Proof of Theorem oddvdsnn0
| Step | Hyp | Ref
| Expression |
| 1 | | 0nn0 12541 |
. . . . 5
⊢ 0 ∈
ℕ0 |
| 2 | | odcl.1 |
. . . . . . 7
⊢ 𝑋 = (Base‘𝐺) |
| 3 | | odcl.2 |
. . . . . . 7
⊢ 𝑂 = (od‘𝐺) |
| 4 | | odid.3 |
. . . . . . 7
⊢ · =
(.g‘𝐺) |
| 5 | | odid.4 |
. . . . . . 7
⊢ 0 =
(0g‘𝐺) |
| 6 | 2, 3, 4, 5 | mndodcong 19560 |
. . . . . 6
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑁 ∈ ℕ0 ∧ 0 ∈
ℕ0) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ (𝑁 − 0) ↔ (𝑁 · 𝐴) = (0 · 𝐴))) |
| 7 | 6 | 3expia 1122 |
. . . . 5
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑁 ∈ ℕ0 ∧ 0 ∈
ℕ0)) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ (𝑁 − 0) ↔ (𝑁 · 𝐴) = (0 · 𝐴)))) |
| 8 | 1, 7 | mpanr2 704 |
. . . 4
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ (𝑁 − 0) ↔ (𝑁 · 𝐴) = (0 · 𝐴)))) |
| 9 | 8 | 3impa 1110 |
. . 3
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ (𝑁 − 0) ↔ (𝑁 · 𝐴) = (0 · 𝐴)))) |
| 10 | | nn0cn 12536 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
| 11 | 10 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈
ℂ) |
| 12 | 11 | subid1d 11609 |
. . . . 5
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑁 − 0) = 𝑁) |
| 13 | 12 | breq2d 5155 |
. . . 4
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) ∥ (𝑁 − 0) ↔ (𝑂‘𝐴) ∥ 𝑁)) |
| 14 | 2, 5, 4 | mulg0 19092 |
. . . . . 6
⊢ (𝐴 ∈ 𝑋 → (0 · 𝐴) = 0 ) |
| 15 | 14 | 3ad2ant2 1135 |
. . . . 5
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (0 · 𝐴) = 0 ) |
| 16 | 15 | eqeq2d 2748 |
. . . 4
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑁 · 𝐴) = (0 · 𝐴) ↔ (𝑁 · 𝐴) = 0 )) |
| 17 | 13, 16 | bibi12d 345 |
. . 3
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (((𝑂‘𝐴) ∥ (𝑁 − 0) ↔ (𝑁 · 𝐴) = (0 · 𝐴)) ↔ ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))) |
| 18 | 9, 17 | sylibd 239 |
. 2
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))) |
| 19 | | simpr 484 |
. . . . 5
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = 0) |
| 20 | 19 | breq1d 5153 |
. . . 4
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → ((𝑂‘𝐴) ∥ 𝑁 ↔ 0 ∥ 𝑁)) |
| 21 | | simpl3 1194 |
. . . . 5
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → 𝑁 ∈
ℕ0) |
| 22 | | nn0z 12638 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℤ) |
| 23 | | 0dvds 16314 |
. . . . 5
⊢ (𝑁 ∈ ℤ → (0
∥ 𝑁 ↔ 𝑁 = 0)) |
| 24 | 21, 22, 23 | 3syl 18 |
. . . 4
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → (0 ∥ 𝑁 ↔ 𝑁 = 0)) |
| 25 | 15 | adantr 480 |
. . . . . 6
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → (0 · 𝐴) = 0 ) |
| 26 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑁 = 0 → (𝑁 · 𝐴) = (0 · 𝐴)) |
| 27 | 26 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑁 = 0 → ((𝑁 · 𝐴) = 0 ↔ (0 · 𝐴) = 0 )) |
| 28 | 25, 27 | syl5ibrcom 247 |
. . . . 5
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → (𝑁 = 0 → (𝑁 · 𝐴) = 0 )) |
| 29 | 2, 3, 4, 5 | odlem2 19557 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ ∧ (𝑁 · 𝐴) = 0 ) → (𝑂‘𝐴) ∈ (1...𝑁)) |
| 30 | 29 | 3com23 1127 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑋 ∧ (𝑁 · 𝐴) = 0 ∧ 𝑁 ∈ ℕ) → (𝑂‘𝐴) ∈ (1...𝑁)) |
| 31 | | elfznn 13593 |
. . . . . . . . . . 11
⊢ ((𝑂‘𝐴) ∈ (1...𝑁) → (𝑂‘𝐴) ∈ ℕ) |
| 32 | | nnne0 12300 |
. . . . . . . . . . 11
⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ≠ 0) |
| 33 | 30, 31, 32 | 3syl 18 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑋 ∧ (𝑁 · 𝐴) = 0 ∧ 𝑁 ∈ ℕ) → (𝑂‘𝐴) ≠ 0) |
| 34 | 33 | 3expia 1122 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑋 ∧ (𝑁 · 𝐴) = 0 ) → (𝑁 ∈ ℕ → (𝑂‘𝐴) ≠ 0)) |
| 35 | 34 | 3ad2antl2 1187 |
. . . . . . . 8
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 · 𝐴) = 0 ) → (𝑁 ∈ ℕ → (𝑂‘𝐴) ≠ 0)) |
| 36 | 35 | necon2bd 2956 |
. . . . . . 7
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 · 𝐴) = 0 ) → ((𝑂‘𝐴) = 0 → ¬ 𝑁 ∈ ℕ)) |
| 37 | | simpl3 1194 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 · 𝐴) = 0 ) → 𝑁 ∈
ℕ0) |
| 38 | | elnn0 12528 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
↔ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
| 39 | 37, 38 | sylib 218 |
. . . . . . . 8
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 · 𝐴) = 0 ) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
| 40 | 39 | ord 865 |
. . . . . . 7
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 · 𝐴) = 0 ) → (¬ 𝑁 ∈ ℕ → 𝑁 = 0)) |
| 41 | 36, 40 | syld 47 |
. . . . . 6
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 · 𝐴) = 0 ) → ((𝑂‘𝐴) = 0 → 𝑁 = 0)) |
| 42 | 41 | impancom 451 |
. . . . 5
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → ((𝑁 · 𝐴) = 0 → 𝑁 = 0)) |
| 43 | 28, 42 | impbid 212 |
. . . 4
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → (𝑁 = 0 ↔ (𝑁 · 𝐴) = 0 )) |
| 44 | 20, 24, 43 | 3bitrd 305 |
. . 3
⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) = 0) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 )) |
| 45 | 44 | ex 412 |
. 2
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) = 0 → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))) |
| 46 | 2, 3 | odcl 19554 |
. . . 4
⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈
ℕ0) |
| 47 | 46 | 3ad2ant2 1135 |
. . 3
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑂‘𝐴) ∈
ℕ0) |
| 48 | | elnn0 12528 |
. . 3
⊢ ((𝑂‘𝐴) ∈ ℕ0 ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0)) |
| 49 | 47, 48 | sylib 218 |
. 2
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0)) |
| 50 | 18, 45, 49 | mpjaod 861 |
1
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 )) |