| Step | Hyp | Ref
| Expression |
| 1 | | eldifi 4131 |
. . 3
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → 𝐷 ∈ ℕ) |
| 2 | | eldifn 4132 |
. . . 4
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → ¬ 𝐷 ∈
◻NN) |
| 3 | 1 | anim1i 615 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (√‘𝐷) ∈ ℚ) → (𝐷 ∈ ℕ ∧ (√‘𝐷) ∈
ℚ)) |
| 4 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑎 = 𝐷 → (√‘𝑎) = (√‘𝐷)) |
| 5 | 4 | eleq1d 2826 |
. . . . . 6
⊢ (𝑎 = 𝐷 → ((√‘𝑎) ∈ ℚ ↔ (√‘𝐷) ∈
ℚ)) |
| 6 | | df-squarenn 42852 |
. . . . . 6
⊢
◻NN = {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈
ℚ} |
| 7 | 5, 6 | elrab2 3695 |
. . . . 5
⊢ (𝐷 ∈
◻NN ↔ (𝐷 ∈ ℕ ∧ (√‘𝐷) ∈
ℚ)) |
| 8 | 3, 7 | sylibr 234 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (√‘𝐷) ∈ ℚ) → 𝐷 ∈
◻NN) |
| 9 | 2, 8 | mtand 816 |
. . 3
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → ¬ (√‘𝐷) ∈ ℚ) |
| 10 | | pellex 42846 |
. . 3
⊢ ((𝐷 ∈ ℕ ∧ ¬
(√‘𝐷) ∈
ℚ) → ∃𝑐
∈ ℕ ∃𝑑
∈ ℕ ((𝑐↑2)
− (𝐷 · (𝑑↑2))) = 1) |
| 11 | 1, 9, 10 | syl2anc 584 |
. 2
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → ∃𝑐 ∈ ℕ ∃𝑑 ∈ ℕ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) |
| 12 | | simpll 767 |
. . . . . 6
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) ∧ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) → 𝐷 ∈ (ℕ ∖
◻NN)) |
| 13 | | nnnn0 12533 |
. . . . . . . 8
⊢ (𝑐 ∈ ℕ → 𝑐 ∈
ℕ0) |
| 14 | 13 | adantr 480 |
. . . . . . 7
⊢ ((𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑐 ∈
ℕ0) |
| 15 | 14 | ad2antlr 727 |
. . . . . 6
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) ∧ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) → 𝑐 ∈ ℕ0) |
| 16 | | nnnn0 12533 |
. . . . . . . 8
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℕ0) |
| 17 | 16 | adantl 481 |
. . . . . . 7
⊢ ((𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑑 ∈
ℕ0) |
| 18 | 17 | ad2antlr 727 |
. . . . . 6
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) ∧ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) → 𝑑 ∈ ℕ0) |
| 19 | | simpr 484 |
. . . . . 6
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) ∧ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) → ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) |
| 20 | | pellqrexplicit 42888 |
. . . . . 6
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝑐 ∈ ℕ0 ∧ 𝑑 ∈ ℕ0)
∧ ((𝑐↑2) −
(𝐷 · (𝑑↑2))) = 1) → (𝑐 + ((√‘𝐷) · 𝑑)) ∈ (Pell1QR‘𝐷)) |
| 21 | 12, 15, 18, 19, 20 | syl31anc 1375 |
. . . . 5
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) ∧ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) → (𝑐 + ((√‘𝐷) · 𝑑)) ∈ (Pell1QR‘𝐷)) |
| 22 | | 1re 11261 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
| 23 | 22 | a1i 11 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 1 ∈
ℝ) |
| 24 | 22, 22 | readdcli 11276 |
. . . . . . . 8
⊢ (1 + 1)
∈ ℝ |
| 25 | 24 | a1i 11 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (1 + 1) ∈
ℝ) |
| 26 | | nnre 12273 |
. . . . . . . . 9
⊢ (𝑐 ∈ ℕ → 𝑐 ∈
ℝ) |
| 27 | 26 | ad2antrl 728 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 𝑐 ∈ ℝ) |
| 28 | 1 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 𝐷 ∈ ℕ) |
| 29 | 28 | nnrpd 13075 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 𝐷 ∈
ℝ+) |
| 30 | 29 | rpsqrtcld 15450 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (√‘𝐷) ∈
ℝ+) |
| 31 | 30 | rpred 13077 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (√‘𝐷) ∈
ℝ) |
| 32 | | nnre 12273 |
. . . . . . . . . 10
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℝ) |
| 33 | 32 | ad2antll 729 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 𝑑 ∈ ℝ) |
| 34 | 31, 33 | remulcld 11291 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) →
((√‘𝐷) ·
𝑑) ∈
ℝ) |
| 35 | 27, 34 | readdcld 11290 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑐 + ((√‘𝐷) · 𝑑)) ∈ ℝ) |
| 36 | 22 | ltp1i 12172 |
. . . . . . . 8
⊢ 1 < (1
+ 1) |
