Theorem pellexlem4 37396

Description: Lemma for pellex 37399. Invoking irrapx1 37392, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.)

Assertion

Ref	Expression
pellexlem4	|- ( ( 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 ) ) ) ) ) } ~~ NN )

Distinct variable group:   y, D, z

Proof of Theorem pellexlem4

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnex 11026	. . . . 5 |- NN e. _V
2	1, 1	xpex 6962	. . . 4 |- ( NN X. NN ) e. _V
3		opabssxp 5193	. . . 4 |- { <. 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		ssdomg 8001	. . . 4 |- ( ( NN X. NN ) e. _V -> ( { <. 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 ) -> { <. 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 ) ) ) ) ) } ~<_ ( NN X. NN ) ) )
5	2, 3, 4	mp2 9	. . 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 ) ) ) ) ) } ~<_ ( NN X. NN )
6		xpnnen 14939	. . 3 |- ( NN X. NN ) ~~ NN
7		domentr 8015	. . 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 ) ) ) ) ) } ~<_ ( NN X. NN ) /\ ( NN X. NN ) ~~ NN ) -> { <. 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 ) ) ) ) ) } ~<_ NN )
8	5, 6, 7	mp2an 708	. 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 ) ) ) ) ) } ~<_ NN
9		nnrp 11842	. . . . . . 7 |- ( D e. NN -> D e. RR+ )
10	9	rpsqrtcld 14150	. . . . . 6 |- ( D e. NN -> ( sqr ` D ) e. RR+ )
11	10	anim1i 592	. . . . 5 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> ( ( sqr ` D ) e. RR+ /\ -. ( sqr ` D ) e. QQ ) )
12		eldif 3584	. . . . 5 |- ( ( sqr ` D ) e. ( RR+ \ QQ ) <-> ( ( sqr ` D ) e. RR+ /\ -. ( sqr ` D ) e. QQ ) )
13	11, 12	sylibr 224	. . . 4 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> ( sqr ` D ) e. ( RR+ \ QQ ) )
14		irrapx1 37392	. . . 4 |- ( ( sqr ` D ) e. ( RR+ \ QQ ) -> { b e. QQ | ( 0 < b /\ ( abs ` ( b - ( sqr ` D ) ) ) < ( (denom ` b ) ^ -u 2 ) ) } ~~ NN )
15		ensym 8005	. . . 4 |- ( { b e. QQ | ( 0 < b /\ ( abs ` ( b - ( sqr ` D ) ) ) < ( (denom ` b ) ^ -u 2 ) ) } ~~ NN -> NN ~~ { b e. QQ | ( 0 < b /\ ( abs ` ( b - ( sqr ` D ) ) ) < ( (denom ` b ) ^ -u 2 ) ) } )
16	13, 14, 15	3syl 18	. . 3 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> NN ~~ { b e. QQ | ( 0 < b /\ ( abs ` ( b - ( sqr ` D ) ) ) < ( (denom ` b ) ^ -u 2 ) ) } )
17		pellexlem3 37395	. . 3 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> { b e. QQ | ( 0 < b /\ ( abs ` ( b - ( sqr ` D ) ) ) < ( (denom ` b ) ^ -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 ) ) ) ) ) } )
18		endomtr 8014	. . 3 |- ( ( NN ~~ { b e. QQ | ( 0 < b /\ ( abs ` ( b - ( sqr ` D ) ) ) < ( (denom ` b ) ^ -u 2 ) ) } /\ { b e. QQ | ( 0 < b /\ ( abs ` ( b - ( sqr ` D ) ) ) < ( (denom ` b ) ^ -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 ) ) ) ) ) } ) -> NN ~<_ { <. 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 ) ) ) ) ) } )
19	16, 17, 18	syl2anc 693	. 2 |- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> NN ~<_ { <. 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 ) ) ) ) ) } )
20		sbth 8080	. 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 ) ) ) ) ) } ~<_ NN /\ NN ~<_ { <. 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 ) ) ) ) ) } ) -> { <. 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 ) ) ) ) ) } ~~ NN )
21	8, 19, 20	sylancr 695	1 |- ( ( 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 ) ) ) ) ) } ~~ NN )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   ~<_ cdom 7953   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   QQcq 11788   RR+crp 11832   ^cexp 12860   sqrcsqrt 13973   abscabs 13974  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-inf2 8538  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-int 4476  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-se 5074  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-isom 5897  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-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  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-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ico 12181  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  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:  pellexlem5  37397