Step |
Hyp |
Ref |
Expression |
1 |
|
elpell1234qr |
|- ( D e. ( NN \ []NN ) -> ( A e. ( Pell1234QR ` D ) <-> ( A e. RR /\ E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) ) |
2 |
|
simp-4r |
|- ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. RR ) |
3 |
|
oveq1 |
|- ( c = a -> ( c + ( ( sqrt ` D ) x. b ) ) = ( a + ( ( sqrt ` D ) x. b ) ) ) |
4 |
3
|
eqeq2d |
|- ( c = a -> ( A = ( c + ( ( sqrt ` D ) x. b ) ) <-> A = ( a + ( ( sqrt ` D ) x. b ) ) ) ) |
5 |
|
oveq1 |
|- ( c = a -> ( c ^ 2 ) = ( a ^ 2 ) ) |
6 |
5
|
oveq1d |
|- ( c = a -> ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) ) |
7 |
6
|
eqeq1d |
|- ( c = a -> ( ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 <-> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) |
8 |
4 7
|
anbi12d |
|- ( c = a -> ( ( A = ( c + ( ( sqrt ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) <-> ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) |
9 |
8
|
rexbidv |
|- ( c = a -> ( E. b e. ZZ ( A = ( c + ( ( sqrt ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) <-> E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) |
10 |
9
|
rspcev |
|- ( ( a e. NN0 /\ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. b e. ZZ ( A = ( c + ( ( sqrt ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) |
11 |
10
|
adantll |
|- ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. b e. ZZ ( A = ( c + ( ( sqrt ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) |
12 |
|
elpell14qr |
|- ( D e. ( NN \ []NN ) -> ( A e. ( Pell14QR ` D ) <-> ( A e. RR /\ E. c e. NN0 E. b e. ZZ ( A = ( c + ( ( sqrt ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) ) |
13 |
12
|
ad4antr |
|- ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. ( Pell14QR ` D ) <-> ( A e. RR /\ E. c e. NN0 E. b e. ZZ ( A = ( c + ( ( sqrt ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) ) |
14 |
2 11 13
|
mpbir2and |
|- ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. ( Pell14QR ` D ) ) |
15 |
14
|
orcd |
|- ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) |
16 |
15
|
exp31 |
|- ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( a e. NN0 -> ( E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) ) |
17 |
|
simp-5r |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. RR ) |
18 |
17
|
renegcld |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A e. RR ) |
19 |
|
simpllr |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u a e. NN0 ) |
20 |
|
znegcl |
|- ( b e. ZZ -> -u b e. ZZ ) |
21 |
20
|
ad2antlr |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u b e. ZZ ) |
22 |
|
simprl |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A = ( a + ( ( sqrt ` D ) x. b ) ) ) |
23 |
22
|
negeqd |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A = -u ( a + ( ( sqrt ` D ) x. b ) ) ) |
24 |
|
zcn |
|- ( a e. ZZ -> a e. CC ) |
25 |
24
|
ad4antlr |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> a e. CC ) |
26 |
|
eldifi |
|- ( D e. ( NN \ []NN ) -> D e. NN ) |
27 |
26
|
nncnd |
|- ( D e. ( NN \ []NN ) -> D e. CC ) |
28 |
27
|
ad5antr |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> D e. CC ) |
29 |
28
|
sqrtcld |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( sqrt ` D ) e. CC ) |
30 |
|
zcn |
|- ( b e. ZZ -> b e. CC ) |
31 |
30
|
ad2antlr |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> b e. CC ) |
32 |
29 31
|
mulcld |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( sqrt ` D ) x. b ) e. CC ) |
33 |
25 32
|
negdid |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u ( a + ( ( sqrt ` D ) x. b ) ) = ( -u a + -u ( ( sqrt ` D ) x. b ) ) ) |
34 |
|
mulneg2 |
|- ( ( ( sqrt ` D ) e. CC /\ b e. CC ) -> ( ( sqrt ` D ) x. -u b ) = -u ( ( sqrt ` D ) x. b ) ) |
35 |
34
|
eqcomd |
|- ( ( ( sqrt ` D ) e. CC /\ b e. CC ) -> -u ( ( sqrt ` D ) x. b ) = ( ( sqrt ` D ) x. -u b ) ) |
36 |
29 31 35
|
syl2anc |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u ( ( sqrt ` D ) x. b ) = ( ( sqrt ` D ) x. -u b ) ) |
37 |
36
|
oveq2d |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u a + -u ( ( sqrt ` D ) x. b ) ) = ( -u a + ( ( sqrt ` D ) x. -u b ) ) ) |
38 |
23 33 37
|
3eqtrd |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A = ( -u a + ( ( sqrt ` D ) x. -u b ) ) ) |
39 |
|
sqneg |
|- ( a e. CC -> ( -u a ^ 2 ) = ( a ^ 2 ) ) |
40 |
25 39
|
syl |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u a ^ 2 ) = ( a ^ 2 ) ) |
41 |
|
sqneg |
|- ( b e. CC -> ( -u b ^ 2 ) = ( b ^ 2 ) ) |
42 |
31 41
|
syl |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u b ^ 2 ) = ( b ^ 2 ) ) |
43 |
42
|
oveq2d |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( D x. ( -u b ^ 2 ) ) = ( D x. ( b ^ 2 ) ) ) |
44 |
40 43
|
oveq12d |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) ) |
45 |
|
simprr |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) |
46 |
44 45
|
eqtrd |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) |
47 |
|
oveq1 |
|- ( c = -u a -> ( c + ( ( sqrt ` D ) x. d ) ) = ( -u a + ( ( sqrt ` D ) x. d ) ) ) |
48 |
47
|
eqeq2d |
|- ( c = -u a -> ( -u A = ( c + ( ( sqrt ` D ) x. d ) ) <-> -u A = ( -u a + ( ( sqrt ` D ) x. d ) ) ) ) |
49 |
|
oveq1 |
|- ( c = -u a -> ( c ^ 2 ) = ( -u a ^ 2 ) ) |
50 |
49
|
oveq1d |
|- ( c = -u a -> ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) ) |
51 |
50
|
eqeq1d |
|- ( c = -u a -> ( ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 <-> ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) |
52 |
48 51
|
anbi12d |
|- ( c = -u a -> ( ( -u A = ( c + ( ( sqrt ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) <-> ( -u A = ( -u a + ( ( sqrt ` D ) x. d ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) |
53 |
|
oveq2 |
|- ( d = -u b -> ( ( sqrt ` D ) x. d ) = ( ( sqrt ` D ) x. -u b ) ) |
54 |
53
|
oveq2d |
|- ( d = -u b -> ( -u a + ( ( sqrt ` D ) x. d ) ) = ( -u a + ( ( sqrt ` D ) x. -u b ) ) ) |
55 |
54
|
eqeq2d |
|- ( d = -u b -> ( -u A = ( -u a + ( ( sqrt ` D ) x. d ) ) <-> -u A = ( -u a + ( ( sqrt ` D ) x. -u b ) ) ) ) |
56 |
|
oveq1 |
|- ( d = -u b -> ( d ^ 2 ) = ( -u b ^ 2 ) ) |
57 |
56
|
oveq2d |
|- ( d = -u b -> ( D x. ( d ^ 2 ) ) = ( D x. ( -u b ^ 2 ) ) ) |
58 |
57
|
oveq2d |
|- ( d = -u b -> ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) ) |
59 |
58
|
eqeq1d |
|- ( d = -u b -> ( ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 <-> ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) |
60 |
55 59
|
anbi12d |
|- ( d = -u b -> ( ( -u A = ( -u a + ( ( sqrt ` D ) x. d ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) <-> ( -u A = ( -u a + ( ( sqrt ` D ) x. -u b ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) ) |
61 |
52 60
|
rspc2ev |
|- ( ( -u a e. NN0 /\ -u b e. ZZ /\ ( -u A = ( -u a + ( ( sqrt ` D ) x. -u b ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. d e. ZZ ( -u A = ( c + ( ( sqrt ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) |
62 |
19 21 38 46 61
|
syl112anc |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. d e. ZZ ( -u A = ( c + ( ( sqrt ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) |
63 |
|
elpell14qr |
|- ( D e. ( NN \ []NN ) -> ( -u A e. ( Pell14QR ` D ) <-> ( -u A e. RR /\ E. c e. NN0 E. d e. ZZ ( -u A = ( c + ( ( sqrt ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) ) |
64 |
63
|
ad5antr |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u A e. ( Pell14QR ` D ) <-> ( -u A e. RR /\ E. c e. NN0 E. d e. ZZ ( -u A = ( c + ( ( sqrt ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) ) |
65 |
18 62 64
|
mpbir2and |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A e. ( Pell14QR ` D ) ) |
66 |
65
|
olcd |
|- ( ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) |
67 |
66
|
ex |
|- ( ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) -> ( ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) |
68 |
67
|
rexlimdva |
|- ( ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) -> ( E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) |
69 |
68
|
ex |
|- ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( -u a e. NN0 -> ( E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) ) |
70 |
|
elznn0 |
|- ( a e. ZZ <-> ( a e. RR /\ ( a e. NN0 \/ -u a e. NN0 ) ) ) |
71 |
70
|
simprbi |
|- ( a e. ZZ -> ( a e. NN0 \/ -u a e. NN0 ) ) |
72 |
71
|
adantl |
|- ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( a e. NN0 \/ -u a e. NN0 ) ) |
73 |
16 69 72
|
mpjaod |
|- ( ( ( D e. ( NN \ []NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) |
74 |
73
|
rexlimdva |
|- ( ( D e. ( NN \ []NN ) /\ A e. RR ) -> ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) |
75 |
74
|
expimpd |
|- ( D e. ( NN \ []NN ) -> ( ( A e. RR /\ E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqrt ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) |
76 |
1 75
|
sylbid |
|- ( D e. ( NN \ []NN ) -> ( A e. ( Pell1234QR ` D ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) ) |
77 |
76
|
imp |
|- ( ( D e. ( NN \ []NN ) /\ A e. ( Pell1234QR ` D ) ) -> ( A e. ( Pell14QR ` D ) \/ -u A e. ( Pell14QR ` D ) ) ) |