| Step | Hyp | Ref
| Expression |
| 1 | | eluz2nn 12924 |
. . . . . 6
⊢ (𝐵 ∈
(ℤ≥‘2) → 𝐵 ∈ ℕ) |
| 2 | 1 | nnrpd 13075 |
. . . . 5
⊢ (𝐵 ∈
(ℤ≥‘2) → 𝐵 ∈
ℝ+) |
| 3 | 2 | 3ad2ant2 1135 |
. . . 4
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)
∧ (𝑋 gcd 𝐵) = 1) → 𝐵 ∈
ℝ+) |
| 4 | | eluz2nn 12924 |
. . . . . 6
⊢ (𝑋 ∈
(ℤ≥‘2) → 𝑋 ∈ ℕ) |
| 5 | 4 | nnrpd 13075 |
. . . . 5
⊢ (𝑋 ∈
(ℤ≥‘2) → 𝑋 ∈
ℝ+) |
| 6 | 5 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)
∧ (𝑋 gcd 𝐵) = 1) → 𝑋 ∈
ℝ+) |
| 7 | | eluz2b3 12964 |
. . . . . 6
⊢ (𝐵 ∈
(ℤ≥‘2) ↔ (𝐵 ∈ ℕ ∧ 𝐵 ≠ 1)) |
| 8 | 7 | simprbi 496 |
. . . . 5
⊢ (𝐵 ∈
(ℤ≥‘2) → 𝐵 ≠ 1) |
| 9 | 8 | 3ad2ant2 1135 |
. . . 4
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)
∧ (𝑋 gcd 𝐵) = 1) → 𝐵 ≠ 1) |
| 10 | 3, 6, 9 | 3jca 1129 |
. . 3
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)
∧ (𝑋 gcd 𝐵) = 1) → (𝐵 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+
∧ 𝐵 ≠
1)) |
| 11 | | relogbcl 26816 |
. . 3
⊢ ((𝐵 ∈ ℝ+
∧ 𝑋 ∈
ℝ+ ∧ 𝐵
≠ 1) → (𝐵
logb 𝑋) ∈
ℝ) |
| 12 | 10, 11 | syl 17 |
. 2
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)
∧ (𝑋 gcd 𝐵) = 1) → (𝐵 logb 𝑋) ∈ ℝ) |
| 13 | | eluz2gt1 12962 |
. . . . . . . . . 10
⊢ (𝑋 ∈
(ℤ≥‘2) → 1 < 𝑋) |
| 14 | 13 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 1 < 𝑋) |
| 15 | 4 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝑋 ∈
ℕ) |
| 16 | 15 | nnrpd 13075 |
. . . . . . . . . 10
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝑋 ∈
ℝ+) |
| 17 | 1 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝐵 ∈
ℕ) |
| 18 | 17 | nnrpd 13075 |
. . . . . . . . . 10
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝐵 ∈
ℝ+) |
| 19 | | eluz2gt1 12962 |
. . . . . . . . . . 11
⊢ (𝐵 ∈
(ℤ≥‘2) → 1 < 𝐵) |
| 20 | 19 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 1 < 𝐵) |
| 21 | | logbgt0b 26836 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ ℝ+
∧ (𝐵 ∈
ℝ+ ∧ 1 < 𝐵)) → (0 < (𝐵 logb 𝑋) ↔ 1 < 𝑋)) |
| 22 | 16, 18, 20, 21 | syl12anc 837 |
. . . . . . . . 9
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ (0 < (𝐵
logb 𝑋) ↔ 1
< 𝑋)) |
| 23 | 14, 22 | mpbird 257 |
. . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 0 < (𝐵
logb 𝑋)) |
| 24 | 23 | anim1ci 616 |
. . . . . . 7
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝐵 logb
𝑋) ∈ ℚ) →
((𝐵 logb 𝑋) ∈ ℚ ∧ 0 <
(𝐵 logb 𝑋))) |
| 25 | | elpq 13017 |
. . . . . . 7
⊢ (((𝐵 logb 𝑋) ∈ ℚ ∧ 0 <
(𝐵 logb 𝑋)) → ∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (𝐵 logb 𝑋) = (𝑚 / 𝑛)) |
| 26 | 24, 25 | syl 17 |
. . . . . 6
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝐵 logb
𝑋) ∈ ℚ) →
∃𝑚 ∈ ℕ
∃𝑛 ∈ ℕ
(𝐵 logb 𝑋) = (𝑚 / 𝑛)) |
| 27 | 26 | ex 412 |
. . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ ((𝐵 logb
𝑋) ∈ ℚ →
∃𝑚 ∈ ℕ
∃𝑛 ∈ ℕ
(𝐵 logb 𝑋) = (𝑚 / 𝑛))) |
| 28 | | oveq2 7439 |
. . . . . . . . . 10
⊢ ((𝑚 / 𝑛) = (𝐵 logb 𝑋) → (𝐵↑𝑐(𝑚 / 𝑛)) = (𝐵↑𝑐(𝐵 logb 𝑋))) |
| 29 | 28 | eqcoms 2745 |
. . . . . . . . 9
⊢ ((𝐵 logb 𝑋) = (𝑚 / 𝑛) → (𝐵↑𝑐(𝑚 / 𝑛)) = (𝐵↑𝑐(𝐵 logb 𝑋))) |
| 30 | | eluzelcn 12890 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈
(ℤ≥‘2) → 𝐵 ∈ ℂ) |
| 31 | 30 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝐵 ∈
ℂ) |
| 32 | | nnne0 12300 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) |
| 33 | 1, 32 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ∈
(ℤ≥‘2) → 𝐵 ≠ 0) |
| 34 | 33, 8 | nelprd 4657 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈
(ℤ≥‘2) → ¬ 𝐵 ∈ {0, 1}) |
| 35 | 34 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ ¬ 𝐵 ∈ {0,
1}) |
| 36 | 31, 35 | eldifd 3962 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝐵 ∈ (ℂ
∖ {0, 1})) |
| 37 | | eluzelcn 12890 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈
(ℤ≥‘2) → 𝑋 ∈ ℂ) |
| 38 | 37 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝑋 ∈
ℂ) |
| 39 | | nnne0 12300 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ ℕ → 𝑋 ≠ 0) |
| 40 | | nelsn 4666 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ≠ 0 → ¬ 𝑋 ∈ {0}) |
| 41 | 4, 39, 40 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈
(ℤ≥‘2) → ¬ 𝑋 ∈ {0}) |
| 42 | 41 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ ¬ 𝑋 ∈
{0}) |
| 43 | 38, 42 | eldifd 3962 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝑋 ∈ (ℂ
∖ {0})) |
| 44 | | cxplogb 26829 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1})
∧ 𝑋 ∈ (ℂ
∖ {0})) → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) |
| 45 | 36, 43, 44 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) |
| 46 | 45 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) |
| 47 | 29, 46 | sylan9eqr 2799 |
. . . . . . . 8
⊢ ((((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
∧ (𝐵 logb
𝑋) = (𝑚 / 𝑛)) → (𝐵↑𝑐(𝑚 / 𝑛)) = 𝑋) |
| 48 | 47 | ex 412 |
. . . . . . 7
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵 logb
𝑋) = (𝑚 / 𝑛) → (𝐵↑𝑐(𝑚 / 𝑛)) = 𝑋)) |
| 49 | | oveq1 7438 |
. . . . . . . 8
⊢ ((𝐵↑𝑐(𝑚 / 𝑛)) = 𝑋 → ((𝐵↑𝑐(𝑚 / 𝑛))↑𝑛) = (𝑋↑𝑛)) |
| 50 | 31 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ 𝐵 ∈
ℂ) |
| 51 | | nncn 12274 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
| 52 | 51 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈
ℂ) |
| 53 | | nncn 12274 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
| 54 | 53 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℂ) |
| 55 | | nnne0 12300 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) |
| 56 | 55 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0) |
| 57 | 52, 54, 56 | 3jca 1129 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) |
| 58 | | divcl 11928 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) → (𝑚 / 𝑛) ∈ ℂ) |
| 59 | 57, 58 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℂ) |
| 60 | 59 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝑚 / 𝑛) ∈
ℂ) |
| 61 | | nnnn0 12533 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
| 62 | 61 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℕ0) |
| 63 | 62 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ 𝑛 ∈
ℕ0) |
| 64 | 50, 60, 63 | 3jca 1129 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝐵 ∈ ℂ
∧ (𝑚 / 𝑛) ∈ ℂ ∧ 𝑛 ∈
ℕ0)) |
| 65 | | cxpmul2 26731 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ℂ ∧ (𝑚 / 𝑛) ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐵↑𝑐((𝑚 / 𝑛) · 𝑛)) = ((𝐵↑𝑐(𝑚 / 𝑛))↑𝑛)) |
| 66 | 65 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℂ ∧ (𝑚 / 𝑛) ∈ ℂ ∧ 𝑛 ∈ ℕ0) → ((𝐵↑𝑐(𝑚 / 𝑛))↑𝑛) = (𝐵↑𝑐((𝑚 / 𝑛) · 𝑛))) |
| 67 | 64, 66 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵↑𝑐(𝑚 / 𝑛))↑𝑛) = (𝐵↑𝑐((𝑚 / 𝑛) · 𝑛))) |
| 68 | 57 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝑚 ∈ ℂ
∧ 𝑛 ∈ ℂ
∧ 𝑛 ≠
0)) |
| 69 | | divcan1 11931 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) → ((𝑚 / 𝑛) · 𝑛) = 𝑚) |
| 70 | 68, 69 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝑚 / 𝑛) · 𝑛) = 𝑚) |
| 71 | 70 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝐵↑𝑐((𝑚 / 𝑛) · 𝑛)) = (𝐵↑𝑐𝑚)) |
| 72 | 33 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝐵 ≠
0) |
| 73 | 72 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ 𝐵 ≠
0) |
| 74 | | nnz 12634 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) |
| 75 | 74 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈
ℤ) |
| 76 | 75 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ 𝑚 ∈
ℤ) |
| 77 | 50, 73, 76 | cxpexpzd 26753 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝐵↑𝑐𝑚) = (𝐵↑𝑚)) |
| 78 | 71, 77 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝐵↑𝑐((𝑚 / 𝑛) · 𝑛)) = (𝐵↑𝑚)) |
| 79 | 67, 78 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵↑𝑐(𝑚 / 𝑛))↑𝑛) = (𝐵↑𝑚)) |
| 80 | 79 | eqeq1d 2739 |
. . . . . . . . 9
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (((𝐵↑𝑐(𝑚 / 𝑛))↑𝑛) = (𝑋↑𝑛) ↔ (𝐵↑𝑚) = (𝑋↑𝑛))) |
| 81 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℕ) |
| 82 | | rplpwr 16595 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝑋 gcd 𝐵) = 1 → ((𝑋↑𝑛) gcd 𝐵) = 1)) |
| 83 | 15, 17, 81, 82 | syl2an3an 1424 |
. . . . . . . . . . 11
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝑋 gcd 𝐵) = 1 → ((𝑋↑𝑛) gcd 𝐵) = 1)) |
| 84 | | oveq1 7438 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑋↑𝑛) = (𝐵↑𝑚) → ((𝑋↑𝑛) gcd 𝐵) = ((𝐵↑𝑚) gcd 𝐵)) |
| 85 | 84 | eqeq1d 2739 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋↑𝑛) = (𝐵↑𝑚) → (((𝑋↑𝑛) gcd 𝐵) = 1 ↔ ((𝐵↑𝑚) gcd 𝐵) = 1)) |
| 86 | 85 | eqcoms 2745 |
. . . . . . . . . . . . . . 15
⊢ ((𝐵↑𝑚) = (𝑋↑𝑛) → (((𝑋↑𝑛) gcd 𝐵) = 1 ↔ ((𝐵↑𝑚) gcd 𝐵) = 1)) |
| 87 | 86 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
∧ (𝐵↑𝑚) = (𝑋↑𝑛)) → (((𝑋↑𝑛) gcd 𝐵) = 1 ↔ ((𝐵↑𝑚) gcd 𝐵) = 1)) |
| 88 | | eluzelz 12888 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈
(ℤ≥‘2) → 𝐵 ∈ ℤ) |
| 89 | 88 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ 𝐵 ∈
ℤ) |
| 90 | | simpl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈
ℕ) |
| 91 | | rpexp 16759 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑚 ∈ ℕ) → (((𝐵↑𝑚) gcd 𝐵) = 1 ↔ (𝐵 gcd 𝐵) = 1)) |
| 92 | 89, 89, 90, 91 | syl2an3an 1424 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (((𝐵↑𝑚) gcd 𝐵) = 1 ↔ (𝐵 gcd 𝐵) = 1)) |
| 93 | | gcdid 16564 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐵 ∈ ℤ → (𝐵 gcd 𝐵) = (abs‘𝐵)) |
| 94 | 88, 93 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐵 ∈
(ℤ≥‘2) → (𝐵 gcd 𝐵) = (abs‘𝐵)) |
| 95 | | eluzelre 12889 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐵 ∈
(ℤ≥‘2) → 𝐵 ∈ ℝ) |
| 96 | | nnnn0 12533 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℕ0) |
| 97 | | nn0ge0 12551 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐵 ∈ ℕ0
→ 0 ≤ 𝐵) |
| 98 | 1, 96, 97 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐵 ∈
(ℤ≥‘2) → 0 ≤ 𝐵) |
| 99 | 95, 98 | absidd 15461 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐵 ∈
(ℤ≥‘2) → (abs‘𝐵) = 𝐵) |
| 100 | 94, 99 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐵 ∈
(ℤ≥‘2) → (𝐵 gcd 𝐵) = 𝐵) |
| 101 | 100 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐵 ∈
(ℤ≥‘2) → ((𝐵 gcd 𝐵) = 1 ↔ 𝐵 = 1)) |
| 102 | 101 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ ((𝐵 gcd 𝐵) = 1 ↔ 𝐵 = 1)) |
| 103 | 102 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵 gcd 𝐵) = 1 ↔ 𝐵 = 1)) |
| 104 | | eqneqall 2951 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐵 = 1 → (𝐵 ≠ 1 → ¬ (𝑋 gcd 𝐵) = 1)) |
| 105 | 8, 104 | syl5com 31 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐵 ∈
(ℤ≥‘2) → (𝐵 = 1 → ¬ (𝑋 gcd 𝐵) = 1)) |
| 106 | 105 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ (𝐵 = 1 → ¬
(𝑋 gcd 𝐵) = 1)) |
| 107 | 106 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (𝐵 = 1 → ¬
(𝑋 gcd 𝐵) = 1)) |
| 108 | 103, 107 | sylbid 240 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵 gcd 𝐵) = 1 → ¬ (𝑋 gcd 𝐵) = 1)) |
| 109 | 92, 108 | sylbid 240 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (((𝐵↑𝑚) gcd 𝐵) = 1 → ¬ (𝑋 gcd 𝐵) = 1)) |
| 110 | 109 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
∧ (𝐵↑𝑚) = (𝑋↑𝑛)) → (((𝐵↑𝑚) gcd 𝐵) = 1 → ¬ (𝑋 gcd 𝐵) = 1)) |
| 111 | 87, 110 | sylbid 240 |
. . . . . . . . . . . . 13
⊢ ((((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
∧ (𝐵↑𝑚) = (𝑋↑𝑛)) → (((𝑋↑𝑛) gcd 𝐵) = 1 → ¬ (𝑋 gcd 𝐵) = 1)) |
| 112 | 111 | ex 412 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵↑𝑚) = (𝑋↑𝑛) → (((𝑋↑𝑛) gcd 𝐵) = 1 → ¬ (𝑋 gcd 𝐵) = 1))) |
| 113 | 112 | com23 86 |
. . . . . . . . . . 11
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (((𝑋↑𝑛) gcd 𝐵) = 1 → ((𝐵↑𝑚) = (𝑋↑𝑛) → ¬ (𝑋 gcd 𝐵) = 1))) |
| 114 | 83, 113 | syld 47 |
. . . . . . . . . 10
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝑋 gcd 𝐵) = 1 → ((𝐵↑𝑚) = (𝑋↑𝑛) → ¬ (𝑋 gcd 𝐵) = 1))) |
| 115 | | ax-1 6 |
. . . . . . . . . 10
⊢ (¬
(𝑋 gcd 𝐵) = 1 → ((𝐵↑𝑚) = (𝑋↑𝑛) → ¬ (𝑋 gcd 𝐵) = 1)) |
| 116 | 114, 115 | pm2.61d1 180 |
. . . . . . . . 9
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵↑𝑚) = (𝑋↑𝑛) → ¬ (𝑋 gcd 𝐵) = 1)) |
| 117 | 80, 116 | sylbid 240 |
. . . . . . . 8
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ (((𝐵↑𝑐(𝑚 / 𝑛))↑𝑛) = (𝑋↑𝑛) → ¬ (𝑋 gcd 𝐵) = 1)) |
| 118 | 49, 117 | syl5 34 |
. . . . . . 7
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵↑𝑐(𝑚 / 𝑛)) = 𝑋 → ¬ (𝑋 gcd 𝐵) = 1)) |
| 119 | 48, 118 | syld 47 |
. . . . . 6
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
∧ (𝑚 ∈ ℕ
∧ 𝑛 ∈ ℕ))
→ ((𝐵 logb
𝑋) = (𝑚 / 𝑛) → ¬ (𝑋 gcd 𝐵) = 1)) |
| 120 | 119 | rexlimdvva 3213 |
. . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ (∃𝑚 ∈
ℕ ∃𝑛 ∈
ℕ (𝐵 logb
𝑋) = (𝑚 / 𝑛) → ¬ (𝑋 gcd 𝐵) = 1)) |
| 121 | 27, 120 | syld 47 |
. . . 4
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ ((𝐵 logb
𝑋) ∈ ℚ →
¬ (𝑋 gcd 𝐵) = 1)) |
| 122 | 121 | con2d 134 |
. . 3
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2))
→ ((𝑋 gcd 𝐵) = 1 → ¬ (𝐵 logb 𝑋) ∈
ℚ)) |
| 123 | 122 | 3impia 1118 |
. 2
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)
∧ (𝑋 gcd 𝐵) = 1) → ¬ (𝐵 logb 𝑋) ∈
ℚ) |
| 124 | 12, 123 | eldifd 3962 |
1
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)
∧ (𝑋 gcd 𝐵) = 1) → (𝐵 logb 𝑋) ∈ (ℝ ∖
ℚ)) |