| 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 2738 | . 2
⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) →
(+g‘𝐺) =
(+g‘𝐺)) | 
| 4 |  | simpl 482 | . 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 2737 | . . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) | 
| 9 | 1, 8 | grpass 18960 | . . . . . . . . 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 1374 | . . . . . . . 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 2737 | . . . . . . . . . . . 12
⊢
(.g‘𝐺) = (.g‘𝐺) | 
| 12 | 1, 11, 8 | mulg2 19101 | . . . . . . . . . . 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 2737 | . . . . . . . . . . . 12
⊢
(0g‘𝐺) = (0g‘𝐺) | 
| 17 | 1, 15, 11, 16 | gexdvdsi 19601 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝐸 ∥ 2) →
(2(.g‘𝐺)𝑦) = (0g‘𝐺)) | 
| 18 | 5, 7, 14, 17 | syl3anc 1373 | . . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)𝑦) = (0g‘𝐺)) | 
| 19 | 13, 18 | eqtr3d 2779 | . . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑦) = (0g‘𝐺)) | 
| 20 | 19 | oveq2d 7447 | . . . . . . . 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 18986 | . . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) | 
| 22 | 5, 6, 21 | syl2anc 584 | . . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) | 
| 23 | 10, 20, 22 | 3eqtrd 2781 | . . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑦) = 𝑥) | 
| 24 | 23 | oveq1d 7446 | . . . . . 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 19101 | . . . . . . 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 19601 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ 2) →
(2(.g‘𝐺)𝑥) = (0g‘𝐺)) | 
| 28 | 5, 6, 14, 27 | syl3anc 1373 | . . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)𝑥) = (0g‘𝐺)) | 
| 29 | 24, 26, 28 | 3eqtr2d 2783 | . . . . 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 18959 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑋) | 
| 31 | 5, 6, 7, 30 | syl3anc 1373 | . . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑋) | 
| 32 | 1, 15, 11, 16 | gexdvdsi 19601 | . . . . . 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 1373 | . . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (2(.g‘𝐺)(𝑥(+g‘𝐺)𝑦)) = (0g‘𝐺)) | 
| 34 | 1, 11, 8 | mulg2 19101 | . . . . . 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 2783 | . . . 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 18960 | . . . . 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 1374 | . . . 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 2779 | . . 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 18959 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑥) ∈ 𝑋) | 
| 41 | 5, 7, 6, 40 | syl3anc 1373 | . . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑥) ∈ 𝑋) | 
| 42 | 1, 8 | grplcan 19018 | . . . 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 1374 | . . 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 232 | . 2
⊢ (((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) | 
| 45 | 2, 3, 4, 44 | isabld 19813 | 1
⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝐺 ∈ Abel) |