| Step | Hyp | Ref
| Expression |
| 1 | | hashscontpow1.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
(1...((odℤ‘𝑅)‘𝑁))) |
| 2 | 1 | elfzelzd 13565 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℤ) |
| 3 | 2 | zred 12722 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 4 | | hashscontpow1.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
(1...((odℤ‘𝑅)‘𝑁))) |
| 5 | 4 | elfzelzd 13565 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℤ) |
| 6 | 5 | zred 12722 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 7 | 3, 6 | resubcld 11691 |
. . . . 5
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 8 | | hashscontpow1.4 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ℕ) |
| 9 | | hashscontpow1.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 10 | 9 | nnzd 12640 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 11 | | hashscontpow1.5 |
. . . . . . 7
⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) |
| 12 | | odzcl 16831 |
. . . . . . 7
⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) →
((odℤ‘𝑅)‘𝑁) ∈ ℕ) |
| 13 | 8, 10, 11, 12 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 →
((odℤ‘𝑅)‘𝑁) ∈ ℕ) |
| 14 | 13 | nnred 12281 |
. . . . 5
⊢ (𝜑 →
((odℤ‘𝑅)‘𝑁) ∈ ℝ) |
| 15 | | elfznn 13593 |
. . . . . . . 8
⊢ (𝐴 ∈
(1...((odℤ‘𝑅)‘𝑁)) → 𝐴 ∈ ℕ) |
| 16 | 4, 15 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℕ) |
| 17 | 16 | nnrpd 13075 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
ℝ+) |
| 18 | 3, 17 | ltsubrpd 13109 |
. . . . 5
⊢ (𝜑 → (𝐵 − 𝐴) < 𝐵) |
| 19 | | elfzle2 13568 |
. . . . . 6
⊢ (𝐵 ∈
(1...((odℤ‘𝑅)‘𝑁)) → 𝐵 ≤ ((odℤ‘𝑅)‘𝑁)) |
| 20 | 1, 19 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐵 ≤ ((odℤ‘𝑅)‘𝑁)) |
| 21 | 7, 3, 14, 18, 20 | ltletrd 11421 |
. . . 4
⊢ (𝜑 → (𝐵 − 𝐴) < ((odℤ‘𝑅)‘𝑁)) |
| 22 | 21 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐵 − 𝐴) < ((odℤ‘𝑅)‘𝑁)) |
| 23 | | odzval 16829 |
. . . . . . 7
⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) →
((odℤ‘𝑅)‘𝑁) = inf({𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}, ℝ, <
)) |
| 24 | 8, 10, 11, 23 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 →
((odℤ‘𝑅)‘𝑁) = inf({𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}, ℝ, <
)) |
| 25 | 24 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) →
((odℤ‘𝑅)‘𝑁) = inf({𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}, ℝ, <
)) |
| 26 | | elrabi 3687 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)} → 𝑗 ∈ ℕ) |
| 27 | 26 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}) → 𝑗 ∈ ℕ) |
| 28 | 27 | nnred 12281 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}) → 𝑗 ∈ ℝ) |
| 29 | 28 | ex 412 |
. . . . . . . 8
⊢ (𝜑 → (𝑗 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)} → 𝑗 ∈ ℝ)) |
| 30 | 29 | ssrdv 3989 |
. . . . . . 7
⊢ (𝜑 → {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)} ⊆
ℝ) |
| 31 | 30 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)} ⊆
ℝ) |
| 32 | | 1red 11262 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℝ) |
| 33 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 = 1) → 𝑥 = 1) |
