| Step |
Hyp |
Ref |
Expression |
| 1 |
|
elpell1234qr |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ↔ ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 2 |
1
|
biimpa |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 3 |
|
pell1234qrre |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → 𝐴 ∈ ℝ ) |
| 4 |
|
pell1234qrne0 |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → 𝐴 ≠ 0 ) |
| 5 |
3 4
|
rereccld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( 1 / 𝐴 ) ∈ ℝ ) |
| 6 |
5
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 1 / 𝐴 ) ∈ ℝ ) |
| 7 |
|
simplrl |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝑎 ∈ ℤ ) |
| 8 |
|
simplrr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝑏 ∈ ℤ ) |
| 9 |
8
|
znegcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝑏 ∈ ℤ ) |
| 10 |
5
|
recnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( 1 / 𝐴 ) ∈ ℂ ) |
| 11 |
10
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 1 / 𝐴 ) ∈ ℂ ) |
| 12 |
|
zcn |
⊢ ( 𝑎 ∈ ℤ → 𝑎 ∈ ℂ ) |
| 13 |
12
|
adantr |
⊢ ( ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) → 𝑎 ∈ ℂ ) |
| 14 |
13
|
ad2antlr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝑎 ∈ ℂ ) |
| 15 |
|
eldifi |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → 𝐷 ∈ ℕ ) |
| 16 |
15
|
nncnd |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → 𝐷 ∈ ℂ ) |
| 17 |
16
|
ad3antrrr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐷 ∈ ℂ ) |
| 18 |
17
|
sqrtcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ 𝐷 ) ∈ ℂ ) |
| 19 |
8
|
zcnd |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝑏 ∈ ℂ ) |
| 20 |
19
|
negcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝑏 ∈ ℂ ) |
| 21 |
18 20
|
mulcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) · - 𝑏 ) ∈ ℂ ) |
| 22 |
14 21
|
addcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∈ ℂ ) |
| 23 |
3
|
recnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → 𝐴 ∈ ℂ ) |
| 24 |
23
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ∈ ℂ ) |
| 25 |
4
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ≠ 0 ) |
| 26 |
18 19
|
sqmuld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) = ( ( ( √ ‘ 𝐷 ) ↑ 2 ) · ( 𝑏 ↑ 2 ) ) ) |
| 27 |
17
|
sqsqrtd |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) ↑ 2 ) = 𝐷 ) |
| 28 |
27
|
oveq1d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( ( √ ‘ 𝐷 ) ↑ 2 ) · ( 𝑏 ↑ 2 ) ) = ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) |
| 29 |
26 28
|
eqtr2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · ( 𝑏 ↑ 2 ) ) = ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) |
| 30 |
29
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) ) |
| 31 |
|
simprr |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) |
| 32 |
18 19
|
mulcld |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) |
| 33 |
|
subsq |
⊢ ( ( 𝑎 ∈ ℂ ∧ ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) → ( ( 𝑎 ↑ 2 ) − ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) = ( ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) · ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
| 34 |
14 32 33
|
syl2anc |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( ( ( √ ‘ 𝐷 ) · 𝑏 ) ↑ 2 ) ) = ( ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) · ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
| 35 |
30 31 34
|
3eqtr3d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 1 = ( ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) · ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
| 36 |
24 25
|
recidd |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 ) |
| 37 |
|
simprl |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 38 |
18 19
|
mulneg2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) · - 𝑏 ) = - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) |
| 39 |
38
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) = ( 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 40 |
14 32
|
negsubd |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) = ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 41 |
39 40
|
eqtrd |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) = ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
| 42 |
37 41
|
oveq12d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 · ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) = ( ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) · ( 𝑎 − ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
| 43 |
35 36 42
|
3eqtr4d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 · ( 1 / 𝐴 ) ) = ( 𝐴 · ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) ) |
| 44 |
11 22 24 25 43
|
mulcanad |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
| 45 |
|
sqneg |
⊢ ( 𝑏 ∈ ℂ → ( - 𝑏 ↑ 2 ) = ( 𝑏 ↑ 2 ) ) |
| 46 |
19 45
|
syl |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( - 𝑏 ↑ 2 ) = ( 𝑏 ↑ 2 ) ) |
| 47 |
46
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · ( - 𝑏 ↑ 2 ) ) = ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) |
| 48 |
47
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) ) |
| 49 |
48 31
|
eqtrd |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) |
| 50 |
|
oveq1 |
⊢ ( 𝑐 = 𝑎 → ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ) |
| 51 |
50
|
eqeq2d |
⊢ ( 𝑐 = 𝑎 → ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ↔ ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ) ) |
| 52 |
|
oveq1 |
⊢ ( 𝑐 = 𝑎 → ( 𝑐 ↑ 2 ) = ( 𝑎 ↑ 2 ) ) |
| 53 |
52
|
oveq1d |
⊢ ( 𝑐 = 𝑎 → ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) ) |
| 54 |
53
|
eqeq1d |
⊢ ( 𝑐 = 𝑎 → ( ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ↔ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) |
| 55 |
51 54
|
anbi12d |
⊢ ( 𝑐 = 𝑎 → ( ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ↔ ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) |
| 56 |
|
oveq2 |
⊢ ( 𝑑 = - 𝑏 → ( ( √ ‘ 𝐷 ) · 𝑑 ) = ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) |
| 57 |
56
|
oveq2d |
⊢ ( 𝑑 = - 𝑏 → ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
| 58 |
57
|
eqeq2d |
⊢ ( 𝑑 = - 𝑏 → ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ↔ ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) ) |
| 59 |
|
oveq1 |
⊢ ( 𝑑 = - 𝑏 → ( 𝑑 ↑ 2 ) = ( - 𝑏 ↑ 2 ) ) |
| 60 |
59
|
oveq2d |
⊢ ( 𝑑 = - 𝑏 → ( 𝐷 · ( 𝑑 ↑ 2 ) ) = ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) |
| 61 |
60
|
oveq2d |
⊢ ( 𝑑 = - 𝑏 → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) ) |
| 62 |
61
|
eqeq1d |
⊢ ( 𝑑 = - 𝑏 → ( ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ↔ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) |
| 63 |
58 62
|
anbi12d |
⊢ ( 𝑑 = - 𝑏 → ( ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ↔ ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 64 |
55 63
|
rspc2ev |
⊢ ( ( 𝑎 ∈ ℤ ∧ - 𝑏 ∈ ℤ ∧ ( ( 1 / 𝐴 ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) |
| 65 |
7 9 44 49 64
|
syl112anc |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) |
| 66 |
6 65
|
jca |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) |
| 67 |
66
|
ex |
⊢ ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) → ( ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 68 |
67
|
rexlimdvva |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 69 |
68
|
adantld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 70 |
2 69
|
mpd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) |
| 71 |
|
elpell1234qr |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( ( 1 / 𝐴 ) ∈ ( Pell1234QR ‘ 𝐷 ) ↔ ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 72 |
71
|
adantr |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( ( 1 / 𝐴 ) ∈ ( Pell1234QR ‘ 𝐷 ) ↔ ( ( 1 / 𝐴 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ ( ( 1 / 𝐴 ) = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 73 |
70 72
|
mpbird |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( 1 / 𝐴 ) ∈ ( Pell1234QR ‘ 𝐷 ) ) |