Theorem pellexlem3 16099

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 9311	. . . 4 |- NN e. _V
2	1, 1	xpex 4891	. . 3 |- ( NN X. NN ) e. _V
3		opabssxp 4849	. . 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 4271	. 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 4134	. . . . . . . 8 |- ( x = a -> ( 0 < x <-> 0 < a ) )
6		fvoveq1 6108	. . . . . . . . 9 |- ( x = a -> ( abs ` ( x - ( sqr ` D ) ) ) = ( abs ` ( a - ( sqr ` D ) ) ) )
7		fveq2 5695	. . . . . . . . . 10 |- ( x = a -> (denom ` x ) = (denom ` a ) )
8	7	oveq1d 6100	. . . . . . . . 9 |- ( x = a -> ( (denom ` x ) ^ -u 2 ) = ( (denom ` a ) ^ -u 2 ) )
9	6, 8	breq12d 4143	. . . . . . . 8 |- ( x = a -> ( ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) <-> ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) )
10	5, 9	anbi12d 477	. . . . . . 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 2982	. . . . . 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 535	. . . . . . . . 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 545	. . . . . . . . 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 12979	. . . . . . . . 9 |- ( ( a e. QQ /\ 0 < a ) -> (numer ` a ) e. NN )
15	12, 13, 14	syl2anc 415	. . . . . . . 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 12969	. . . . . . . . 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 531	. . . . . . . 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 533	. . . . . . . 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 16097	. . . . . . . 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 1281	. . . . . . 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 546	. . . . . . . . 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 12974	. . . . . . . . . . . . 13 |- ( a e. QQ -> a = ( (numer ` a ) / (denom ` a ) ) )
25	24	oveq1d 6100	. . . . . . . . . . . 12 |- ( a e. QQ -> ( a - ( sqr ` D ) ) = ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) )
26	25	fveq2d 5699	. . . . . . . . . . 11 |- ( a e. QQ -> ( abs ` ( a - ( sqr ` D ) ) ) = ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) )
27	26	breq1d 4140	. . . . . . . . . 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 16098	. . . . . . . 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 1281	. . . . . . 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 12968	. . . . . . . 8 |- ( a e. QQ -> (numer ` a ) e. ZZ )
35		eleq1 2301	. . . . . . . . . . 11 |- ( y = (numer ` a ) -> ( y e. NN <-> (numer ` a ) e. NN ) )
36	35	anbi1d 469	. . . . . . . . . 10 |- ( y = (numer ` a ) -> ( ( y e. NN /\ z e. NN ) <-> ( (numer ` a ) e. NN /\ z e. NN ) ) )
37		oveq1 6092	. . . . . . . . . . . . 13 |- ( y = (numer ` a ) -> ( y ^ 2 ) = ( (numer ` a ) ^ 2 ) )
38	37	oveq1d 6100	. . . . . . . . . . . 12 |- ( y = (numer ` a ) -> ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) )
39	38	neeq1d 2438	. . . . . . . . . . 11 |- ( y = (numer ` a ) -> ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 ) )
40	38	fveq2d 5699	. . . . . . . . . . . 12 |- ( y = (numer ` a ) -> ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) )
41	40	breq1d 4140	. . . . . . . . . . 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 477	. . . . . . . . . 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 477	. . . . . . . . 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 2301	. . . . . . . . . . 11 |- ( z = (denom ` a ) -> ( z e. NN <-> (denom ` a ) e. NN ) )
45	44	anbi2d 468	. . . . . . . . . 10 |- ( z = (denom ` a ) -> ( ( (numer ` a ) e. NN /\ z e. NN ) <-> ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) ) )
46		oveq1 6092	. . . . . . . . . . . . . 14 |- ( z = (denom ` a ) -> ( z ^ 2 ) = ( (denom ` a ) ^ 2 ) )
47	46	oveq2d 6101	. . . . . . . . . . . . 13 |- ( z = (denom ` a ) -> ( D x. ( z ^ 2 ) ) = ( D x. ( (denom ` a ) ^ 2 ) ) )
48	47	oveq2d 6101	. . . . . . . . . . . 12 |- ( z = (denom ` a ) -> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) )
49	48	neeq1d 2438	. . . . . . . . . . 11 |- ( z = (denom ` a ) -> ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 ) )
50	48	fveq2d 5699	. . . . . . . . . . . 12 |- ( z = (denom ` a ) -> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) )
51	50	breq1d 4140	. . . . . . . . . . 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 477	. . . . . . . . . 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 477	. . . . . . . . 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 4410	. . . . . . . 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 415	. . . . . . 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 3333	. . . . . 6 |- { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } C_ QQ
61		simprl 535	. . . . . 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 3246	. . . . 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 537	. . . . . 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 3246	. . . . 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 4378	. . . . . . . . 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 415	. . . . . . . 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 535	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> (numer ` a ) = (numer ` b ) )
69		simprr 537	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> (denom ` a ) = (denom ` b ) )
70	68, 69	oveq12d 6103	. . . . . . . . 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 531	. . . . . . . . . 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 533	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> b e. QQ )
74		qeqnumdivden 12974	. . . . . . . . . 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 2281	. . . . . . . 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 5695	. . . . . . 7 |- ( a = b -> (numer ` a ) = (numer ` b ) )
80		fveq2 5695	. . . . . . 7 |- ( a = b -> (denom ` a ) = (denom ` b ) )
81	79, 80	opeq12d 3912	. . . . . 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 415	. . . 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 7059	. 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 ) ) ) ) ) } )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   <.cop 3712   class class class wbr 4130   {copab 4191   X. cxp 4772   ` cfv 5377  (class class class)co 6085   ~<_ cdom 7021   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   -ucneg 8498   / cdiv 9003   NNcn 9305   2c2 9356   ZZcz 9646   QQcq 10021   ^cexp 10977   sqrcsqrt 11764   abscabs 11765  numercnumer 12961  denomcdenom 12962

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-dom 7024  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-fl 10707  df-mod 10762  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-dvds 12557  df-gcd 12733  df-numer 12963  df-denom 12964

This theorem is used by: (None)