| Step |
Hyp |
Ref |
Expression |
| 1 |
|
elpell14qr |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ↔ ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 2 |
1
|
biimpa |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 3 |
|
simplrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝑏 ∈ ℤ ) |
| 4 |
|
elznn0 |
⊢ ( 𝑏 ∈ ℤ ↔ ( 𝑏 ∈ ℝ ∧ ( 𝑏 ∈ ℕ0 ∨ - 𝑏 ∈ ℕ0 ) ) ) |
| 5 |
3 4
|
sylib |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑏 ∈ ℝ ∧ ( 𝑏 ∈ ℕ0 ∨ - 𝑏 ∈ ℕ0 ) ) ) |
| 6 |
5
|
simprd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑏 ∈ ℕ0 ∨ - 𝑏 ∈ ℕ0 ) ) |
| 7 |
|
simplr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → 𝐴 ∈ ℝ ) |
| 8 |
7
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ 𝑏 ∈ ℕ0 ) → 𝐴 ∈ ℝ ) |
| 9 |
|
simprl |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → 𝑎 ∈ ℕ0 ) |
| 10 |
9
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ 𝑏 ∈ ℕ0 ) → 𝑎 ∈ ℕ0 ) |
| 11 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ 𝑏 ∈ ℕ0 ) → 𝑏 ∈ ℕ0 ) |
| 12 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ 𝑏 ∈ ℕ0 ) → ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) |
| 13 |
|
rsp2e |
⊢ ( ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0 ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) |
| 14 |
10 11 12 13
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ 𝑏 ∈ ℕ0 ) → ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) |
| 15 |
8 14
|
jca |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ 𝑏 ∈ ℕ0 ) → ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 16 |
15
|
ex |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑏 ∈ ℕ0 → ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 17 |
|
elpell1qr |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ↔ ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 18 |
17
|
ad4antr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ↔ ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 19 |
16 18
|
sylibrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑏 ∈ ℕ0 → 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ) ) |
| 20 |
7
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → 𝐴 ∈ ℝ ) |
| 21 |
|
pell14qrne0 |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐴 ≠ 0 ) |
| 22 |
21
|
ad4antr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → 𝐴 ≠ 0 ) |
| 23 |
20 22
|
rereccld |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( 1 / 𝐴 ) ∈ ℝ ) |
| 24 |
9
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → 𝑎 ∈ ℕ0 ) |
| 25 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → - 𝑏 ∈ ℕ0 ) |
| 26 |
|
pell14qrre |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐴 ∈ ℝ ) |
| 27 |
26
|
recnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐴 ∈ ℂ ) |
| 28 |
27 21
|
reccld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 1 / 𝐴 ) ∈ ℂ ) |
| 29 |
28
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 1 / 𝐴 ) ∈ ℂ ) |
| 30 |
|
nn0cn |
⊢ ( 𝑎 ∈ ℕ0 → 𝑎 ∈ ℂ ) |
| 31 |
30
|
ad2antrl |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → 𝑎 ∈ ℂ ) |
| 32 |
|
eldifi |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → 𝐷 ∈ ℕ ) |
| 33 |
32
|
nncnd |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → 𝐷 ∈ ℂ ) |
| 34 |
33
|
ad3antrrr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → 𝐷 ∈ ℂ ) |
| 35 |
34
|
sqrtcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( √ ‘ 𝐷 ) ∈ ℂ ) |
| 36 |
|
zcn |
⊢ ( 𝑏 ∈ ℤ → 𝑏 ∈ ℂ ) |
| 37 |
36
|
ad2antll |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → 𝑏 ∈ ℂ ) |
| 38 |
37
|
negcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → - 𝑏 ∈ ℂ ) |
| 39 |
35 38
|
mulcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( √ ‘ 𝐷 ) · - 𝑏 ) ∈ ℂ ) |
| 40 |
31 39
|
addcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∈ ℂ ) |
| 41 |
40
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∈ ℂ ) |
| 42 |
27
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ∈ ℂ ) |
| 43 |
21
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ≠ 0 ) |
| 44 |
27 21
|
recidd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 ) |
| 45 |
44
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 ) |
| 46 |
|
simprr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) |
| 47 |
45 46
|
eqtr4d |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 · ( 1 / 𝐴 ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) ) |
| 48 |
31
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → 𝑎 ∈ ℂ ) |
| 49 |
35 37
|
mulcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) |
| 50 |
49
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) |
| 51 |
|
subsq |
⊢ ( ( 𝑎 ∈ ℂ ∧ ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) → ( ( 𝑎 ↑ 2 ) − ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) = ( ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) · ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
| 52 |
48 50 51
|
syl2anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → ( ( 𝑎 ↑ 2 ) − ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) = ( ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) · ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
| 53 |
35 37
|
sqmuld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) = ( ( ( √ ‘ 𝐷 ) ↑ 2 ) · ( 𝑏 ↑ 2 ) ) ) |
| 54 |
34
|
sqsqrtd |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( √ ‘ 𝐷 ) ↑ 2 ) = 𝐷 ) |
| 55 |
54
|
oveq1d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( ( √ ‘ 𝐷 ) ↑ 2 ) · ( 𝑏 ↑ 2 ) ) = ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) |
| 56 |
53 55
|
eqtr2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( 𝐷 · ( 𝑏 ↑ 2 ) ) = ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) |
| 57 |
56
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) ) |
| 58 |
57
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) ) |
| 59 |
|
simpr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 60 |
35 37
|
