Theorem pell1234qrmulcl 37419

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

Assertion

Ref	Expression
pell1234qrmulcl	|- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) /\ B e. (Pell1234QR ` D ) ) -> ( A x. B ) e. (Pell1234QR ` D ) )

Proof of Theorem pell1234qrmulcl

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

Step	Hyp	Ref	Expression
1		remulcl 10021	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
2	1	ad5antlr 771	. . . . . . . . . 10 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( A x. B ) e. RR )
3		simprl 794	. . . . . . . . . . . . . 14 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> a e. ZZ )
4	3	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> a e. ZZ )
5		simplrl 800	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> c e. ZZ )
6	4, 5	zmulcld 11488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( a x. c ) e. ZZ )
7		eldifi 3732	. . . . . . . . . . . . . . . 16 |- ( D e. ( NN \ ◻NN ) -> D e. NN )
8	7	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> D e. NN )
9	8	nnzd 11481	. . . . . . . . . . . . . 14 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> D e. ZZ )
10	9	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> D e. ZZ )
11		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> d e. ZZ )
12		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> b e. ZZ )
13	12	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> b e. ZZ )
14	11, 13	zmulcld 11488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( d x. b ) e. ZZ )
15	10, 14	zmulcld 11488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( D x. ( d x. b ) ) e. ZZ )
16	6, 15	zaddcld 11486	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a x. c ) + ( D x. ( d x. b ) ) ) e. ZZ )
17	4, 11	zmulcld 11488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( a x. d ) e. ZZ )
18	5, 13	zmulcld 11488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( c x. b ) e. ZZ )
19	17, 18	zaddcld 11486	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a x. d ) + ( c x. b ) ) e. ZZ )
20		simprl 794	. . . . . . . . . . . . . 14 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( 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 ) ) )
21	20	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> A = ( a + ( ( sqr ` D ) x. b ) ) )
22		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> B = ( c + ( ( sqr ` D ) x. d ) ) )
23	21, 22	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( A x. B ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( c + ( ( sqr ` D ) x. d ) ) ) )
24		zcn 11382	. . . . . . . . . . . . . . 15 |- ( a e. ZZ -> a e. CC )
25	24	ad2antrl 764	. . . . . . . . . . . . . 14 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> a e. CC )
26	25	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> a e. CC )
27	8	nncnd 11036	. . . . . . . . . . . . . . . 16 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> D e. CC )
28	27	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> D e. CC )
29	28	sqrtcld 14176	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( sqr ` D ) e. CC )
30		zcn 11382	. . . . . . . . . . . . . . . 16 |- ( b e. ZZ -> b e. CC )
31	30	ad2antll 765	. . . . . . . . . . . . . . 15 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> b e. CC )
32	31	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> b e. CC )
33	29, 32	mulcld 10060	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. b ) e. CC )
34		zcn 11382	. . . . . . . . . . . . . . 15 |- ( c e. ZZ -> c e. CC )
35	34	adantr 481	. . . . . . . . . . . . . 14 |- ( ( c e. ZZ /\ d e. ZZ ) -> c e. CC )
36	35	ad2antlr 763	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> c e. CC )
37		zcn 11382	. . . . . . . . . . . . . . . 16 |- ( d e. ZZ -> d e. CC )
38	37	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( c e. ZZ /\ d e. ZZ ) -> d e. CC )
39	38	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> d e. CC )
40	29, 39	mulcld 10060	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. d ) e. CC )
41	26, 33, 36, 40	muladdd 10489	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a + ( ( sqr ` D ) x. b ) ) x. ( c + ( ( sqr ` D ) x. d ) ) ) = ( ( ( a x. c ) + ( ( ( sqr ` D ) x. d ) x. ( ( sqr ` D ) x. b ) ) ) + ( ( a x. ( ( sqr ` D ) x. d ) ) + ( c x. ( ( sqr ` D ) x. b ) ) ) ) )
42	29, 39, 29, 32	mul4d 10248	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) x. d ) x. ( ( sqr ` D ) x. b ) ) = ( ( ( sqr ` D ) x. ( sqr ` D ) ) x. ( d x. b ) ) )
43	28	msqsqrtd 14179	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. ( sqr ` D ) ) = D )
44	43	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) x. ( sqr ` D ) ) x. ( d x. b ) ) = ( D x. ( d x. b ) ) )
45	42, 44	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) x. d ) x. ( ( sqr ` D ) x. b ) ) = ( D x. ( d x. b ) ) )
46	45	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a x. c ) + ( ( ( sqr ` D ) x. d ) x. ( ( sqr ` D ) x. b ) ) ) = ( ( a x. c ) + ( D x. ( d x. b ) ) ) )
47	26, 29, 39	mul12d 10245	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( a x. ( ( sqr ` D ) x. d ) ) = ( ( sqr ` D ) x. ( a x. d ) ) )
48	36, 29, 32	mul12d 10245	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( c x. ( ( sqr ` D ) x. b ) ) = ( ( sqr ` D ) x. ( c x. b ) ) )
49	47, 48	oveq12d 6668	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a x. ( ( sqr ` D ) x. d ) ) + ( c x. ( ( sqr ` D ) x. b ) ) ) = ( ( ( sqr ` D ) x. ( a x. d ) ) + ( ( sqr ` D ) x. ( c x. b ) ) ) )
50	26, 39	mulcld 10060	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( a x. d ) e. CC )
51	36, 32	mulcld 10060	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( c x. b ) e. CC )
52	29, 50, 51	adddid 10064	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) = ( ( ( sqr ` D ) x. ( a x. d ) ) + ( ( sqr ` D ) x. ( c x. b ) ) ) )
53	49, 52	eqtr4d 2659	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a x. ( ( sqr ` D ) x. d ) ) + ( c x. ( ( sqr ` D ) x. b ) ) ) = ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) )
54	46, 53	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a x. c ) + ( ( ( sqr ` D ) x. d ) x. ( ( sqr ` D ) x. b ) ) ) + ( ( a x. ( ( sqr ` D ) x. d ) ) + ( c x. ( ( sqr ` D ) x. b ) ) ) ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) )
55	23, 41, 54	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) )
56	50, 51	addcld 10059	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a x. d ) + ( c x. b ) ) e. CC )
57	29, 56	sqmuld 13020	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ^ 2 ) = ( ( ( sqr ` D ) ^ 2 ) x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) )
58	28	sqsqrtd 14178	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) ^ 2 ) = D )
59	58	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) ^ 2 ) x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) = ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) )
60	57, 59	eqtr2d 2657	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) = ( ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ^ 2 ) )
61	60	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) ) = ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ^ 2 ) ) )
62	26, 36	mulcld 10060	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( a x. c ) e. CC )
63	39, 32	mulcld 10060	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( d x. b ) e. CC )
64	28, 63	mulcld 10060	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( D x. ( d x. b ) ) e. CC )
65	62, 64	addcld 10059	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a x. c ) + ( D x. ( d x. b ) ) ) e. CC )
66	29, 56	mulcld 10060	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) e. CC )
67		subsq 12972	. . . . . . . . . . . . . 14 |- ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) e. CC /\ ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) e. CC ) -> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ^ 2 ) ) = ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) x. ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) - ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) ) )
68	65, 66, 67	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ^ 2 ) ) = ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) x. ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) - ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) ) )
69	41, 54	eqtr2d 2657	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( c + ( ( sqr ` D ) x. d ) ) ) )
70	26, 33, 36, 40	mulsubd 10490	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a - ( ( sqr ` D ) x. b ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) = ( ( ( a x. c ) + ( ( ( sqr ` D ) x. d ) x. ( ( sqr ` D ) x. b ) ) ) - ( ( a x. ( ( sqr ` D ) x. d ) ) + ( c x. ( ( sqr ` D ) x. b ) ) ) ) )
71	46, 53	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a x. c ) + ( ( ( sqr ` D ) x. d ) x. ( ( sqr ` D ) x. b ) ) ) - ( ( a x. ( ( sqr ` D ) x. d ) ) + ( c x. ( ( sqr ` D ) x. b ) ) ) ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) - ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) )
72	70, 71	eqtr2d 2657	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) - ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) = ( ( a - ( ( sqr ` D ) x. b ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) )
73	69, 72	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) x. ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) - ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) ) = ( ( ( a + ( ( sqr ` D ) x. b ) ) x. ( c + ( ( sqr ` D ) x. d ) ) ) x. ( ( a - ( ( sqr ` D ) x. b ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) ) )
74	61, 68, 73	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) ) = ( ( ( a + ( ( sqr ` D ) x. b ) ) x. ( c + ( ( sqr ` D ) x. d ) ) ) x. ( ( a - ( ( sqr ` D ) x. b ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) ) )
75	26, 33	addcld 10059	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( a + ( ( sqr ` D ) x. b ) ) e. CC )
76	36, 40	addcld 10059	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( c + ( ( sqr ` D ) x. d ) ) e. CC )
77	26, 33	subcld 10392	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( a - ( ( sqr ` D ) x. b ) ) e. CC )
78	36, 40	subcld 10392	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( c - ( ( sqr ` D ) x. d ) ) e. CC )
79	75, 76, 77, 78	mul4d 10248	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a + ( ( sqr ` D ) x. b ) ) x. ( c + ( ( sqr ` D ) x. d ) ) ) x. ( ( a - ( ( sqr ` D ) x. b ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) ) = ( ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) x. ( ( c + ( ( sqr ` D ) x. d ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) ) )
