Theorem pell14qrdich 37433

Description: A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.)

Assertion

Ref	Expression
pell14qrdich	|- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A e. (Pell1QR ` D ) \/ ( 1 / A ) e. (Pell1QR ` D ) ) )

Proof of Theorem pell14qrdich

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

Step	Hyp	Ref	Expression
1		elpell14qr 37413	. . 3 |- ( D e. ( NN \ ◻NN ) -> ( A e. (Pell14QR ` D ) <-> ( A e. RR /\ E. a e. NN0 E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) )
2	1	biimpa 501	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A e. RR /\ E. a e. NN0 E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) )
3		simplrr 801	. . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> b e. ZZ )
4		elznn0 11392	. . . . . . . 8 |- ( b e. ZZ <-> ( b e. RR /\ ( b e. NN0 \/ -u b e. NN0 ) ) )
5	3, 4	sylib 208	. . . . . . 7 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( b e. RR /\ ( b e. NN0 \/ -u b e. NN0 ) ) )
6	5	simprd 479	. . . . . 6 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( b e. NN0 \/ -u b e. NN0 ) )
7		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> A e. RR )
8	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ b e. NN0 ) -> A e. RR )
9		simprl 794	. . . . . . . . . . . 12 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> a e. NN0 )
10	9	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ b e. NN0 ) -> a e. NN0 )
11		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ b e. NN0 ) -> b e. NN0 )
12		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ b e. NN0 ) -> ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) )
13		rsp2e 3004	. . . . . . . . . . 11 |- ( ( a e. NN0 /\ b e. NN0 /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> E. a e. NN0 E. b e. NN0 ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) )
14	10, 11, 12, 13	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ b e. NN0 ) -> E. a e. NN0 E. b e. NN0 ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) )
15	8, 14	jca 554	. . . . . . . . 9 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ b e. NN0 ) -> ( A e. RR /\ E. a e. NN0 E. b e. NN0 ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) )
16	15	ex 450	. . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( b e. NN0 -> ( A e. RR /\ E. a e. NN0 E. b e. NN0 ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) )
17		elpell1qr 37411	. . . . . . . . 9 |- ( D e. ( NN \ ◻NN ) -> ( A e. (Pell1QR ` D ) <-> ( A e. RR /\ E. a e. NN0 E. b e. NN0 ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) )
18	17	ad4antr 768	. . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. (Pell1QR ` D ) <-> ( A e. RR /\ E. a e. NN0 E. b e. NN0 ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) ) )
19	16, 18	sylibrd 249	. . . . . . 7 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( b e. NN0 -> A e. (Pell1QR ` D ) ) )
20	7	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> A e. RR )
21		pell14qrne0 37422	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A =/= 0 )
22	21	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> A =/= 0 )
23	20, 22	rereccld 10852	. . . . . . . . . 10 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( 1 / A ) e. RR )
24	9	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> a e. NN0 )
25		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> -u b e. NN0 )
26		pell14qrre 37421	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A e. RR )
27	26	recnd 10068	. . . . . . . . . . . . . . . . 17 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A e. CC )
28	27, 21	reccld 10794	. . . . . . . . . . . . . . . 16 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( 1 / A ) e. CC )
29	28	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( 1 / A ) e. CC )
30		nn0cn 11302	. . . . . . . . . . . . . . . . . 18 |- ( a e. NN0 -> a e. CC )
31	30	ad2antrl 764	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> a e. CC )
32		eldifi 3732	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. ( NN \ ◻NN ) -> D e. NN )
33	32	nncnd 11036	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. ( NN \ ◻NN ) -> D e. CC )
34	33	ad3antrrr 766	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> D e. CC )
35	34	sqrtcld 14176	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( sqr ` D ) e. CC )
36		zcn 11382	. . . . . . . . . . . . . . . . . . . 20 |- ( b e. ZZ -> b e. CC )
37	36	ad2antll 765	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> b e. CC )
38	37	negcld 10379	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> -u b e. CC )
39	35, 38	mulcld 10060	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( sqr ` D ) x. -u b ) e. CC )
40	31, 39	addcld 10059	. . . . . . . . . . . . . . . 16 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) e. CC )
41	40	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) e. CC )
42	27	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A e. CC )
43	21	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> A =/= 0 )
44	27, 21	recidd 10796	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A x. ( 1 / A ) ) = 1 )
45	44	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A x. ( 1 / A ) ) = 1 )
46		simprr 796	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( 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 )
47	45, 46	eqtr4d 2659	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A x. ( 1 / A ) ) = ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) )
48	31	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> a e. CC )
49	35, 37	mulcld 10060	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( sqr ` D ) x. b ) e. CC )
50	49	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> ( ( sqr ` D ) x. b ) e. CC )
51		subsq 12972	. . . . . . . . . . . . . . . . . . 19 |- ( ( 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 ) ) ) )
52	48, 50, 51	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) )
53	35, 37	sqmuld 13020	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( ( sqr ` D ) x. b ) ^ 2 ) = ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) )
54	34	sqsqrtd 14178	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( sqr ` D ) ^ 2 ) = D )
55	54	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) = ( D x. ( b ^ 2 ) ) )
56	53, 55	eqtr2d 2657	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( D x. ( b ^ 2 ) ) = ( ( ( sqr ` D ) x. b ) ^ 2 ) )
57	56	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) )
58	57	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) )
59		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> A = ( a + ( ( sqr ` D ) x. b ) ) )
60	35, 37	mulneg2d 10484	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( sqr ` D ) x. -u b ) = -u ( ( sqr ` D ) x. b ) )
61	60	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) = ( a + -u ( ( sqr ` D ) x. b ) ) )
62		negsub 10329	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( a e. CC /\ ( ( sqr ` D ) x. b ) e. CC ) -> ( a + -u ( ( sqr ` D ) x. b ) ) = ( a - ( ( sqr ` D ) x. b ) ) )
63	62	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( a e. CC /\ ( ( sqr ` D ) x. b ) e. CC ) -> ( a - ( ( sqr ` D ) x. b ) ) = ( a + -u ( ( sqr ` D ) x. b ) ) )
64	31, 49, 63	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( a - ( ( sqr ` D ) x. b ) ) = ( a + -u ( ( sqr ` D ) x. b ) ) )
65	61, 64	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) = ( a - ( ( sqr ` D ) x. b ) ) )
66	65	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> ( a + ( ( sqr ` D ) x. -u b ) ) = ( a - ( ( sqr ` D ) x. b ) ) )
67	59, 66	oveq12d 6668	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> ( A x. ( a + ( ( sqr ` D ) x. -u b ) ) ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) )
68	52, 58, 67	3eqtr4d 2666	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ A = ( a + ( ( sqr ` D ) x. b ) ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( A x. ( a + ( ( sqr ` D ) x. -u b ) ) ) )
69	68	adantrr 753	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( 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 ) ) ) = ( A x. ( a + ( ( sqr ` D ) x. -u b ) ) ) )
70	47, 69	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ 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 ) ) ) )
71	29, 41, 42, 43, 70	mulcanad 10662	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) )
72	71	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) )
73	37	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> b e. CC )
74		sqneg 12923	. . . . . . . . . . . . . . . . 17 |- ( b e. CC -> ( -u b ^ 2 ) = ( b ^ 2 ) )
75	73, 74	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( -u b ^ 2 ) = ( b ^ 2 ) )
76	75	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( D x. ( -u b ^ 2 ) ) = ( D x. ( b ^ 2 ) ) )
77	76	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) )
78		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 )