| 34 | 33 | breq1d 5153 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 = 1) → (𝑥 ≤ 𝑦 ↔ 1 ≤ 𝑦)) |
| 35 | 34 | ralbidv 3178 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 = 1) → (∀𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}1 ≤ 𝑦)) |
| 36 | | elrabi 3687 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)} → 𝑦 ∈ ℕ) |
| 37 | 36 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}) → 𝑦 ∈ ℕ) |
| 38 | 37 | nnge1d 12314 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}) → 1 ≤ 𝑦) |
| 39 | 38 | ralrimiva 3146 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}1 ≤ 𝑦) |
| 40 | 32, 35, 39 | rspcedvd 3624 |
. . . . . . 7
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}𝑥 ≤ 𝑦) |
| 41 | 40 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}𝑥 ≤ 𝑦) |
| 42 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑖 = (𝐵 − 𝐴) → (𝑁↑𝑖) = (𝑁↑(𝐵 − 𝐴))) |
| 43 | 42 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑖 = (𝐵 − 𝐴) → ((𝑁↑𝑖) − 1) = ((𝑁↑(𝐵 − 𝐴)) − 1)) |
| 44 | 43 | breq2d 5155 |
. . . . . . 7
⊢ (𝑖 = (𝐵 − 𝐴) → (𝑅 ∥ ((𝑁↑𝑖) − 1) ↔ 𝑅 ∥ ((𝑁↑(𝐵 − 𝐴)) − 1))) |
| 45 | 2 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝐵 ∈ ℤ) |
| 46 | 5 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝐴 ∈ ℤ) |
| 47 | 45, 46 | zsubcld 12727 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐵 − 𝐴) ∈ ℤ) |
| 48 | | hashscontpow1.8 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 < 𝐵) |
| 49 | 6, 3 | posdifd 11850 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
| 50 | 48, 49 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
| 51 | 50 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 0 < (𝐵 − 𝐴)) |
| 52 | 47, 51 | jca 511 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → ((𝐵 − 𝐴) ∈ ℤ ∧ 0 < (𝐵 − 𝐴))) |
| 53 | | elnnz 12623 |
. . . . . . . 8
⊢ ((𝐵 − 𝐴) ∈ ℕ ↔ ((𝐵 − 𝐴) ∈ ℤ ∧ 0 < (𝐵 − 𝐴))) |
| 54 | 52, 53 | sylibr 234 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐵 − 𝐴) ∈ ℕ) |
| 55 | 8 | nnzd 12640 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ ℤ) |
| 56 | 55 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝑅 ∈ ℤ) |
| 57 | 10 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝑁 ∈ ℤ) |
| 58 | 16 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈
ℕ0) |
| 59 | 58 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝐴 ∈
ℕ0) |
| 60 | 57, 59 | zexpcld 14128 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝑁↑𝐴) ∈ ℤ) |
| 61 | 54 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐵 − 𝐴) ∈
ℕ0) |
| 62 | 57, 61 | zexpcld 14128 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝑁↑(𝐵 − 𝐴)) ∈ ℤ) |
| 63 | | 1zzd 12648 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 1 ∈
ℤ) |
| 64 | 62, 63 | zsubcld 12727 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → ((𝑁↑(𝐵 − 𝐴)) − 1) ∈
ℤ) |
| 65 | 56, 60, 64 | 3jca 1129 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝑅 ∈ ℤ ∧ (𝑁↑𝐴) ∈ ℤ ∧ ((𝑁↑(𝐵 − 𝐴)) − 1) ∈
ℤ)) |
| 66 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) |
| 67 | 66 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐿‘(𝑁↑𝐵)) = (𝐿‘(𝑁↑𝐴))) |
| 68 | 8 | nnnn0d 12587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ∈
ℕ0) |