| 37 | 36 | a1i 11 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 1 < (1 +
1)) |
| 38 | | nnge1 12294 |
. . . . . . . . 9
⊢ (𝑐 ∈ ℕ → 1 ≤
𝑐) |
| 39 | 38 | ad2antrl 728 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 1 ≤ 𝑐) |
| 40 | | 1t1e1 12428 |
. . . . . . . . 9
⊢ (1
· 1) = 1 |
| 41 | | nnge1 12294 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ ℕ → 1 ≤
𝐷) |
| 42 | | sq1 14234 |
. . . . . . . . . . . . . 14
⊢
(1↑2) = 1 |
| 43 | 42 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ ℕ →
(1↑2) = 1) |
| 44 | | nncn 12274 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ ℕ → 𝐷 ∈
ℂ) |
| 45 | 44 | sqsqrtd 15478 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ ℕ →
((√‘𝐷)↑2)
= 𝐷) |
| 46 | 41, 43, 45 | 3brtr4d 5175 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ ℕ →
(1↑2) ≤ ((√‘𝐷)↑2)) |
| 47 | 22 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ ℕ → 1 ∈
ℝ) |
| 48 | | nnrp 13046 |
. . . . . . . . . . . . . . 15
⊢ (𝐷 ∈ ℕ → 𝐷 ∈
ℝ+) |
| 49 | 48 | rpsqrtcld 15450 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ ℕ →
(√‘𝐷) ∈
ℝ+) |
| 50 | 49 | rpred 13077 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ ℕ →
(√‘𝐷) ∈
ℝ) |
| 51 | | 0le1 11786 |
. . . . . . . . . . . . . 14
⊢ 0 ≤
1 |
| 52 | 51 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ ℕ → 0 ≤
1) |
| 53 | 49 | rpge0d 13081 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ ℕ → 0 ≤
(√‘𝐷)) |
| 54 | 47, 50, 52, 53 | le2sqd 14296 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ ℕ → (1 ≤
(√‘𝐷) ↔
(1↑2) ≤ ((√‘𝐷)↑2))) |
| 55 | 46, 54 | mpbird 257 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ ℕ → 1 ≤
(√‘𝐷)) |
| 56 | 28, 55 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 1 ≤
(√‘𝐷)) |
| 57 | | nnge1 12294 |
. . . . . . . . . . 11
⊢ (𝑑 ∈ ℕ → 1 ≤
𝑑) |
| 58 | 57 | ad2antll 729 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 1 ≤ 𝑑) |
| 59 | 23, 51 | jctir 520 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (1 ∈ ℝ
∧ 0 ≤ 1)) |
| 60 | | lemul12a 12125 |
. . . . . . . . . . 11
⊢ ((((1
∈ ℝ ∧ 0 ≤ 1) ∧ (√‘𝐷) ∈ ℝ) ∧ ((1 ∈ ℝ
∧ 0 ≤ 1) ∧ 𝑑
∈ ℝ)) → ((1 ≤ (√‘𝐷) ∧ 1 ≤ 𝑑) → (1 · 1) ≤
((√‘𝐷) ·
𝑑))) |
| 61 | 59, 31, 59, 33, 60 | syl22anc 839 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((1 ≤
(√‘𝐷) ∧ 1
≤ 𝑑) → (1 ·
1) ≤ ((√‘𝐷)
· 𝑑))) |
| 62 | 56, 58, 61 | mp2and 699 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (1 · 1) ≤
((√‘𝐷) ·
𝑑)) |
| 63 | 40, 62 | eqbrtrrid 5179 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 1 ≤
((√‘𝐷) ·
𝑑)) |
| 64 | 23, 23, 27, 34, 39, 63 | le2addd 11882 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (1 + 1) ≤ (𝑐 + ((√‘𝐷) · 𝑑))) |
| 65 | 23, 25, 35, 37, 64 | ltletrd 11421 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → 1 < (𝑐 + ((√‘𝐷) · 𝑑))) |
| 66 | 65 | adantr 480 |
. . . . 5
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) ∧ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) → 1 < (𝑐 + ((√‘𝐷) · 𝑑))) |
| 67 | | breq2 5147 |
. . . . . 6
⊢ (𝑥 = (𝑐 + ((√‘𝐷) · 𝑑)) → (1 < 𝑥 ↔ 1 < (𝑐 + ((√‘𝐷) · 𝑑)))) |
| 68 | 67 | rspcev 3622 |
. . . . 5
⊢ (((𝑐 + ((√‘𝐷) · 𝑑)) ∈ (Pell1QR‘𝐷) ∧ 1 < (𝑐 + ((√‘𝐷) · 𝑑))) → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥) |
| 69 | 21, 66, 68 | syl2anc 584 |
. . . 4
⊢ (((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) ∧ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1) → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥) |
| 70 | 69 | ex 412 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ (𝑐 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1 → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥)) |
| 71 | 70 | rexlimdvva 3213 |
. 2
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (∃𝑐 ∈ ℕ ∃𝑑 ∈ ℕ ((𝑐↑2) − (𝐷 · (𝑑↑2))) = 1 → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥)) |
| 72 | 11, 71 | mpd 15 |
1
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥) |