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Theorem pell1234qrdich 37425
Description: A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell1234qrdich |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) )
Proof of Theorem pell1234qrdich
Dummy variables a b c d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpell1234qr 37415 . . 3 |- ( D e. ( NN \ ◻NN ) -> ( A e. (Pell1234QR ` D ) <-> ( A e. RR /\ E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) )
2 simp-4r 807 . . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. RR )
3 oveq1 6657 . . . . . . . . . . . . . 14 |- ( c = a -> ( c + ( ( sqr ` D ) x. b ) ) = ( a + ( ( sqr ` D ) x. b ) ) )
43eqeq2d 2632 . . . . . . . . . . . . 13 |- ( c = a -> ( A = ( c + ( ( sqr ` D ) x. b ) ) <-> A = ( a + ( ( sqr ` D ) x. b ) ) ) )
5 oveq1 6657 . . . . . . . . . . . . . . 15 |- ( c = a -> ( c ^ 2 ) = ( a ^ 2 ) )
65oveq1d 6665 . . . . . . . . . . . . . 14 |- ( c = a -> ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) )
76eqeq1d 2624 . . . . . . . . . . . . 13 |- ( c = a -> ( ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 <-> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) )
84, 7anbi12d 747 . . . . . . . . . . . 12 |- ( c = a -> ( ( A = ( c + ( ( sqr ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) <-> ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) )
98rexbidv 3052 . . . . . . . . . . 11 |- ( c = a -> ( E. b e. ZZ ( A = ( c + ( ( sqr ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) <-> E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) )
109rspcev 3309 . . . . . . . . . 10 |- ( ( a e. NN0 /\ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. b e. ZZ ( A = ( c + ( ( sqr ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) )
1110adantll 750 . . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. b e. ZZ ( A = ( c + ( ( sqr ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) )
12 elpell14qr 37413 . . . . . . . . . 10 |- ( D e. ( NN \ ◻NN ) -> ( A e. (Pell14QR ` D ) <-> ( A e. RR /\ E. c e. NN0 E. b e. ZZ ( A = ( c + ( ( sqr ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) )
1312ad4antr 768 . . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqr ` 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 + ( ( sqr ` D ) x. b ) ) /\ ( ( c ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) )
142, 11, 13mpbir2and 957 . . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. (Pell14QR ` D ) )
1514orcd 407 . . . . . . 7 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ a e. NN0 ) /\ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) )
1615exp31 630 . . . . . 6 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( a e. NN0 -> ( E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) ) )
17 simp-5r 809 . . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. RR )
1817renegcld 10457 . . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A e. RR )
19 simpllr 799 . . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u a e. NN0 )
20 znegcl 11412 . . . . . . . . . . . . 13 |- ( b e. ZZ -> -u b e. ZZ )
2120ad2antlr 763 . . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u b e. ZZ )
22 simprl 794 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A = ( a + ( ( sqr ` D ) x. b ) ) )
2322negeqd 10275 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A = -u ( a + ( ( sqr ` D ) x. b ) ) )
24 zcn 11382 . . . . . . . . . . . . . . 15 |- ( a e. ZZ -> a e. CC )
2524ad4antlr 769 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> a e. CC )
26 eldifi 3732 . . . . . . . . . . . . . . . . . 18 |- ( D e. ( NN \ ◻NN ) -> D e. NN )
2726nncnd 11036 . . . . . . . . . . . . . . . . 17 |- ( D e. ( NN \ ◻NN ) -> D e. CC )
2827ad5antr 770 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> D e. CC )
2928sqrtcld 14176 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( sqr ` D ) e. CC )
30 zcn 11382 . . . . . . . . . . . . . . . 16 |- ( b e. ZZ -> b e. CC )
3130ad2antlr 763 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> b e. CC )
3229, 31mulcld 10060 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. b ) e. CC )
3325, 32negdid 10405 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u ( a + ( ( sqr ` D ) x. b ) ) = ( -u a + -u ( ( sqr ` D ) x. b ) ) )
34 mulneg2 10467 . . . . . . . . . . . . . . . 16 |- ( ( ( sqr ` D ) e. CC /\ b e. CC ) -> ( ( sqr ` D ) x. -u b ) = -u ( ( sqr ` D ) x. b ) )
3534eqcomd 2628 . . . . . . . . . . . . . . 15 |- ( ( ( sqr ` D ) e. CC /\ b e. CC ) -> -u ( ( sqr ` D ) x. b ) = ( ( sqr ` D ) x. -u b ) )
3629, 31, 35syl2anc 693 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u ( ( sqr ` D ) x. b ) = ( ( sqr ` D ) x. -u b ) )
3736oveq2d 6666 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u a + -u ( ( sqr ` D ) x. b ) ) = ( -u a + ( ( sqr ` D ) x. -u b ) ) )
3823, 33, 373eqtrd 2660 . . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A = ( -u a + ( ( sqr ` D ) x. -u b ) ) )
39 sqneg 12923 . . . . . . . . . . . . . . 15 |- ( a e. CC -> ( -u a ^ 2 ) = ( a ^ 2 ) )
4025, 39syl 17 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u a ^ 2 ) = ( a ^ 2 ) )
41 sqneg 12923 . . . . . . . . . . . . . . . 16 |- ( b e. CC -> ( -u b ^ 2 ) = ( b ^ 2 ) )
4231, 41syl 17 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u b ^ 2 ) = ( b ^ 2 ) )