| 69 | 68 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝑅 ∈
ℕ0) |
| 70 | | elfznn 13593 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈
(1...((odℤ‘𝑅)‘𝑁)) → 𝐵 ∈ ℕ) |
| 71 | 1, 70 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 ∈ ℕ) |
| 72 | 71 | nnnn0d 12587 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵 ∈
ℕ0) |
| 73 | 10, 72 | zexpcld 14128 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁↑𝐵) ∈ ℤ) |
| 74 | 73 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝑁↑𝐵) ∈ ℤ) |
| 75 | | hashscontpow1.7 |
. . . . . . . . . . . . 13
⊢ 𝑌 =
(ℤ/nℤ‘𝑅) |
| 76 | | hashscontpow1.6 |
. . . . . . . . . . . . 13
⊢ 𝐿 = (ℤRHom‘𝑌) |
| 77 | 75, 76 | zndvds 21568 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ ℕ0
∧ (𝑁↑𝐵) ∈ ℤ ∧ (𝑁↑𝐴) ∈ ℤ) → ((𝐿‘(𝑁↑𝐵)) = (𝐿‘(𝑁↑𝐴)) ↔ 𝑅 ∥ ((𝑁↑𝐵) − (𝑁↑𝐴)))) |
| 78 | 69, 74, 60, 77 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → ((𝐿‘(𝑁↑𝐵)) = (𝐿‘(𝑁↑𝐴)) ↔ 𝑅 ∥ ((𝑁↑𝐵) − (𝑁↑𝐴)))) |
| 79 | 67, 78 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝑅 ∥ ((𝑁↑𝐵) − (𝑁↑𝐴))) |
| 80 | 10, 58 | zexpcld 14128 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁↑𝐴) ∈ ℤ) |
| 81 | 80 | zcnd 12723 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁↑𝐴) ∈ ℂ) |
| 82 | 2, 5 | zsubcld 12727 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℤ) |
| 83 | | 0red 11264 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 0 ∈
ℝ) |
| 84 | 83, 7, 50 | ltled 11409 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴)) |
| 85 | 82, 84 | jca 511 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐵 − 𝐴) ∈ ℤ ∧ 0 ≤ (𝐵 − 𝐴))) |
| 86 | | elnn0z 12626 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐵 − 𝐴) ∈ ℕ0 ↔ ((𝐵 − 𝐴) ∈ ℤ ∧ 0 ≤ (𝐵 − 𝐴))) |
| 87 | 85, 86 | sylibr 234 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐵 − 𝐴) ∈
ℕ0) |
| 88 | 10, 87 | zexpcld 14128 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁↑(𝐵 − 𝐴)) ∈ ℤ) |
| 89 | 88 | zcnd 12723 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁↑(𝐵 − 𝐴)) ∈ ℂ) |
| 90 | | 1cnd 11256 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℂ) |
| 91 | 81, 89, 90 | subdid 11719 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁↑𝐴) · ((𝑁↑(𝐵 − 𝐴)) − 1)) = (((𝑁↑𝐴) · (𝑁↑(𝐵 − 𝐴))) − ((𝑁↑𝐴) · 1))) |
| 92 | 6 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 93 | 3 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 94 | 92, 93 | pncan3d 11623 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
| 95 | 94 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐵 = (𝐴 + (𝐵 − 𝐴))) |
| 96 | 95 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁↑𝐵) = (𝑁↑(𝐴 + (𝐵 − 𝐴)))) |
| 97 | 9 | nncnd 12282 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 98 | 97, 87, 58 | expaddd 14188 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁↑(𝐴 + (𝐵 − 𝐴))) = ((𝑁↑𝐴) · (𝑁↑(𝐵 − 𝐴)))) |
| 99 | 96, 98 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁↑𝐵) = ((𝑁↑𝐴) · (𝑁↑(𝐵 − 𝐴)))) |
| 100 | 99 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁↑𝐴) · (𝑁↑(𝐵 − 𝐴))) = (𝑁↑𝐵)) |
| 101 | 81 | mulridd 11278 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁↑𝐴) · 1) = (𝑁↑𝐴)) |
| 102 | 100, 101 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝑁↑𝐴) · (𝑁↑(𝐵 − 𝐴))) − ((𝑁↑𝐴) · 1)) = ((𝑁↑𝐵) − (𝑁↑𝐴))) |
| 103 | 91, 102 | eqtr2d 2778 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁↑𝐵) − (𝑁↑𝐴)) = ((𝑁↑𝐴) · ((𝑁↑(𝐵 − 𝐴)) − 1))) |
| 104 | 103 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → ((𝑁↑𝐵) − (𝑁↑𝐴)) = ((𝑁↑𝐴) · ((𝑁↑(𝐵 − 𝐴)) − 1))) |
| 105 | 79, 104 | breqtrd 5169 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝑅 ∥ ((𝑁↑𝐴) · ((𝑁↑(𝐵 − 𝐴)) − 1))) |
| 106 | 55, 80 | gcdcomd 16551 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑅 gcd (𝑁↑𝐴)) = ((𝑁↑𝐴) gcd 𝑅)) |
| 107 | | rpexp 16759 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (((𝑁↑𝐴) gcd 𝑅) = 1 ↔ (𝑁 gcd 𝑅) = 1)) |
| 108 | 10, 55, 16, 107 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝑁↑𝐴) gcd 𝑅) = 1 ↔ (𝑁 gcd 𝑅) = 1)) |
| 109 | 11, 108 | mpbird 257 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁↑𝐴) gcd 𝑅) = 1) |
| 110 | 106, 109 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 gcd (𝑁↑𝐴)) = 1) |
| 111 | 110 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝑅 gcd (𝑁↑𝐴)) = 1) |
| 112 | 105, 111 | jca 511 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝑅 ∥ ((𝑁↑𝐴) · ((𝑁↑(𝐵 − 𝐴)) − 1)) ∧ (𝑅 gcd (𝑁↑𝐴)) = 1)) |
| 113 | | coprmdvds 16690 |
. . . . . . . . 9
⊢ ((𝑅 ∈ ℤ ∧ (𝑁↑𝐴) ∈ ℤ ∧ ((𝑁↑(𝐵 − 𝐴)) − 1) ∈ ℤ) → ((𝑅 ∥ ((𝑁↑𝐴) · ((𝑁↑(𝐵 − 𝐴)) − 1)) ∧ (𝑅 gcd (𝑁↑𝐴)) = 1) → 𝑅 ∥ ((𝑁↑(𝐵 − 𝐴)) − 1))) |
| 114 | 113 | imp 406 |
. . . . . . . 8
⊢ (((𝑅 ∈ ℤ ∧ (𝑁↑𝐴) ∈ ℤ ∧ ((𝑁↑(𝐵 − 𝐴)) − 1) ∈ ℤ) ∧ (𝑅 ∥ ((𝑁↑𝐴) · ((𝑁↑(𝐵 − 𝐴)) − 1)) ∧ (𝑅 gcd (𝑁↑𝐴)) = 1)) → 𝑅 ∥ ((𝑁↑(𝐵 − 𝐴)) − 1)) |
| 115 | 65, 112, 114 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → 𝑅 ∥ ((𝑁↑(𝐵 − 𝐴)) − 1)) |
| 116 | 44, 54, 115 | elrabd 3694 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐵 − 𝐴) ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}) |
| 117 | | infrelb 12253 |
. . . . . 6
⊢ (({𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)} ⊆ ℝ ∧
∃𝑥 ∈ ℝ
∀𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}𝑥 ≤ 𝑦 ∧ (𝐵 − 𝐴) ∈ {𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}) → inf({𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}, ℝ, < ) ≤ (𝐵 − 𝐴)) |
| 118 | 31, 41, 116, 117 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → inf({𝑖 ∈ ℕ ∣ 𝑅 ∥ ((𝑁↑𝑖) − 1)}, ℝ, < ) ≤ (𝐵 − 𝐴)) |
| 119 | 25, 118 | eqbrtrd 5165 |
. . . 4
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) →
((odℤ‘𝑅)‘𝑁) ≤ (𝐵 − 𝐴)) |
| 120 | 13 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) →
((odℤ‘𝑅)‘𝑁) ∈ ℕ) |
| 121 | 120 | nnred 12281 |
. . . . 5
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) →
((odℤ‘𝑅)‘𝑁) ∈ ℝ) |
| 122 | 7 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → (𝐵 − 𝐴) ∈ ℝ) |
| 123 | 121, 122 | lenltd 11407 |
. . . 4
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) →
(((odℤ‘𝑅)‘𝑁) ≤ (𝐵 − 𝐴) ↔ ¬ (𝐵 − 𝐴) < ((odℤ‘𝑅)‘𝑁))) |
| 124 | 119, 123 | mpbid 232 |
. . 3
⊢ ((𝜑 ∧ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) → ¬ (𝐵 − 𝐴) < ((odℤ‘𝑅)‘𝑁)) |
| 125 | 22, 124 | pm2.65da 817 |
. 2
⊢ (𝜑 → ¬ (𝐿‘(𝑁↑𝐴)) = (𝐿‘(𝑁↑𝐵))) |
| 126 | 125 | neqned 2947 |
1
⊢ (𝜑 → (𝐿‘(𝑁↑𝐴)) ≠ (𝐿‘(𝑁↑𝐵))) |