80		subsq 12972	. . . . . . . . . . . . . . 15 |- ( ( 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 ) ) ) )
81	26, 33, 80	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) = ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) )
82		subsq 12972	. . . . . . . . . . . . . . 15 |- ( ( c e. CC /\ ( ( sqr ` D ) x. d ) e. CC ) -> ( ( c ^ 2 ) - ( ( ( sqr ` D ) x. d ) ^ 2 ) ) = ( ( c + ( ( sqr ` D ) x. d ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) )
83	36, 40, 82	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( c ^ 2 ) - ( ( ( sqr ` D ) x. d ) ^ 2 ) ) = ( ( c + ( ( sqr ` D ) x. d ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) )
84	81, 83	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) x. ( ( c ^ 2 ) - ( ( ( sqr ` D ) x. d ) ^ 2 ) ) ) = ( ( ( a + ( ( sqr ` D ) x. b ) ) x. ( a - ( ( sqr ` D ) x. b ) ) ) x. ( ( c + ( ( sqr ` D ) x. d ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) ) )
85	29, 32	sqmuld 13020	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) x. b ) ^ 2 ) = ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) )
86	85	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) = ( ( a ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) ) )
87	29, 39	sqmuld 13020	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) x. d ) ^ 2 ) = ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) )
88	87	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( c ^ 2 ) - ( ( ( sqr ` D ) x. d ) ^ 2 ) ) = ( ( c ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) ) )
89	86, 88	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a ^ 2 ) - ( ( ( sqr ` D ) x. b ) ^ 2 ) ) x. ( ( c ^ 2 ) - ( ( ( sqr ` D ) x. d ) ^ 2 ) ) ) = ( ( ( a ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) ) x. ( ( c ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) ) ) )
90	79, 84, 89	3eqtr2d 2662	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a + ( ( sqr ` D ) x. b ) ) x. ( c + ( ( sqr ` D ) x. d ) ) ) x. ( ( a - ( ( sqr ` D ) x. b ) ) x. ( c - ( ( sqr ` D ) x. d ) ) ) ) = ( ( ( a ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) ) x. ( ( c ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) ) ) )
91	58	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) = ( D x. ( b ^ 2 ) ) )
92	91	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) ) = ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) )
93	58	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) = ( D x. ( d ^ 2 ) ) )
94	93	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( c ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) ) = ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) )
95	92, 94	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) ) x. ( ( c ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) ) ) = ( ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) x. ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) ) )
96		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( 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 )
97	96	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 )
98		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 )
99	97, 98	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) x. ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) ) = ( 1 x. 1 ) )
100		1t1e1 11175	. . . . . . . . . . . . . 14 |- ( 1 x. 1 ) = 1
101	100	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( 1 x. 1 ) = 1 )
102	95, 99, 101	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( a ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( b ^ 2 ) ) ) x. ( ( c ^ 2 ) - ( ( ( sqr ` D ) ^ 2 ) x. ( d ^ 2 ) ) ) ) = 1 )
103	74, 90, 102	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) ) = 1 )
104		oveq1 6657	. . . . . . . . . . . . . 14 |- ( e = ( ( a x. c ) + ( D x. ( d x. b ) ) ) -> ( e + ( ( sqr ` D ) x. f ) ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. f ) ) )
105	104	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( e = ( ( a x. c ) + ( D x. ( d x. b ) ) ) -> ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) <-> ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. f ) ) ) )
106		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( e = ( ( a x. c ) + ( D x. ( d x. b ) ) ) -> ( e ^ 2 ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) )
107	106	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( e = ( ( a x. c ) + ( D x. ( d x. b ) ) ) -> ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( f ^ 2 ) ) ) )
108	107	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( e = ( ( a x. c ) + ( D x. ( d x. b ) ) ) -> ( ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 <-> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) )
109	105, 108	anbi12d 747	. . . . . . . . . . . 12 |- ( e = ( ( a x. c ) + ( D x. ( d x. b ) ) ) -> ( ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) <-> ( ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. f ) ) /\ ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) )
110		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( ( sqr ` D ) x. f ) = ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) )
