Theorem pellexlem3 15959

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 9260	. . . 4 |- NN e. _V
2	1, 1	xpex 4871	. . 3 |- ( NN X. NN ) e. _V
3		opabssxp 4829	. . 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 4253	. 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 4118	. . . . . . . 8 |- ( x = a -> ( 0 < x <-> 0 < a ) )
6		fvoveq1 6081	. . . . . . . . 9 |- ( x = a -> ( abs ` ( x - ( sqr ` D ) ) ) = ( abs ` ( a - ( sqr ` D ) ) ) )
7		fveq2 5675	. . . . . . . . . 10 |- ( x = a -> (denom ` x ) = (denom ` a ) )
8	7	oveq1d 6073	. . . . . . . . 9 |- ( x = a -> ( (denom ` x ) ^ -u 2 ) = ( (denom ` a ) ^ -u 2 ) )
9	6, 8	breq12d 4127	. . . . . . . 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 2976	. . . . . 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 12921	. . . . . . . . 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 12911	. . . . . . . . 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 15957	. . . . . . . 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 12916	. . . . . . . . . . . . 13 |- ( a e. QQ -> a = ( (numer ` a ) / (denom ` a ) ) )
25	24	oveq1d 6073	. . . . . . . . . . . 12 |- ( a e. QQ -> ( a - ( sqr ` D ) ) = ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) )
26	25	fveq2d 5679	. . . . . . . . . . 11 |- ( a e. QQ -> ( abs ` ( a - ( sqr ` D ) ) ) = ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) )
27	26	breq1d 4124	. . . . . . . . . 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 15958	. . . . . . . 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 12910	. . . . . . . 8 |- ( a e. QQ -> (numer ` a ) e. ZZ )
35		eleq1 2297	. . . . . . . . . . 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 6065	. . . . . . . . . . . . 13 |- ( y = (numer ` a ) -> ( y ^ 2 ) = ( (numer ` a ) ^ 2 ) )
38	37	oveq1d 6073	. . . . . . . . . . . 12 |- ( y = (numer ` a ) -> ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) )
39	38	neeq1d 2432	. . . . . . . . . . 11 |- ( y = (numer ` a ) -> ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 ) )
40	38	fveq2d 5679	. . . . . . . . . . . 12 |- ( y = (numer ` a ) -> ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) )
41	40	breq1d 4124	. . . . . . . . . . 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 2297	. . . . . . . . . . 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 6065	. . . . . . . . . . . . . 14 |- ( z = (denom ` a ) -> ( z ^ 2 ) = ( (denom ` a ) ^ 2 ) )
47	46	oveq2d 6074	. . . . . . . . . . . . 13 |- ( z = (denom ` a ) -> ( D x. ( z ^ 2 ) ) = ( D x. ( (denom ` a ) ^ 2 ) ) )
48	47	oveq2d 6074	. . . . . . . . . . . 12 |- ( z = (denom ` a ) -> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) )
49	48	neeq1d 2432	. . . . . . . . . . 11 |- ( z = (denom ` a ) -> ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 ) )
50	48	fveq2d 5679	. . . . . . . . . . . 12 |- ( z = (denom ` a ) -> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) )
51	50	breq1d 4124	. . . . . . . . . . 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 4391	. . . . . . . 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 3327	. . . . . 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 3240	. . . . 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 3240	. . . . 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 4359	. . . . . . . . 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 6076	. . . . . . . . 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 12916	. . . . . . . . . 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 2277	. . . . . . . 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 5675	. . . . . . 7 |- ( a = b -> (numer ` a ) = (numer ` b ) )
80		fveq2 5675	. . . . . . 7 |- ( a = b -> (denom ` a ) = (denom ` b ) )
81	79, 80	opeq12d 3896	. . . . . 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 7025	. 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 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   <.cop 3697   class class class wbr 4114   {copab 4175   X. cxp 4752   ` cfv 5357  (class class class)co 6058   ~<_ cdom 6987   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8460   -ucneg 8461   / cdiv 8963   NNcn 9254   2c2 9305   ZZcz 9594   QQcq 9969   ^cexp 10924   sqrcsqrt 11706   abscabs 11707  numercnumer 12903  denomcdenom 12904

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-dom 6990  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-numer 12905  df-denom 12906

This theorem is referenced by: (None)