Step |
Hyp |
Ref |
Expression |
0 |
|
cpell1234qr |
⊢ Pell1234QR |
1 |
|
vx |
⊢ 𝑥 |
2 |
|
cn |
⊢ ℕ |
3 |
|
csquarenn |
⊢ ◻NN |
4 |
2 3
|
cdif |
⊢ ( ℕ ∖ ◻NN ) |
5 |
|
vy |
⊢ 𝑦 |
6 |
|
cr |
⊢ ℝ |
7 |
|
vz |
⊢ 𝑧 |
8 |
|
cz |
⊢ ℤ |
9 |
|
vw |
⊢ 𝑤 |
10 |
5
|
cv |
⊢ 𝑦 |
11 |
7
|
cv |
⊢ 𝑧 |
12 |
|
caddc |
⊢ + |
13 |
|
csqrt |
⊢ √ |
14 |
1
|
cv |
⊢ 𝑥 |
15 |
14 13
|
cfv |
⊢ ( √ ‘ 𝑥 ) |
16 |
|
cmul |
⊢ · |
17 |
9
|
cv |
⊢ 𝑤 |
18 |
15 17 16
|
co |
⊢ ( ( √ ‘ 𝑥 ) · 𝑤 ) |
19 |
11 18 12
|
co |
⊢ ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) |
20 |
10 19
|
wceq |
⊢ 𝑦 = ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) |
21 |
|
cexp |
⊢ ↑ |
22 |
|
c2 |
⊢ 2 |
23 |
11 22 21
|
co |
⊢ ( 𝑧 ↑ 2 ) |
24 |
|
cmin |
⊢ − |
25 |
17 22 21
|
co |
⊢ ( 𝑤 ↑ 2 ) |
26 |
14 25 16
|
co |
⊢ ( 𝑥 · ( 𝑤 ↑ 2 ) ) |
27 |
23 26 24
|
co |
⊢ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) |
28 |
|
c1 |
⊢ 1 |
29 |
27 28
|
wceq |
⊢ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) = 1 |
30 |
20 29
|
wa |
⊢ ( 𝑦 = ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) ∧ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) = 1 ) |
31 |
30 9 8
|
wrex |
⊢ ∃ 𝑤 ∈ ℤ ( 𝑦 = ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) ∧ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) = 1 ) |
32 |
31 7 8
|
wrex |
⊢ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ ( 𝑦 = ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) ∧ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) = 1 ) |
33 |
32 5 6
|
crab |
⊢ { 𝑦 ∈ ℝ ∣ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ ( 𝑦 = ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) ∧ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) = 1 ) } |
34 |
1 4 33
|
cmpt |
⊢ ( 𝑥 ∈ ( ℕ ∖ ◻NN ) ↦ { 𝑦 ∈ ℝ ∣ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ ( 𝑦 = ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) ∧ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) = 1 ) } ) |
35 |
0 34
|
wceq |
⊢ Pell1234QR = ( 𝑥 ∈ ( ℕ ∖ ◻NN ) ↦ { 𝑦 ∈ ℝ ∣ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ ( 𝑦 = ( 𝑧 + ( ( √ ‘ 𝑥 ) · 𝑤 ) ) ∧ ( ( 𝑧 ↑ 2 ) − ( 𝑥 · ( 𝑤 ↑ 2 ) ) ) = 1 ) } ) |