Theorem pellexlem3 15839

Description: Lemma for pellex . To each good rational approximation of ( sqr ` D ), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)

Assertion

Ref	Expression
pellexlem3	|- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ~<_ { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } )

Distinct variable group:   x, D, y, z

Proof of Theorem pellexlem3

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

Step	Hyp	Ref	Expression
1		nnex 9242	. . . 4 |- NN e. _V
2	1, 1	xpex 4865	. . 3 |- ( NN X. NN ) e. _V
3		opabssxp 4823	. . 3 |- { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } C_ ( NN X. NN )
4	2, 3	ssexi 4247	. 2 |- { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } e. _V
5		breq2 4112	. . . . . . . 8 |- ( x = a -> ( 0 < x <-> 0 < a ) )
6		fvoveq1 6072	. . . . . . . . 9 |- ( x = a -> ( abs ` ( x - ( sqr ` D ) ) ) = ( abs ` ( a - ( sqr ` D ) ) ) )
7		fveq2 5669	. . . . . . . . . 10 |- ( x = a -> (denom ` x ) = (denom ` a ) )
8	7	oveq1d 6064	. . . . . . . . 9 |- ( x = a -> ( (denom ` x ) ^ -u 2 ) = ( (denom ` a ) ^ -u 2 ) )
9	6, 8	breq12d 4121	. . . . . . . 8 |- ( x = a -> ( ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) <-> ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) )
10	5, 9	anbi12d 473	. . . . . . 7 |- ( x = a -> ( ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) <-> ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) )
11	10	elrab 2972	. . . . . 6 |- ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } <-> ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) )
12		simprl 531	. . . . . . . . 9 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> a e. QQ )
13		simprrl 541	. . . . . . . . 9 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> 0 < a )
14		qgt0numnn 12892	. . . . . . . . 9 |- ( ( a e. QQ /\ 0 < a ) -> (numer ` a ) e. NN )
15	12, 13, 14	syl2anc 411	. . . . . . . 8 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> (numer ` a ) e. NN )
16		qdencl 12882	. . . . . . . . 9 |- ( a e. QQ -> (denom ` a ) e. NN )
17	12, 16	syl 14	. . . . . . . 8 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> (denom ` a ) e. NN )
18	15, 17	jca 306	. . . . . . 7 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) )
19		simpll 527	. . . . . . . 8 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> D e. NN )
20		simplr 529	. . . . . . . 8 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> -. ( sqr ` D ) e. QQ )
21		pellexlem1 15837	. . . . . . . 8 |- ( ( ( D e. NN /\ (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ -. ( sqr ` D ) e. QQ ) -> ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 )
22	19, 15, 17, 20, 21	syl31anc 1277	. . . . . . 7 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 )
23		simprrr 542	. . . . . . . . 9 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) )
24		qeqnumdivden 12887	. . . . . . . . . . . . 13 |- ( a e. QQ -> a = ( (numer ` a ) / (denom ` a ) ) )
25	24	oveq1d 6064	. . . . . . . . . . . 12 |- ( a e. QQ -> ( a - ( sqr ` D ) ) = ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) )
26	25	fveq2d 5673	. . . . . . . . . . 11 |- ( a e. QQ -> ( abs ` ( a - ( sqr ` D ) ) ) = ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) )
27	26	breq1d 4118	. . . . . . . . . 10 |- ( a e. QQ -> ( ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) <-> ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) )
28	12, 27	syl 14	. . . . . . . . 9 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) <-> ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) )
29	23, 28	mpbid 147	. . . . . . . 8 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) )
30		pellexlem2 15838	. . . . . . . 8 |- ( ( ( D e. NN /\ (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) -> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) )
31	19, 15, 17, 29, 30	syl31anc 1277	. . . . . . 7 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) )
32	18, 22, 31	jca32 310	. . . . . 6 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) )
33	11, 32	sylan2b 287	. . . . 5 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) -> ( ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) )