79	77, 78	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 )
80	72, 79	jca 554	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) /\ ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) )
81		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( c = -u b -> ( ( sqr ` D ) x. c ) = ( ( sqr ` D ) x. -u b ) )
82	81	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( c = -u b -> ( a + ( ( sqr ` D ) x. c ) ) = ( a + ( ( sqr ` D ) x. -u b ) ) )
83	82	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( c = -u b -> ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) <-> ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) ) )
84		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( c = -u b -> ( c ^ 2 ) = ( -u b ^ 2 ) )
85	84	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( c = -u b -> ( D x. ( c ^ 2 ) ) = ( D x. ( -u b ^ 2 ) ) )
86	85	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( c = -u b -> ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) )
87	86	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( c = -u b -> ( ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 <-> ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) )
88	83, 87	anbi12d 747	. . . . . . . . . . . . 13 |- ( c = -u b -> ( ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) <-> ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) /\ ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) )
89	88	rspcev 3309	. . . . . . . . . . . 12 |- ( ( -u b e. NN0 /\ ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. -u b ) ) /\ ( ( a ^ 2 ) - ( D x. ( -u b ^ 2 ) ) ) = 1 ) ) -> E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) )
90	25, 80, 89	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) )
91		rspe 3003	. . . . . . . . . . 11 |- ( ( a e. NN0 /\ E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) ) -> E. a e. NN0 E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) )
92	24, 90, 91	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> E. a e. NN0 E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) )
93	23, 92	jca 554	. . . . . . . . 9 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ -u b e. NN0 ) -> ( ( 1 / A ) e. RR /\ E. a e. NN0 E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) ) )
94	93	ex 450	. . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u b e. NN0 -> ( ( 1 / A ) e. RR /\ E. a e. NN0 E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) ) ) )
95		elpell1qr 37411	. . . . . . . . 9 |- ( D e. ( NN \ ◻NN ) -> ( ( 1 / A ) e. (Pell1QR ` D ) <-> ( ( 1 / A ) e. RR /\ E. a e. NN0 E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) ) ) )
96	95	ad4antr 768	. . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( 1 / A ) e. (Pell1QR ` D ) <-> ( ( 1 / A ) e. RR /\ E. a e. NN0 E. c e. NN0 ( ( 1 / A ) = ( a + ( ( sqr ` D ) x. c ) ) /\ ( ( a ^ 2 ) - ( D x. ( c ^ 2 ) ) ) = 1 ) ) ) )
97	94, 96	sylibrd 249	. . . . . . 7 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( -u b e. NN0 -> ( 1 / A ) e. (Pell1QR ` D ) ) )
98	19, 97	orim12d 883	. . . . . 6 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( ( b e. NN0 \/ -u b e. NN0 ) -> ( A e. (Pell1QR ` D ) \/ ( 1 / A ) e. (Pell1QR ` D ) ) ) )
99	6, 98	mpd 15	. . . . 5 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. (Pell1QR ` D ) \/ ( 1 / A ) e. (Pell1QR ` D ) ) )
100	99	ex 450	. . . 4 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) /\ ( a e. NN0 /\ b e. ZZ ) ) -> ( ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell1QR ` D ) \/ ( 1 / A ) e. (Pell1QR ` D ) ) ) )
101	100	rexlimdvva 3038	. . 3 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ A e. RR ) -> ( E. a e. NN0 E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( A e. (Pell1QR ` D ) \/ ( 1 / A ) e. (Pell1QR ` D ) ) ) )
102	101	expimpd 629	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( A e. RR /\ E. a e. NN0 E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) -> ( A e. (Pell1QR ` D ) \/ ( 1 / A ) e. (Pell1QR ` D ) ) ) )
103	2, 102	mpd 15	1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A e. (Pell1QR ` D ) \/ ( 1 / A ) e. (Pell1QR ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ 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   NN0cn0 11292   ZZcz 11377   ^cexp 12860   sqrcsqrt 13973  ◻_NNcsquarenn 37400  Pell1QRcpell1qr 37401  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-pell1qr 37406  df-pell14qr 37407  df-pell1234qr 37408

This theorem is referenced by:  elpell1qr2  37436