1, 8 | grpcl 18227 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑋) |
31 | 5, 6, 7, 30 | syl3anc 1372 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑋) |
32 | 1, 15, 11, 16 | gexdvdsi 18826 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑋 ∧ 𝐸 ∥ 2) →
(2(.g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = (0g‘𝐺)) |
33 | 5, 31, 14, 32 | syl3anc 1372 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = (0g‘𝐺)) |
34 | 1, 11, 8 | mulg2 18355 |
. . . . . 6
⊢ ((𝑥(+g‘𝐺)𝑦) ∈ 𝑋 → (2(.g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑥(+g‘𝐺)𝑦))) |
35 | 31, 34 | syl 17 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑥(+g‘𝐺)𝑦))) |
36 | 29, 33, 35 | 3eqtr2d 2779 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦)(+g‘𝐺)𝑥) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑥(+g‘𝐺)𝑦))) |
37 | 1, 8 | grpass 18228 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ ((𝑥(+g‘𝐺)𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦)(+g‘𝐺)𝑥) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑦(+g‘𝐺)𝑥))) |
38 | 5, 31, 7, 6, 37 | syl13anc 1373 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦)(+g‘𝐺)𝑥) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑦(+g‘𝐺)𝑥))) |
39 | 36, 38 | eqtr3d 2775 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑦(+g‘𝐺)𝑥))) |
40 | 1, 8 | grpcl 18227 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑥) ∈ 𝑋) |
41 | 5, 7, 6, 40 | syl3anc 1372 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑥) ∈ 𝑋) |
42 | 1, 8 | grplcan 18279 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧ ((𝑥(+g‘𝐺)𝑦) ∈ 𝑋 ∧ (𝑦(+g‘𝐺)𝑥) ∈ 𝑋 ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑋)) → (((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑦(+g‘𝐺)𝑥)) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
43 | 5, 31, 41, 31, 42 | syl13anc 1373 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)(𝑦(+g‘𝐺)𝑥)) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
44 | 39, 43 | mpbid 235 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
45 | 2, 3, 4, 44 | isabld 19038 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝐺 ∈ Abel) |