Proof of Theorem mndodconglem
| Step | Hyp | Ref
| Expression |
| 1 | | mndodconglem.2 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑋) |
| 2 | | mndodconglem.3 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑂‘𝐴) ∈ ℕ) |
| 3 | 2 | nnred 12281 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) |
| 4 | 3 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → (𝑂‘𝐴) ∈ ℂ) |
| 5 | | mndodconglem.4 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 6 | 5 | nn0red 12588 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 7 | 6 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 8 | | mndodconglem.5 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 9 | 8 | nn0red 12588 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 10 | 9 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 11 | 4, 7, 10 | addsubassd 11640 |
. . . . . . . 8
⊢ (𝜑 → (((𝑂‘𝐴) + 𝑀) − 𝑁) = ((𝑂‘𝐴) + (𝑀 − 𝑁))) |
| 12 | 2 | nnzd 12640 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑂‘𝐴) ∈ ℤ) |
| 13 | 5 | nn0zd 12639 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 14 | 12, 13 | zaddcld 12726 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑂‘𝐴) + 𝑀) ∈ ℤ) |
| 15 | 14 | zred 12722 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑂‘𝐴) + 𝑀) ∈ ℝ) |
| 16 | | mndodconglem.7 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 < (𝑂‘𝐴)) |
| 17 | | nn0addge1 12572 |
. . . . . . . . . . 11
⊢ (((𝑂‘𝐴) ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + 𝑀)) |
| 18 | 3, 5, 17 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + 𝑀)) |
| 19 | 9, 3, 15, 16, 18 | ltletrd 11421 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 < ((𝑂‘𝐴) + 𝑀)) |
| 20 | 8 | nn0zd 12639 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 21 | | znnsub 12663 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ ((𝑂‘𝐴) + 𝑀) ∈ ℤ) → (𝑁 < ((𝑂‘𝐴) + 𝑀) ↔ (((𝑂‘𝐴) + 𝑀) − 𝑁) ∈ ℕ)) |
| 22 | 20, 14, 21 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 < ((𝑂‘𝐴) + 𝑀) ↔ (((𝑂‘𝐴) + 𝑀) − 𝑁) ∈ ℕ)) |
| 23 | 19, 22 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → (((𝑂‘𝐴) + 𝑀) − 𝑁) ∈ ℕ) |
| 24 | 11, 23 | eqeltrrd 2842 |
. . . . . . 7
⊢ (𝜑 → ((𝑂‘𝐴) + (𝑀 − 𝑁)) ∈ ℕ) |
| 25 | 4, 7, 10 | addsub12d 11643 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑂‘𝐴) + (𝑀 − 𝑁)) = (𝑀 + ((𝑂‘𝐴) − 𝑁))) |
| 26 | 25 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → (((𝑂‘𝐴) + (𝑀 − 𝑁)) · 𝐴) = ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴)) |
| 27 | | mndodconglem.8 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑀 · 𝐴) = (𝑁 · 𝐴)) |
| 28 | 27 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 · 𝐴)(+g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴)) = ((𝑁 · 𝐴)(+g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴))) |
| 29 | | mndodconglem.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 30 | | znnsub 12663 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑁 < (𝑂‘𝐴) ↔ ((𝑂‘𝐴) − 𝑁) ∈ ℕ)) |
| 31 | 20, 12, 30 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 < (𝑂‘𝐴) ↔ ((𝑂‘𝐴) − 𝑁) ∈ ℕ)) |
| 32 | 16, 31 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑂‘𝐴) − 𝑁) ∈ ℕ) |
| 33 | 32 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑂‘𝐴) − 𝑁) ∈
ℕ0) |
| 34 | | odcl.1 |
