Theorem pell1234qrreccl 37418

Description: General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.)

Assertion

Ref	Expression
pell1234qrreccl	|- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( 1 / A ) e. (Pell1234QR ` D ) )

Proof of Theorem pell1234qrreccl

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpell1234qr 37415	. . . 4 |- ( 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	1	biimpa 501	. . 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 ) ) )
3		pell1234qrre 37416	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> A e. RR )
4		pell1234qrne0 37417	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> A =/= 0 )
5	3, 4	rereccld 10852	. . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( 1 / A ) e. RR )
6	5	ad2antrr 762	. . . . . . 7 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( 1 / A ) e. RR )
7		simplrl 800	. . . . . . . 8 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> a e. ZZ )
8		simplrr 801	. . . . . . . . 9 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> b e. ZZ )
9	8	znegcld 11484	. . . . . . . 8 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u b e. ZZ )
10	5	recnd 10068	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( 1 / A ) e. CC )
11	10	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( 1 / A ) e. CC )
12		zcn 11382	. . . . . . . . . . . 12 |- ( a e. ZZ -> a e. CC )
13	12	adantr 481	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ ) -> a e. CC )
14	13	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> a e. CC )
15		eldifi 3732	. . . . . . . . . . . . . 14 |- ( D e. ( NN \ ◻NN ) -> D e. NN )
16	15	nncnd 11036	. . . . . . . . . . . . 13 |- ( D e. ( NN \ ◻NN ) -> D e. CC )
17	16	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> D e. CC )
18	17	sqrtcld 14176	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( sqr ` D ) e. CC )
19	8	zcnd 11483	. . . . . . . . . . . 12 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> b e. CC )
20	19	negcld 10379	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> -u b e. CC )
21	18, 20	mulcld 10060	. . . . . . . . . 10 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. -u b ) e. CC )
22	14, 21	addcld 10059	. . . . . . . . 9 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) e. CC )
23	3	recnd 10068	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> A e. CC )
24	23	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. CC )
25	4	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A =/= 0 )
26	18, 19	sqmuld 13020	. . . . . . . . . . . . 13 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) x. b ) ^ 2 ) = ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) )
27	17	sqsqrtd 14178	. . . . . . . . . . . . . 14 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) ^ 2 ) = D )
28	27	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) = ( D x. ( b ^ 2 ) ) )
29	26, 28	eqtr2d 2657	. . . . . . . . . . . 12 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( D x. ( b ^ 2 ) ) = ( ( ( sqr ` D ) x. b ) ^ 2 ) )
30	29	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) )
31		simprr 796	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 )
32	18, 19	mulcld 10060	. . . . . . . . . . . 12 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. b ) e. CC )
33		subsq 12972	. . . . . . . . . . . 12 |- ( ( a e. CC /\ ( ( sqr ` D ) x. b ) e. CC ) -> ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) )
34	14, 32, 33	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) )
35	30, 31, 34	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> 1 = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) )
36	24, 25	recidd 10796	. . . . . . . . . 10 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A x. ( 1 / A ) ) = 1 )
37		simprl 794	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A = ( a + ( ( sqr ` D ) x. b ) ) )
38	18, 19	mulneg2d 10484	. . . . . . . . . . . . 13 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. -u b ) = -u ( ( sqr ` D ) x. b ) )
39	38	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) = ( a + -u ( ( sqr ` D ) x. b ) ) )
40	14, 32	negsubd 10398	. . . . . . . . . . . 12 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( a + -u ( ( sqr ` D ) x. b ) ) = ( a - ( ( sqr ` D ) x. b ) ) )
41	39, 40	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) = ( a - ( ( sqr ` D ) x. b ) ) )
42	37, 41	oveq12d 6668	. . . . . . . . . 10 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A x. ( a + ( ( sqr ` D ) x. -u b ) ) ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) )
43	35, 36, 42	3eqtr4d 2666	. . . . . . . . 9 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A x. ( 1 / A ) ) = ( A x. ( a + ( ( sqr ` D ) x. -u b ) ) ) )
