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 |
|
simp-4r |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ 𝑎 ∈ ℕ0 ) ∧ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ∈ ℝ ) |
3 |
|
oveq1 |
⊢ ( 𝑐 = 𝑎 → ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
4 |
3
|
eqeq2d |
⊢ ( 𝑐 = 𝑎 → ( 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ↔ 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) ) |
5 |
|
oveq1 |
⊢ ( 𝑐 = 𝑎 → ( 𝑐 ↑ 2 ) = ( 𝑎 ↑ 2 ) ) |
6 |
5
|
oveq1d |
⊢ ( 𝑐 = 𝑎 → ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) ) |
7 |
6
|
eqeq1d |
⊢ ( 𝑐 = 𝑎 → ( ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ↔ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) |
8 |
4 7
|
anbi12d |
⊢ ( 𝑐 = 𝑎 → ( ( 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ↔ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
9 |
8
|
rexbidv |
⊢ ( 𝑐 = 𝑎 → ( ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ↔ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
10 |
9
|
rspcev |
⊢ ( ( 𝑎 ∈ ℕ0 ∧ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑐 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) |
11 |
10
|
adantll |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ 𝑎 ∈ ℕ0 ) ∧ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑐 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) |
12 |
|
elpell14qr |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ↔ ( 𝐴 ∈ ℝ ∧ ∃ 𝑐 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
13 |
12
|
ad4antr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ 𝑎 ∈ ℕ0 ) ∧ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ↔ ( 𝐴 ∈ ℝ ∧ ∃ 𝑐 ∈ ℕ0 ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
14 |
2 11 13
|
mpbir2and |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ 𝑎 ∈ ℕ0 ) ∧ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) |
15 |
14
|
orcd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ 𝑎 ∈ ℕ0 ) ∧ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) |
16 |
15
|
exp31 |
⊢ ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) → ( 𝑎 ∈ ℕ0 → ( ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) ) |
17 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ∈ ℝ ) |
18 |
17
|
renegcld |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝐴 ∈ ℝ ) |
19 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝑎 ∈ ℕ0 ) |
20 |
|
znegcl |
⊢ ( 𝑏 ∈ ℤ → - 𝑏 ∈ ℤ ) |
21 |
20
|
ad2antlr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝑏 ∈ ℤ ) |
22 |
|
simprl |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
23 |
22
|
negeqd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝐴 = - ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
24 |
|
zcn |
⊢ ( 𝑎 ∈ ℤ → 𝑎 ∈ ℂ ) |
25 |
24
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝑎 ∈ ℂ ) |
26 |
|
eldifi |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → 𝐷 ∈ ℕ ) |
27 |
26
|
nncnd |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → 𝐷 ∈ ℂ ) |
28 |
27
|
ad5antr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝐷 ∈ ℂ ) |
29 |
28
|
sqrtcld |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ 𝐷 ) ∈ ℂ ) |
30 |
|
zcn |
⊢ ( 𝑏 ∈ ℤ → 𝑏 ∈ ℂ ) |
31 |
30
|
ad2antlr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → 𝑏 ∈ ℂ ) |
32 |
29 31
|
mulcld |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) · 𝑏 ) ∈ ℂ ) |
33 |
25 32
|
negdid |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) = ( - 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ) |
34 |
|
mulneg2 |
⊢ ( ( ( √ ‘ 𝐷 ) ∈ ℂ ∧ 𝑏 ∈ ℂ ) → ( ( √ ‘ 𝐷 ) · - 𝑏 ) = - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) |
35 |
34
|
eqcomd |
⊢ ( ( ( √ ‘ 𝐷 ) ∈ ℂ ∧ 𝑏 ∈ ℂ ) → - ( ( √ ‘ 𝐷 ) · 𝑏 ) = ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) |
36 |
29 31 35
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - ( ( √ ‘ 𝐷 ) · 𝑏 ) = ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) |
37 |
36
|
oveq2d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( - 𝑎 + - ( ( √ ‘ 𝐷 ) · 𝑏 ) ) = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
38 |
23 33 37
|
3eqtrd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
39 |
|
sqneg |
⊢ ( 𝑎 ∈ ℂ → ( - 𝑎 ↑ 2 ) = ( 𝑎 ↑ 2 ) ) |
40 |
25 39
|
syl |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( - 𝑎 ↑ 2 ) = ( 𝑎 ↑ 2 ) ) |
41 |
|
sqneg |
⊢ ( 𝑏 ∈ ℂ → ( - 𝑏 ↑ 2 ) = ( 𝑏 ↑ 2 ) ) |
42 |
31 41
|
syl |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( - 𝑏 ↑ 2 ) = ( 𝑏 ↑ 2 ) ) |
43 |
42
|