111	110	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. f ) ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) )
112	111	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. f ) ) <-> ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) ) )
113		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( f ^ 2 ) = ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) )
114	113	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( D x. ( f ^ 2 ) ) = ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) )
115	114	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) ) )
116	115	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 <-> ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) ) = 1 ) )
117	112, 116	anbi12d 747	. . . . . . . . . . . 12 |- ( f = ( ( a x. d ) + ( c x. b ) ) -> ( ( ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. f ) ) /\ ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) <-> ( ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) /\ ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) ) = 1 ) ) )
118	109, 117	rspc2ev 3324	. . . . . . . . . . 11 |- ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) e. ZZ /\ ( ( a x. d ) + ( c x. b ) ) e. ZZ /\ ( ( A x. B ) = ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) + ( ( sqr ` D ) x. ( ( a x. d ) + ( c x. b ) ) ) ) /\ ( ( ( ( a x. c ) + ( D x. ( d x. b ) ) ) ^ 2 ) - ( D x. ( ( ( a x. d ) + ( c x. b ) ) ^ 2 ) ) ) = 1 ) ) -> E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) )
119	16, 19, 55, 103, 118	syl112anc 1330	. . . . . . . . . 10 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) )
120	2, 119	jca 554	. . . . . . . . 9 |- ( ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) /\ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) )
121	120	ex 450	. . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) /\ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( c e. ZZ /\ d e. ZZ ) ) -> ( ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) -> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) ) )
122	121	rexlimdvva 3038	. . . . . . 7 |- ( ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( 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 ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) -> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) ) )
123	122	ex 450	. . . . . 6 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) /\ ( 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 ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) -> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) ) ) )
124	123	rexlimdvva 3038	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) -> ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) -> ( E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) -> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) ) ) )
125	124	impd 447	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ ( A e. RR /\ B e. RR ) ) -> ( ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) /\ E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) -> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) ) )
126	125	expimpd 629	. . 3 |- ( D e. ( NN \ ◻NN ) -> ( ( ( A e. RR /\ B e. RR ) /\ ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) /\ E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) -> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) ) )
127		elpell1234qr 37415	. . . . 5 |- ( 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 ) ) ) )
128		elpell1234qr 37415	. . . . 5 |- ( D e. ( NN \ ◻NN ) -> ( B e. (Pell1234QR ` D ) <-> ( B e. RR /\ E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
129	127, 128	anbi12d 747	. . . 4 |- ( D e. ( NN \ ◻NN ) -> ( ( A e. (Pell1234QR ` D ) /\ B 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 ) ) /\ ( B e. RR /\ E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) ) )
130		an4 865	. . . 4 |- ( ( ( A e. RR /\ E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) ) /\ ( B e. RR /\ E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) <-> ( ( A e. RR /\ B e. RR ) /\ ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) /\ E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) )
131	129, 130	syl6bb 276	. . 3 |- ( D e. ( NN \ ◻NN ) -> ( ( A e. (Pell1234QR ` D ) /\ B e. (Pell1234QR ` D ) ) <-> ( ( A e. RR /\ B e. RR ) /\ ( E. a e. ZZ E. b e. ZZ ( A = ( a + ( ( sqr ` D ) x. b ) ) /\ ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) /\ E. c e. ZZ E. d e. ZZ ( B = ( c + ( ( sqr ` D ) x. d ) ) /\ ( ( c ^ 2 ) - ( D x. ( d ^ 2 ) ) ) = 1 ) ) ) ) )
132		elpell1234qr 37415	. . 3 |- ( D e. ( NN \ ◻NN ) -> ( ( A x. B ) e. (Pell1234QR ` D ) <-> ( ( A x. B ) e. RR /\ E. e e. ZZ E. f e. ZZ ( ( A x. B ) = ( e + ( ( sqr ` D ) x. f ) ) /\ ( ( e ^ 2 ) - ( D x. ( f ^ 2 ) ) ) = 1 ) ) ) )
133	126, 131, 132	3imtr4d 283	. 2 |- ( D e. ( NN \ ◻NN ) -> ( ( A e. (Pell1234QR ` D ) /\ B e. (Pell1234QR ` D ) ) -> ( A x. B ) e. (Pell1234QR ` D ) ) )
134	133	3impib 1262	1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell1234QR ` D ) /\ B e. (Pell1234QR ` D ) ) -> ( A x. B ) e. (Pell1234QR ` D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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   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:  pell14qrmulcl  37427