34		qnumcl 12881	. . . . . . . 8 |- ( a e. QQ -> (numer ` a ) e. ZZ )
35		eleq1 2295	. . . . . . . . . . 11 |- ( y = (numer ` a ) -> ( y e. NN <-> (numer ` a ) e. NN ) )
36	35	anbi1d 465	. . . . . . . . . 10 |- ( y = (numer ` a ) -> ( ( y e. NN /\ z e. NN ) <-> ( (numer ` a ) e. NN /\ z e. NN ) ) )
37		oveq1 6056	. . . . . . . . . . . . 13 |- ( y = (numer ` a ) -> ( y ^ 2 ) = ( (numer ` a ) ^ 2 ) )
38	37	oveq1d 6064	. . . . . . . . . . . 12 |- ( y = (numer ` a ) -> ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) )
39	38	neeq1d 2430	. . . . . . . . . . 11 |- ( y = (numer ` a ) -> ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 ) )
40	38	fveq2d 5673	. . . . . . . . . . . 12 |- ( y = (numer ` a ) -> ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) )
41	40	breq1d 4118	. . . . . . . . . . 11 |- ( y = (numer ` a ) -> ( ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) <-> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) )
42	39, 41	anbi12d 473	. . . . . . . . . 10 |- ( y = (numer ` a ) -> ( ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) <-> ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) )
43	36, 42	anbi12d 473	. . . . . . . . 9 |- ( y = (numer ` a ) -> ( ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) <-> ( ( (numer ` a ) e. NN /\ z e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) ) )
44		eleq1 2295	. . . . . . . . . . 11 |- ( z = (denom ` a ) -> ( z e. NN <-> (denom ` a ) e. NN ) )
45	44	anbi2d 464	. . . . . . . . . 10 |- ( z = (denom ` a ) -> ( ( (numer ` a ) e. NN /\ z e. NN ) <-> ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) ) )
46		oveq1 6056	. . . . . . . . . . . . . 14 |- ( z = (denom ` a ) -> ( z ^ 2 ) = ( (denom ` a ) ^ 2 ) )
47	46	oveq2d 6065	. . . . . . . . . . . . 13 |- ( z = (denom ` a ) -> ( D x. ( z ^ 2 ) ) = ( D x. ( (denom ` a ) ^ 2 ) ) )
48	47	oveq2d 6065	. . . . . . . . . . . 12 |- ( z = (denom ` a ) -> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) )
49	48	neeq1d 2430	. . . . . . . . . . 11 |- ( z = (denom ` a ) -> ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 ) )
50	48	fveq2d 5673	. . . . . . . . . . . 12 |- ( z = (denom ` a ) -> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) )
51	50	breq1d 4118	. . . . . . . . . . 11 |- ( z = (denom ` a ) -> ( ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) <-> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) )
52	49, 51	anbi12d 473	. . . . . . . . . 10 |- ( z = (denom ` a ) -> ( ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) <-> ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) )
53	45, 52	anbi12d 473	. . . . . . . . 9 |- ( z = (denom ` a ) -> ( ( ( (numer ` a ) e. NN /\ z e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) <-> ( ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) ) )
54	43, 53	opelopabg 4385	. . . . . . . 8 |- ( ( (numer ` a ) e. ZZ /\ (denom ` a ) e. NN ) -> ( <. (numer ` a ) , (denom ` a ) >. e. { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } <-> ( ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) ) )
55	34, 16, 54	syl2anc 411	. . . . . . 7 |- ( a e. QQ -> ( <. (numer ` a ) , (denom ` a ) >. e. { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } <-> ( ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) ) )
56	12, 55	syl 14	. . . . . 6 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> ( <. (numer ` a ) , (denom ` a ) >. e. { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } <-> ( ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) ) )
57	11, 56	sylan2b 287	. . . . 5 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) -> ( <. (numer ` a ) , (denom ` a ) >. e. { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } <-> ( ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) ) )
58	33, 57	mpbird 167	. . . 4 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) -> <. (numer ` a ) , (denom ` a ) >. e. { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } )
59	58	ex 115	. . 3 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } -> <. (numer ` a ) , (denom ` a ) >. e. { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } ) )