oveq2d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · ( - 𝑏 ↑ 2 ) ) = ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) |
44 |
40 43
|
oveq12d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) ) |
45 |
|
simprr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) |
46 |
44 45
|
eqtrd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) |
47 |
|
oveq1 |
⊢ ( 𝑐 = - 𝑎 → ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ) |
48 |
47
|
eqeq2d |
⊢ ( 𝑐 = - 𝑎 → ( - 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ↔ - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ) ) |
49 |
|
oveq1 |
⊢ ( 𝑐 = - 𝑎 → ( 𝑐 ↑ 2 ) = ( - 𝑎 ↑ 2 ) ) |
50 |
49
|
oveq1d |
⊢ ( 𝑐 = - 𝑎 → ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) ) |
51 |
50
|
eqeq1d |
⊢ ( 𝑐 = - 𝑎 → ( ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ↔ ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) |
52 |
48 51
|
anbi12d |
⊢ ( 𝑐 = - 𝑎 → ( ( - 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ↔ ( - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) |
53 |
|
oveq2 |
⊢ ( 𝑑 = - 𝑏 → ( ( √ ‘ 𝐷 ) · 𝑑 ) = ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) |
54 |
53
|
oveq2d |
⊢ ( 𝑑 = - 𝑏 → ( - 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) |
55 |
54
|
eqeq2d |
⊢ ( 𝑑 = - 𝑏 → ( - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ↔ - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ) ) |
56 |
|
oveq1 |
⊢ ( 𝑑 = - 𝑏 → ( 𝑑 ↑ 2 ) = ( - 𝑏 ↑ 2 ) ) |
57 |
56
|
oveq2d |
⊢ ( 𝑑 = - 𝑏 → ( 𝐷 · ( 𝑑 ↑ 2 ) ) = ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) |
58 |
57
|
oveq2d |
⊢ ( 𝑑 = - 𝑏 → ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) ) |
59 |
58
|
eqeq1d |
⊢ ( 𝑑 = - 𝑏 → ( ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ↔ ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) |
60 |
55 59
|
anbi12d |
⊢ ( 𝑑 = - 𝑏 → ( ( - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ↔ ( - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∧ ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
61 |
52 60
|
rspc2ev |
⊢ ( ( - 𝑎 ∈ ℕ0 ∧ - 𝑏 ∈ ℤ ∧ ( - 𝐴 = ( - 𝑎 + ( ( √ ‘ 𝐷 ) · - 𝑏 ) ) ∧ ( ( - 𝑎 ↑ 2 ) − ( 𝐷 · ( - 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑐 ∈ ℕ0 ∃ 𝑑 ∈ ℤ ( - 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) |
62 |
19 21 38 46 61
|
syl112anc |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ∃ 𝑐 ∈ ℕ0 ∃ 𝑑 ∈ ℤ ( - 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) |
63 |
|
elpell14qr |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ↔ ( - 𝐴 ∈ ℝ ∧ ∃ 𝑐 ∈ ℕ0 ∃ 𝑑 ∈ ℤ ( - 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) ) |
64 |
63
|
ad5antr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ↔ ( - 𝐴 ∈ ℝ ∧ ∃ 𝑐 ∈ ℕ0 ∃ 𝑑 ∈ ℤ ( - 𝐴 = ( 𝑐 + ( ( √ ‘ 𝐷 ) · 𝑑 ) ) ∧ ( ( 𝑐 ↑ 2 ) − ( 𝐷 · ( 𝑑 ↑ 2 ) ) ) = 1 ) ) ) ) |
65 |
18 62 64
|
mpbir2and |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) |
66 |
65
|
olcd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) ∧ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) |
67 |
66
|
ex |
⊢ ( ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) ∧ 𝑏 ∈ ℤ ) → ( ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) |
68 |
67
|
rexlimdva |
⊢ ( ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) ∧ - 𝑎 ∈ ℕ0 ) → ( ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) |
69 |
68
|
ex |
⊢ ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) → ( - 𝑎 ∈ ℕ0 → ( ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) ) |
70 |
|
elznn0 |
⊢ ( 𝑎 ∈ ℤ ↔ ( 𝑎 ∈ ℝ ∧ ( 𝑎 ∈ ℕ0 ∨ - 𝑎 ∈ ℕ0 ) ) ) |
71 |
70
|
simprbi |
⊢ ( 𝑎 ∈ ℤ → ( 𝑎 ∈ ℕ0 ∨ - 𝑎 ∈ ℕ0 ) ) |
72 |
71
|
adantl |
⊢ ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) → ( 𝑎 ∈ ℕ0 ∨ - 𝑎 ∈ ℕ0 ) ) |
73 |
16 69 72
|
mpjaod |
⊢ ( ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) ∧ 𝑎 ∈ ℤ ) → ( ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) |
74 |
73
|
rexlimdva |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℝ ) → ( ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) |
75 |
74
|
expimpd |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( ( 𝐴 ∈ ℝ ∧ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ( 𝐴 = ( 𝑎 + ( ( √ ‘ 𝐷 ) · 𝑏 ) ) ∧ ( ( 𝑎 ↑ 2 ) − ( 𝐷 · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) |
76 |
1 75
|
sylbid |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) ) |
77 |
76
|
imp |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell1234QR ‘ 𝐷 ) ) → ( 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∨ - 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) ) |