4342oveq2d 6666 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( D x. ( -u b ^ 2 ) ) = ( D x. ( b ^ 2 ) ) )
4440, 43oveq12d 6668 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` 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 796 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 )
4644, 45eqtrd 2656 . . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 )
47 oveq1 6657 . . . . . . . . . . . . . . 15 |- ( c = -u a -> ( c + ( ( sqr ` D ) x. d ) ) = ( -u a + ( ( sqr ` D ) x. d ) ) )
4847eqeq2d 2632 . . . . . . . . . . . . . 14 |- ( c = -u a -> ( -u A = ( c + ( ( sqr ` D ) x. d ) ) <-> -u A = ( -u a + ( ( sqr ` D ) x. d ) ) ) )
49 oveq1 6657 . . . . . . . . . . . . . . . 16 |- ( c = -u a -> ( c ^ 2 ) = ( -u a ^ 2 ) )
5049oveq1d 6665 . . . . . . . . . . . . . . 15 |- ( c = -u a -> ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) )
5150eqeq1d 2624 . . . . . . . . . . . . . 14 |- ( c = -u a -> ( ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 <-> ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) )
5248, 51anbi12d 747 . . . . . . . . . . . . 13 |- ( c = -u a -> ( ( -u A = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) <-> ( -u A = ( -u a + ( ( sqr ` D ) x. d ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) )
53 oveq2 6658 . . . . . . . . . . . . . . . 16 |- ( d = -u b -> ( ( sqr ` D ) x. d ) = ( ( sqr ` D ) x. -u b ) )
5453oveq2d 6666 . . . . . . . . . . . . . . 15 |- ( d = -u b -> ( -u a + ( ( sqr ` D ) x. d ) ) = ( -u a + ( ( sqr ` D ) x. -u b ) ) )
5554eqeq2d 2632 . . . . . . . . . . . . . 14 |- ( d = -u b -> ( -u A = ( -u a + ( ( sqr ` D ) x. d ) ) <-> -u A = ( -u a + ( ( sqr ` D ) x. -u b ) ) ) )
56 oveq1 6657 . . . . . . . . . . . . . . . . 17 |- ( d = -u b -> ( d ^ 2 ) = ( -u b ^ 2 ) )
5756oveq2d 6666 . . . . . . . . . . . . . . . 16 |- ( d = -u b -> ( D x. ( d ^ 2 ) ) = ( D x. ( -u b ^ 2 ) ) )
5857oveq2d 6666 . . . . . . . . . . . . . . 15 |- ( d = -u b -> ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) )
5958eqeq1d 2624 . . . . . . . . . . . . . 14 |- ( d = -u b -> ( ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 <-> ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) )
6055, 59anbi12d 747 . . . . . . . . . . . . 13 |- ( d = -u b -> ( ( -u A = ( -u a + ( ( sqr ` D ) x. d ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) <-> ( -u A = ( -u a + ( ( sqr ` D ) x. -u b ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) )
6152, 60rspc2ev 3324 . . . . . . . . . . . 12 |- ( ( -u a e. NN0 /\ -u b e. ZZ /\ ( -u A = ( -u a + ( ( sqr ` D ) x. -u b ) ) /\ ( ( -u a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. d e. ZZ ( -u A = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) )
6219, 21, 38, 46, 61syl112anc 1330 . . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 E. d e. ZZ ( -u A = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) )
63 elpell14qr 37413 . . . . . . . . . . . 12 |- ( D e. ( NN \ ◻NN ) -> ( -u A e. (Pell14QR ` D ) <-> ( -u A e. RR /\ E. c e. NN0 E. d e. ZZ ( -u A = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
6463ad5antr 770 . . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` 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 + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
6518, 62, 64mpbir2and 957 . . . . . . . . . 10 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u A e. (Pell14QR ` D ) )
6665olcd 408 . . . . . . . . 9 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) )
6766ex 450 . . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) /\ b e. ZZ ) -> ( ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) )
6867rexlimdva 3031 . . . . . . 7 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) /\ -u a e. NN0 ) -> ( E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) )
6968ex 450 . . . . . 6 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( -u a e. NN0 -> ( E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) ) )
70 elznn0 11392 . . . . . . . 8 |- ( a e. ZZ <-> ( a e. RR /\ ( a e. NN0 \/ -u a e. NN0 ) ) )
7170simprbi 480 . . . . . . 7 |- ( a e. ZZ -> ( a e. NN0 \/ -u a e. NN0 ) )
7271adantl 482 . . . . . 6 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( a e. NN0 \/ -u a e. NN0 ) )
7316, 69, 72mpjaod 396 . . . . 5 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) /\ a e. ZZ ) -> ( E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) )
7473rexlimdva 3031 . . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR ) -> ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) )
7574expimpd 629 . . 3 |- ( D e. ( NN \ ◻NN ) -> ( ( A e. RR /\ E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) )
761, 75sylbid 230 . 2 |- ( D e. ( NN \ ◻NN ) -> ( A e. (Pell1234QR ` D ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) ) )
7776imp 445 1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( A e. (Pell14QR ` D ) \/ -u A e. (Pell14QR ` D ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   sqrcsqrt 13973  ◻NNcsquarenn 37400  Pell1234QRcpell1234qr 37402  Pell14QRcpell14qr 37403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-pell14qr 37407  df-pell1234qr 37408
This theorem is referenced by:  elpell14qr2  37426
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