60		ssrab2 3322	. . . . . 6 |- { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } C_ QQ
61		simprl 531	. . . . . 6 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } /\ b e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) ) -> a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } )
62	60, 61	sselid 3235	. . . . 5 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } /\ b e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) ) -> a e. QQ )
63		simprr 533	. . . . . 6 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } /\ b e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) ) -> b e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } )
64	60, 63	sselid 3235	. . . . 5 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } /\ b e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) ) -> b e. QQ )
65		opthg 4353	. . . . . . . . 9 |- ( ( (numer ` a ) e. ZZ /\ (denom ` a ) e. NN ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) )
66	34, 16, 65	syl2anc 411	. . . . . . . 8 |- ( a e. QQ -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) )
67	66	adantr 276	. . . . . . 7 |- ( ( a e. QQ /\ b e. QQ ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) )
68		simprl 531	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> (numer ` a ) = (numer ` b ) )
69		simprr 533	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> (denom ` a ) = (denom ` b ) )
70	68, 69	oveq12d 6067	. . . . . . . . 9 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> ( (numer ` a ) / (denom ` a ) ) = ( (numer ` b ) / (denom ` b ) ) )
71		simpll 527	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> a e. QQ )
72	71, 24	syl 14	. . . . . . . . 9 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> a = ( (numer ` a ) / (denom ` a ) ) )
73		simplr 529	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> b e. QQ )
74		qeqnumdivden 12887	. . . . . . . . . 10 |- ( b e. QQ -> b = ( (numer ` b ) / (denom ` b ) ) )
75	73, 74	syl 14	. . . . . . . . 9 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> b = ( (numer ` b ) / (denom ` b ) ) )
76	70, 72, 75	3eqtr4d 2275	. . . . . . . 8 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> a = b )
77	76	ex 115	. . . . . . 7 |- ( ( a e. QQ /\ b e. QQ ) -> ( ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) -> a = b ) )
78	67, 77	sylbid 150	. . . . . 6 |- ( ( a e. QQ /\ b e. QQ ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. -> a = b ) )
79		fveq2 5669	. . . . . . 7 |- ( a = b -> (numer ` a ) = (numer ` b ) )
80		fveq2 5669	. . . . . . 7 |- ( a = b -> (denom ` a ) = (denom ` b ) )
81	79, 80	opeq12d 3890	. . . . . 6 |- ( a = b -> <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. )
82	78, 81	impbid1 142	. . . . 5 |- ( ( a e. QQ /\ b e. QQ ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> a = b ) )
83	62, 64, 82	syl2anc 411	. . . 4 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } /\ b e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> a = b ) )
84	83	ex 115	. . 3 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> ( ( a e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } /\ b e. { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> a = b ) ) )
85	59, 84	dom2d 7011	. 2 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> ( { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } e. _V -> { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ~<_ { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } ) )
86	4, 85	mpi 15	1 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ~<_ { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   {crab 2524   _Vcvv 2812   <.cop 3691   class class class wbr 4108   {copab 4169   X. cxp 4746   ` cfv 5351  (class class class)co 6049   ~<_ cdom 6973   0cc0 8126   1c1 8127   + caddc 8129   x. cmul 8131   < clt 8307   - cmin 8443   -ucneg 8444   / cdiv 8945   NNcn 9236   2c2 9287   ZZcz 9576   QQcq 9950   ^cexp 10899   sqrcsqrt 11677   abscabs 11678  numercnumer 12874  denomcdenom 12875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-dom 6976  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-gcd 12646  df-numer 12876  df-denom 12877

This theorem is referenced by: (None)