. . . . . . . . . . . 12
⊢ 𝑋 = (Base‘𝐺) |
| 35 | | odid.3 |
. . . . . . . . . . . 12
⊢ · =
(.g‘𝐺) |
| 36 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 37 | 34, 35, 36 | mulgnn0dir 19122 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0
∧ ((𝑂‘𝐴) − 𝑁) ∈ ℕ0 ∧ 𝐴 ∈ 𝑋)) → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑀 · 𝐴)(+g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴))) |
| 38 | 29, 5, 33, 1, 37 | syl13anc 1374 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑀 · 𝐴)(+g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴))) |
| 39 | 34, 35, 36 | mulgnn0dir 19122 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ (𝑁 ∈ ℕ0
∧ ((𝑂‘𝐴) − 𝑁) ∈ ℕ0 ∧ 𝐴 ∈ 𝑋)) → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑁 · 𝐴)(+g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴))) |
| 40 | 29, 8, 33, 1, 39 | syl13anc 1374 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑁 · 𝐴)(+g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴))) |
| 41 | 28, 38, 40 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴)) |
| 42 | 10, 4 | pncan3d 11623 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 + ((𝑂‘𝐴) − 𝑁)) = (𝑂‘𝐴)) |
| 43 | 42 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑂‘𝐴) · 𝐴)) |
| 44 | | odcl.2 |
. . . . . . . . . . . 12
⊢ 𝑂 = (od‘𝐺) |
| 45 | | odid.4 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘𝐺) |
| 46 | 34, 44, 35, 45 | odid 19556 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑋 → ((𝑂‘𝐴) · 𝐴) = 0 ) |
| 47 | 1, 46 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑂‘𝐴) · 𝐴) = 0 ) |
| 48 | 43, 47 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = 0 ) |
| 49 | 41, 48 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = 0 ) |
| 50 | 26, 49 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (((𝑂‘𝐴) + (𝑀 − 𝑁)) · 𝐴) = 0 ) |
| 51 | 34, 44, 35, 45 | odlem2 19557 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑋 ∧ ((𝑂‘𝐴) + (𝑀 − 𝑁)) ∈ ℕ ∧ (((𝑂‘𝐴) + (𝑀 − 𝑁)) · 𝐴) = 0 ) → (𝑂‘𝐴) ∈ (1...((𝑂‘𝐴) + (𝑀 − 𝑁)))) |
| 52 | 1, 24, 50, 51 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝑂‘𝐴) ∈ (1...((𝑂‘𝐴) + (𝑀 − 𝑁)))) |
| 53 | | elfzle2 13568 |
. . . . . 6
⊢ ((𝑂‘𝐴) ∈ (1...((𝑂‘𝐴) + (𝑀 − 𝑁))) → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + (𝑀 − 𝑁))) |
| 54 | 52, 53 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + (𝑀 − 𝑁))) |
| 55 | 13, 20 | zsubcld 12727 |
. . . . . . 7
⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℤ) |
| 56 | 55 | zred 12722 |
. . . . . 6
⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℝ) |
| 57 | 3, 56 | addge01d 11851 |
. . . . 5
⊢ (𝜑 → (0 ≤ (𝑀 − 𝑁) ↔ (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + (𝑀 − 𝑁)))) |
| 58 | 54, 57 | mpbird 257 |
. . . 4
⊢ (𝜑 → 0 ≤ (𝑀 − 𝑁)) |
| 59 | 6, 9 | subge0d 11853 |
. . . 4
⊢ (𝜑 → (0 ≤ (𝑀 − 𝑁) ↔ 𝑁 ≤ 𝑀)) |
| 60 | 58, 59 | mpbid 232 |
. . 3
⊢ (𝜑 → 𝑁 ≤ 𝑀) |
| 61 | 6, 9 | letri3d 11403 |
. . . 4
⊢ (𝜑 → (𝑀 = 𝑁 ↔ (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))) |
| 62 | 61 | biimprd 248 |
. . 3
⊢ (𝜑 → ((𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 𝑀 = 𝑁)) |
| 63 | 60, 62 | mpan2d 694 |
. 2
⊢ (𝜑 → (𝑀 ≤ 𝑁 → 𝑀 = 𝑁)) |
| 64 | 63 | imp 406 |
1
⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑀 = 𝑁) |