| Step | Hyp | Ref
| Expression |
| 1 | | odf1.1 |
. . . . . . . 8
⊢ 𝑋 = (Base‘𝐺) |
| 2 | | odf1.3 |
. . . . . . . 8
⊢ · =
(.g‘𝐺) |
| 3 | 1, 2 | mulgcl 19109 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋) |
| 4 | 3 | 3expa 1119 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋) |
| 5 | 4 | an32s 652 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋) |
| 6 | | odf1.4 |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) |
| 7 | 5, 6 | fmptd 7134 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:ℤ⟶𝑋) |
| 8 | 7 | adantr 480 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ⟶𝑋) |
| 9 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴)) |
| 10 | | ovex 7464 |
. . . . . . . . 9
⊢ (𝑥 · 𝐴) ∈ V |
| 11 | 9, 6, 10 | fvmpt3i 7021 |
. . . . . . . 8
⊢ (𝑦 ∈ ℤ → (𝐹‘𝑦) = (𝑦 · 𝐴)) |
| 12 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → (𝑥 · 𝐴) = (𝑧 · 𝐴)) |
| 13 | 12, 6, 10 | fvmpt3i 7021 |
. . . . . . . 8
⊢ (𝑧 ∈ ℤ → (𝐹‘𝑧) = (𝑧 · 𝐴)) |
| 14 | 11, 13 | eqeqan12d 2751 |
. . . . . . 7
⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝑦 · 𝐴) = (𝑧 · 𝐴))) |
| 15 | 14 | adantl 481 |
. . . . . 6
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝑦 · 𝐴) = (𝑧 · 𝐴))) |
| 16 | | simplr 769 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑂‘𝐴) = 0) |
| 17 | 16 | breq1d 5153 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑦 − 𝑧) ↔ 0 ∥ (𝑦 − 𝑧))) |
| 18 | | odf1.2 |
. . . . . . . . 9
⊢ 𝑂 = (od‘𝐺) |
| 19 | | eqid 2737 |
. . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 20 | 1, 18, 2, 19 | odcong 19567 |
. . . . . . . 8
⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑦 − 𝑧) ↔ (𝑦 · 𝐴) = (𝑧 · 𝐴))) |
| 21 | 20 | ad4ant124 1174 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑦 − 𝑧) ↔ (𝑦 · 𝐴) = (𝑧 · 𝐴))) |
| 22 | | zsubcl 12659 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑦 − 𝑧) ∈ ℤ) |
| 23 | 22 | adantl 481 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 − 𝑧) ∈ ℤ) |
| 24 | | 0dvds 16314 |
. . . . . . . 8
⊢ ((𝑦 − 𝑧) ∈ ℤ → (0 ∥ (𝑦 − 𝑧) ↔ (𝑦 − 𝑧) = 0)) |
| 25 | 23, 24 | syl 17 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (0 ∥ (𝑦 − 𝑧) ↔ (𝑦 − 𝑧) = 0)) |
| 26 | 17, 21, 25 | 3bitr3d 309 |
. . . . . 6
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 · 𝐴) = (𝑧 · 𝐴) ↔ (𝑦 − 𝑧) = 0)) |
| 27 | | zcn 12618 |
. . . . . . . 8
⊢ (𝑦 ∈ ℤ → 𝑦 ∈
ℂ) |
| 28 | | zcn 12618 |
. . . . . . . 8
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℂ) |
| 29 | | subeq0 11535 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑦 − 𝑧) = 0 ↔ 𝑦 = 𝑧)) |
| 30 | 27, 28, 29 | syl2an 596 |
. . . . . . 7
⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 − 𝑧) = 0 ↔ 𝑦 = 𝑧)) |
| 31 | 30 | adantl 481 |
. . . . . 6
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 − 𝑧) = 0 ↔ 𝑦 = 𝑧)) |
| 32 | 15, 26, 31 | 3bitrd 305 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ 𝑦 = 𝑧)) |
| 33 | 32 | biimpd 229 |
. . . 4
⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
| 34 | 33 | ralrimivva 3202 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) → ∀𝑦 ∈ ℤ ∀𝑧 ∈ ℤ ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
| 35 | | dff13 7275 |
. . 3
⊢ (𝐹:ℤ–1-1→𝑋 ↔ (𝐹:ℤ⟶𝑋 ∧ ∀𝑦 ∈ ℤ ∀𝑧 ∈ ℤ ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))) |
| 36 | 8, 34, 35 | sylanbrc 583 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ–1-1→𝑋) |
| 37 | 1, 18, 2, 19 | odid 19556 |
. . . . . 6
⊢ (𝐴 ∈ 𝑋 → ((𝑂‘𝐴) · 𝐴) = (0g‘𝐺)) |
| 38 | 1, 19, 2 | mulg0 19092 |
. . . . . 6
⊢ (𝐴 ∈ 𝑋 → (0 · 𝐴) = (0g‘𝐺)) |
| 39 | 37, 38 | eqtr4d 2780 |
. . . . 5
⊢ (𝐴 ∈ 𝑋 → ((𝑂‘𝐴) · 𝐴) = (0 · 𝐴)) |
| 40 | 39 | ad2antlr 727 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → ((𝑂‘𝐴) · 𝐴) = (0 · 𝐴)) |
| 41 | 1, 18 | odcl 19554 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈
ℕ0) |
| 42 | 41 | ad2antlr 727 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → (𝑂‘𝐴) ∈
ℕ0) |
| 43 | 42 | nn0zd 12639 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → (𝑂‘𝐴) ∈ ℤ) |
| 44 | | oveq1 7438 |
. . . . . 6
⊢ (𝑥 = (𝑂‘𝐴) → (𝑥 · 𝐴) = ((𝑂‘𝐴) · 𝐴)) |
| 45 | 44, 6, 10 | fvmpt3i 7021 |
. . . . 5
⊢ ((𝑂‘𝐴) ∈ ℤ → (𝐹‘(𝑂‘𝐴)) = ((𝑂‘𝐴) · 𝐴)) |
| 46 | 43, 45 | syl 17 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → (𝐹‘(𝑂‘𝐴)) = ((𝑂‘𝐴) · 𝐴)) |
| 47 | | 0zd 12625 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → 0 ∈ ℤ) |
| 48 | | oveq1 7438 |
. . . . . 6
⊢ (𝑥 = 0 → (𝑥 · 𝐴) = (0 · 𝐴)) |
| 49 | 48, 6, 10 | fvmpt3i 7021 |
. . . . 5
⊢ (0 ∈
ℤ → (𝐹‘0)
= (0 · 𝐴)) |
| 50 | 47, 49 | syl 17 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → (𝐹‘0) = (0 · 𝐴)) |
| 51 | 40, 46, 50 | 3eqtr4d 2787 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → (𝐹‘(𝑂‘𝐴)) = (𝐹‘0)) |
| 52 | | simpr 484 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → 𝐹:ℤ–1-1→𝑋) |
| 53 | | f1fveq 7282 |
. . . 4
⊢ ((𝐹:ℤ–1-1→𝑋 ∧ ((𝑂‘𝐴) ∈ ℤ ∧ 0 ∈ ℤ))
→ ((𝐹‘(𝑂‘𝐴)) = (𝐹‘0) ↔ (𝑂‘𝐴) = 0)) |
| 54 | 52, 43, 47, 53 | syl12anc 837 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → ((𝐹‘(𝑂‘𝐴)) = (𝐹‘0) ↔ (𝑂‘𝐴) = 0)) |
| 55 | 51, 54 | mpbid 232 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:ℤ–1-1→𝑋) → (𝑂‘𝐴) = 0) |
| 56 | 36, 55 | impbida 801 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) = 0 ↔ 𝐹:ℤ–1-1→𝑋)) |