Theorem pellexlem3 37395

Description: Lemma for pellex 37399. 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 11026	. . . 4 |- NN e. _V
2	1, 1	xpex 6962	. . 3 |- ( NN X. NN ) e. _V
3		opabssxp 5193	. . 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 4803	. 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		simprl 794	. . . . . . . 8 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> a e. QQ )
6		simprrl 804	. . . . . . . 8 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> 0 < a )
7		qgt0numnn 15459	. . . . . . . 8 |- ( ( a e. QQ /\ 0 < a ) -> (numer ` a ) e. NN )
8	5, 6, 7	syl2anc 693	. . . . . . 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 )
9		qdencl 15449	. . . . . . . 8 |- ( a e. QQ -> (denom ` a ) e. NN )
10	5, 9	syl 17	. . . . . . 7 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> (denom ` a ) e. NN )
11	8, 10	jca 554	. . . . . 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 ) )
12		simpll 790	. . . . . . 7 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> D e. NN )
13		simplr 792	. . . . . . 7 |- ( ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) /\ ( a e. QQ /\ ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) ) -> -. ( sqr ` D ) e. QQ )
14		pellexlem1 37393	. . . . . . 7 |- ( ( ( D e. NN /\ (numer ` a ) e. NN /\ (denom ` a ) e. NN ) /\ -. ( sqr ` D ) e. QQ ) -> ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 )
15	12, 8, 10, 13, 14	syl31anc 1329	. . . . . 6 |- ( ( ( 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 )
16		simprrr 805	. . . . . . . 8 |- ( ( ( 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 ) )
17		qeqnumdivden 15454	. . . . . . . . . . . 12 |- ( a e. QQ -> a = ( (numer ` a ) / (denom ` a ) ) )
18	17	oveq1d 6665	. . . . . . . . . . 11 |- ( a e. QQ -> ( a - ( sqr ` D ) ) = ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) )
19	18	fveq2d 6195	. . . . . . . . . 10 |- ( a e. QQ -> ( abs ` ( a - ( sqr ` D ) ) ) = ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) )
20	19	breq1d 4663	. . . . . . . . 9 |- ( a e. QQ -> ( ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) <-> ( abs ` ( ( (numer ` a ) / (denom ` a ) ) - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) )
21	5, 20	syl 17	. . . . . . . 8 |- ( ( ( 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 ) ) )
22	16, 21	mpbid 222	. . . . . . 7 |- ( ( ( 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 ) )
23		pellexlem2 37394	. . . . . . 7 |- ( ( ( 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 ) ) ) )
24	12, 8, 10, 22, 23	syl31anc 1329	. . . . . 6 |- ( ( ( 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 ) ) ) )
25	11, 15, 24	jca32 558	. . . . 5 |- ( ( ( 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 ) ) ) ) ) )
26	25	ex 450	. . . 4 |- ( ( 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 ) ) ) ) ) ) )
27		breq2 4657	. . . . . 6 |- ( x = a -> ( 0 < x <-> 0 < a ) )
28		oveq1 6657	. . . . . . . 8 |- ( x = a -> ( x - ( sqr ` D ) ) = ( a - ( sqr ` D ) ) )
29	28	fveq2d 6195	. . . . . . 7 |- ( x = a -> ( abs ` ( x - ( sqr ` D ) ) ) = ( abs ` ( a - ( sqr ` D ) ) ) )
30		fveq2 6191	. . . . . . . 8 |- ( x = a -> (denom ` x ) = (denom ` a ) )
31	30	oveq1d 6665	. . . . . . 7 |- ( x = a -> ( (denom ` x ) ^ -u 2 ) = ( (denom ` a ) ^ -u 2 ) )
32	29, 31	breq12d 4666	. . . . . 6 |- ( x = a -> ( ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) <-> ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) )
33	27, 32	anbi12d 747	. . . . 5 |- ( x = a -> ( ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) <-> ( 0 < a /\ ( abs ` ( a - ( sqr ` D ) ) ) < ( (denom ` a ) ^ -u 2 ) ) ) )
34	33	elrab 3363	. . . 4 |- ( 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 ) ) ) )
35		fvex 6201	. . . . 5 |- (numer ` a ) e. _V
36		fvex 6201	. . . . 5 |- (denom ` a ) e. _V
37		eleq1 2689	. . . . . . 7 |- ( y = (numer ` a ) -> ( y e. NN <-> (numer ` a ) e. NN ) )
38	37	anbi1d 741	. . . . . 6 |- ( y = (numer ` a ) -> ( ( y e. NN /\ z e. NN ) <-> ( (numer ` a ) e. NN /\ z e. NN ) ) )
39		oveq1 6657	. . . . . . . . 9 |- ( y = (numer ` a ) -> ( y ^ 2 ) = ( (numer ` a ) ^ 2 ) )
40	39	oveq1d 6665	. . . . . . . 8 |- ( y = (numer ` a ) -> ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) )
41	40	neeq1d 2853	. . . . . . 7 |- ( y = (numer ` a ) -> ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 ) )
42	40	fveq2d 6195	. . . . . . . 8 |- ( y = (numer ` a ) -> ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) )
43	42	breq1d 4663	. . . . . . 7 |- ( 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 ) ) ) ) )
44	41, 43	anbi12d 747	. . . . . 6 |- ( 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 ) ) ) ) ) )
45	38, 44	anbi12d 747	. . . . 5 |- ( 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 ) ) ) ) ) ) )
46		eleq1 2689	. . . . . . 7 |- ( z = (denom ` a ) -> ( z e. NN <-> (denom ` a ) e. NN ) )
47	46	anbi2d 740	. . . . . 6 |- ( z = (denom ` a ) -> ( ( (numer ` a ) e. NN /\ z e. NN ) <-> ( (numer ` a ) e. NN /\ (denom ` a ) e. NN ) ) )
48		oveq1 6657	. . . . . . . . . 10 |- ( z = (denom ` a ) -> ( z ^ 2 ) = ( (denom ` a ) ^ 2 ) )
49	48	oveq2d 6666	. . . . . . . . 9 |- ( z = (denom ` a ) -> ( D x. ( z ^ 2 ) ) = ( D x. ( (denom ` a ) ^ 2 ) ) )
50	49	oveq2d 6666	. . . . . . . 8 |- ( z = (denom ` a ) -> ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) = ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) )
51	50	neeq1d 2853	. . . . . . 7 |- ( z = (denom ` a ) -> ( ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 <-> ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) =/= 0 ) )
52	50	fveq2d 6195	. . . . . . . 8 |- ( z = (denom ` a ) -> ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) = ( abs ` ( ( (numer ` a ) ^ 2 ) - ( D x. ( (denom ` a ) ^ 2 ) ) ) ) )
53	52	breq1d 4663	. . . . . . 7 |- ( 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 ) ) ) ) )
54	51, 53	anbi12d 747	. . . . . 6 |- ( 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 ) ) ) ) ) )
55	47, 54	anbi12d 747	. . . . 5 |- ( 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 ) ) ) ) ) ) )
56	35, 36, 45, 55	opelopab 4997	. . . 4 |- ( <. (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	26, 34, 56	3imtr4g 285	. . 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 ) ) ) ) ) } ) )
58		ssrab2 3687	. . . . . 6 |- { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } C_ QQ
59		simprl 794	. . . . . 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 ) ) } )
60	58, 59	sseldi 3601	. . . . 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 )
61		simprr 796	. . . . . 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 ) ) } )
62	58, 61	sseldi 3601	. . . . 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 )
63	35, 36	opth 4945	. . . . . . 7 |- ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) )
64		simprl 794	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> (numer ` a ) = (numer ` b ) )
65		simprr 796	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> (denom ` a ) = (denom ` b ) )
66	64, 65	oveq12d 6668	. . . . . . . . 9 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> ( (numer ` a ) / (denom ` a ) ) = ( (numer ` b ) / (denom ` b ) ) )
67		simpll 790	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> a e. QQ )
68	67, 17	syl 17	. . . . . . . . 9 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> a = ( (numer ` a ) / (denom ` a ) ) )
69		simplr 792	. . . . . . . . . 10 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> b e. QQ )
70		qeqnumdivden 15454	. . . . . . . . . 10 |- ( b e. QQ -> b = ( (numer ` b ) / (denom ` b ) ) )
71	69, 70	syl 17	. . . . . . . . 9 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> b = ( (numer ` b ) / (denom ` b ) ) )
72	66, 68, 71	3eqtr4d 2666	. . . . . . . 8 |- ( ( ( a e. QQ /\ b e. QQ ) /\ ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) ) -> a = b )
73	72	ex 450	. . . . . . 7 |- ( ( a e. QQ /\ b e. QQ ) -> ( ( (numer ` a ) = (numer ` b ) /\ (denom ` a ) = (denom ` b ) ) -> a = b ) )
74	63, 73	syl5bi 232	. . . . . 6 |- ( ( a e. QQ /\ b e. QQ ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. -> a = b ) )
75		fveq2 6191	. . . . . . 7 |- ( a = b -> (numer ` a ) = (numer ` b ) )
76		fveq2 6191	. . . . . . 7 |- ( a = b -> (denom ` a ) = (denom ` b ) )
77	75, 76	opeq12d 4410	. . . . . 6 |- ( a = b -> <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. )
78	74, 77	impbid1 215	. . . . 5 |- ( ( a e. QQ /\ b e. QQ ) -> ( <. (numer ` a ) , (denom ` a ) >. = <. (numer ` b ) , (denom ` b ) >. <-> a = b ) )
79	60, 62, 78	syl2anc 693	. . . 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 ) )
80	79	ex 450	. . 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 ) ) )
81	57, 80	dom2d 7996	. 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 ) ) ) ) ) } ) )
82	4, 81	mpi 20	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ~<_ cdom 7953   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   QQcq 11788   ^cexp 12860   sqrcsqrt 13973   abscabs 13974  numercnumer 15441  denomcdenom 15442

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-rep 4771  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-1st 7168  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-inf 8349  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-q 11789  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444

This theorem is referenced by:  pellexlem4  37396