mulneg2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( √ ‘ 𝐷 ) · - 𝑏 ) = - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) |
| 61 |
60
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) = ( 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 62 |
|
negsub |
⊢ ( ( 𝑎 ∈ ℂ ∧ ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) → ( 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) = ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 63 |
62
|
eqcomd |
⊢ ( ( 𝑎 ∈ ℂ ∧ ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) → ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) = ( 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 64 |
31 49 63
|
syl2anc |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) = ( 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 65 |
61 64
|
eqtr4d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) = ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 66 |
65
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) = ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 67 |
59 66
|
oveq12d |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → ( 𝐴 · ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) = ( ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) · ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
| 68 |
52 58 67
|
3eqtr4d |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = ( 𝐴 · ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) ) |
| 69 |
68
|
adantrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = ( 𝐴 · ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) ) |
| 70 |
47 69
|
eqtrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 · ( 1 / 𝐴 ) ) = ( 𝐴 · ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) ) |
| 71 |
29 41 42 43 70
|
mulcanad |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
| 72 |
71
|
adantr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
| 73 |
37
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → 𝑏 ∈ ℂ ) |
| 74 |
|
sqneg |
⊢ ( 𝑏 ∈ ℂ → ( - 𝑏 ↑ 2 ) = ( 𝑏 ↑ 2 ) ) |
| 75 |
73 74
|
syl |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( - 𝑏 ↑ 2 ) = ( 𝑏 ↑ 2 ) ) |
| 76 |
75
|
oveq2d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( 𝐷 · ( - 𝑏 ↑ 2 ) ) = ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) |
| 77 |
76
|
oveq2d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) ) |
| 78 |
|
simplrr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) |
| 79 |
77 78
|
eqtrd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) |
| 80 |
72 79
|
jca |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) |
| 81 |
|
oveq2 |
⊢ ( 𝑐 = - 𝑏 → ( ( √ ‘ 𝐷 ) · 𝑐 ) = ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) |
| 82 |
81
|
oveq2d |
⊢ ( 𝑐 = - 𝑏 → ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
| 83 |
82
|
eqeq2d |
⊢ ( 𝑐 = - 𝑏 → ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ↔ ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) ) |
| 84 |
|
oveq1 |
⊢ ( 𝑐 = - 𝑏 → ( 𝑐 ↑ 2 ) = ( - 𝑏 ↑ 2 ) ) |
| 85 |
84
|
oveq2d |
⊢ ( 𝑐 = - 𝑏 → ( 𝐷 · ( 𝑐 ↑ 2 ) ) = ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) |
| 86 |
85
|
oveq2d |
⊢ ( 𝑐 = - 𝑏 → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) ) |
| 87 |
86
|
eqeq1d |
⊢ ( 𝑐 = - 𝑏 → ( ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ↔ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) |
| 88 |
83 87
|
anbi12d |
⊢ ( 𝑐 = - 𝑏 → ( ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ↔ ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 89 |
88
|
rspcev |
⊢ ( ( - 𝑏 ∈ ℕ0 ∧ ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) |
| 90 |
25 80 89
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) |
| 91 |
|
rspe |
⊢ ( ( 𝑎 ∈ ℕ0 ∧ ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑎 ∈ ℕ0 ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) |
| 92 |
24 90 91
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ∃ 𝑎 ∈ ℕ0 ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) |
| 93 |
23 92
|
jca |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ∧ - 𝑏 ∈ ℕ0 ) → ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) ) |
| 94 |
93
|
ex |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( - 𝑏 ∈ ℕ0 → ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 95 |
|
elpell1qr |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ↔ ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 96 |
95
|
ad4antr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ↔ ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑐 ∈ ℕ0 ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑐 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑐 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 97 |
94 96
|
sylibrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( - 𝑏 ∈ ℕ0 → ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ) ) |
| 98 |
19 97
|
orim12d |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑏 ∈ ℕ0 ∨ - 𝑏 ∈ ℕ0 ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ∨ ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ) ) ) |
| 99 |
6 98
|
mpd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ∨ ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ) ) |
| 100 |
99
|
ex |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) ∧ ( 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℤ ) ) → ( ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ∨ ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ) ) ) |
| 101 |
100
|
rexlimdvva |
⊢ ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ∧ 𝐴 ∈ ℝ ) → ( ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ∨ ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ) ) ) |
| 102 |
101
|
expimpd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ∨ ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ) ) ) |
| 103 |
2 102
|
mpd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 ∈ ( Pell1QR ‘ 𝐷 ) ∨ ( 1 / 𝐴 ) ∈ ( Pell1QR ‘ 𝐷 ) ) ) |