44	11, 22, 24, 25, 43	mulcanad 10662	. . . . . . . 8 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) )
45		sqneg 12923	. . . . . . . . . . . 12 |- ( b e. CC -> ( -u b ^ 2 ) = ( b ^ 2 ) )
46	19, 45	syl 17	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u b ^ 2 ) = ( b ^ 2 ) )
47	46	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ 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 ) ) )
48	47	oveq2d 6666	. . . . . . . . 9 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) )
49	48, 31	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 )
50		oveq1 6657	. . . . . . . . . . 11 |- ( c = a -> ( c + ( ( sqr ` D ) x. d ) ) = ( a + ( ( sqr ` D ) x. d ) ) )
51	50	eqeq2d 2632	. . . . . . . . . 10 |- ( c = a -> ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) <-> ( 1 / A ) = ( a + ( ( sqr ` D ) x. d ) ) ) )
52		oveq1 6657	. . . . . . . . . . . 12 |- ( c = a -> ( c ^ 2 ) = ( a ^ 2 ) )
53	52	oveq1d 6665	. . . . . . . . . . 11 |- ( c = a -> ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) )
54	53	eqeq1d 2624	. . . . . . . . . 10 |- ( c = a -> ( ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 <-> ( ( a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) )
55	51, 54	anbi12d 747	. . . . . . . . 9 |- ( c = a -> ( ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) <-> ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. d ) ) /\ ( ( a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) )
56		oveq2 6658	. . . . . . . . . . . 12 |- ( d = -u b -> ( ( sqr ` D ) x. d ) = ( ( sqr ` D ) x. -u b ) )
57	56	oveq2d 6666	. . . . . . . . . . 11 |- ( d = -u b -> ( a + ( ( sqr ` D ) x. d ) ) = ( a + ( ( sqr ` D ) x. -u b ) ) )
58	57	eqeq2d 2632	. . . . . . . . . 10 |- ( d = -u b -> ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. d ) ) <-> ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) ) )
59		oveq1 6657	. . . . . . . . . . . . 13 |- ( d = -u b -> ( d ^ 2 ) = ( -u b ^ 2 ) )
60	59	oveq2d 6666	. . . . . . . . . . . 12 |- ( d = -u b -> ( D x. ( d ^ 2 ) ) = ( D x. ( -u b ^ 2 ) ) )
61	60	oveq2d 6666	. . . . . . . . . . 11 |- ( d = -u b -> ( ( a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) )
62	61	eqeq1d 2624	. . . . . . . . . 10 |- ( d = -u b -> ( ( ( a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 <-> ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) )
63	58, 62	anbi12d 747	. . . . . . . . 9 |- ( d = -u b -> ( ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. d ) ) /\ ( ( a ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) <-> ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) /\ ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) )
64	55, 63	rspc2ev 3324	. . . . . . . 8 |- ( ( a e. ZZ /\ -u b e. ZZ /\ ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) /\ ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) -> E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) )
65	7, 9, 44, 49, 64	syl112anc 1330	. . . . . . 7 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) )
66	6, 65	jca 554	. . . . . 6 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( 1 / A ) e. RR /\ E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) )
67	66	ex 450	. . . . 5 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> ( ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( ( 1 / A ) e. RR /\ E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
68	67	rexlimdvva 3038	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( ( 1 / A ) e. RR /\ E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
69	68	adantld 483	. . 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 ) ) -> ( ( 1 / A ) e. RR /\ E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
70	2, 69	mpd 15	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( ( 1 / A ) e. RR /\ E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) )
71		elpell1234qr 37415	. . 3 |- ( D e. ( NN \ ◻NN ) -> ( ( 1 / A ) e. (Pell1234QR ` D ) <-> ( ( 1 / A ) e. RR /\ E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
72	71	adantr 481	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( ( 1 / A ) e. (Pell1234QR ` D ) <-> ( ( 1 / A ) e. RR /\ E. c e. ZZ E. d e. ZZ ( ( 1 / A ) = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
73	70, 72	mpbird 247	1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) ) -> ( 1 / A ) e. (Pell1234QR ` D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   ZZcz 11377   ^cexp 12860   sqrcsqrt 13973  ◻_NNcsquarenn 37400  Pell1234QRcpell1234qr 37402

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-pell1234qr 37408

This theorem is referenced by:  pell14qrreccl  37428