Theorem pellexlem1 16097

Description: Lemma for pellex . Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.)

Assertion

Ref	Expression
pellexlem1	|- ( ( ( D e. NN /\ A e. NN /\ B e. NN ) /\ -. ( sqr ` D ) e. QQ ) -> ( ( A ^ 2 ) - ( D x. ( B ^ 2 ) ) ) =/= 0 )

Proof of Theorem pellexlem1

Step	Hyp	Ref	Expression
1		nncn 9313	. . . . . . 7 |- ( A e. NN -> A e. CC )
2	1	3ad2ant2 1050	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> A e. CC )
3	2	sqcld 11111	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A ^ 2 ) e. CC )
4		nncn 9313	. . . . . . 7 |- ( D e. NN -> D e. CC )
5	4	3ad2ant1 1049	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> D e. CC )
6		nncn 9313	. . . . . . . 8 |- ( B e. NN -> B e. CC )
7	6	3ad2ant3 1051	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B e. CC )
8	7	sqcld 11111	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( B ^ 2 ) e. CC )
9	5, 8	mulcld 8346	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( D x. ( B ^ 2 ) ) e. CC )
10	3, 9	subeq0ad 8647	. . . 4 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( ( A ^ 2 ) - ( D x. ( B ^ 2 ) ) ) = 0 <-> ( A ^ 2 ) = ( D x. ( B ^ 2 ) ) ) )
11		nnap0 9334	. . . . . . . 8 |- ( B e. NN -> B # 0 )
12	11	3ad2ant3 1051	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B # 0 )
13		sqap0 11045	. . . . . . . 8 |- ( B e. CC -> ( ( B ^ 2 ) # 0 <-> B # 0 ) )
14	7, 13	syl 14	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( B ^ 2 ) # 0 <-> B # 0 ) )
15	12, 14	mpbird 167	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( B ^ 2 ) # 0 )
16	3, 5, 8, 15	divmulap3d 9156	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) = D <-> ( A ^ 2 ) = ( D x. ( B ^ 2 ) ) ) )
17		sqdivap 11042	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
18	17	fveq2d 5699	. . . . . . . . 9 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( sqr ` ( ( A / B ) ^ 2 ) ) = ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) )
19	2, 7, 12, 18	syl3anc 1278	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A / B ) ^ 2 ) ) = ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) )
20		nnre 9312	. . . . . . . . . . 11 |- ( A e. NN -> A e. RR )
21	20	3ad2ant2 1050	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> A e. RR )
22		nnre 9312	. . . . . . . . . . 11 |- ( B e. NN -> B e. RR )
23	22	3ad2ant3 1051	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B e. RR )
24	21, 23, 12	redivclapd 9166	. . . . . . . . 9 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A / B ) e. RR )
25		nnnn0 9572	. . . . . . . . . . . 12 |- ( A e. NN -> A e. NN0 )
26	25	nn0ge0d 9625	. . . . . . . . . . 11 |- ( A e. NN -> 0 <_ A )
27	26	3ad2ant2 1050	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 <_ A )
28		nngt0 9330	. . . . . . . . . . 11 |- ( B e. NN -> 0 < B )
29	28	3ad2ant3 1051	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 < B )
30		divge0 9204	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
31	21, 27, 23, 29, 30	syl22anc 1279	. . . . . . . . 9 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 <_ ( A / B ) )
32	24, 31	sqrtsqd 11933	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A / B ) ^ 2 ) ) = ( A / B ) )
33	19, 32	eqtr3d 2273	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) = ( A / B ) )
34		nnq 10035	. . . . . . . . 9 |- ( A e. NN -> A e. QQ )
35	34	3ad2ant2 1050	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> A e. QQ )
36		nnq 10035	. . . . . . . . 9 |- ( B e. NN -> B e. QQ )
37	36	3ad2ant3 1051	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B e. QQ )
38		nnne0 9333	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
39	38	3ad2ant3 1051	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B =/= 0 )
40		qdivcl 10045	. . . . . . . 8 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
41	35, 37, 39, 40	syl3anc 1278	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A / B ) e. QQ )
42	33, 41	eqeltrd 2315	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) e. QQ )
43		fveq2 5695	. . . . . . 7 |- ( ( ( A ^ 2 ) / ( B ^ 2 ) ) = D -> ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) = ( sqr ` D ) )
44	43	eleq1d 2307	. . . . . 6 |- ( ( ( A ^ 2 ) / ( B ^ 2 ) ) = D -> ( ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) e. QQ <-> ( sqr ` D ) e. QQ ) )
45	42, 44	syl5ibcom 155	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) = D -> ( sqr ` D ) e. QQ ) )
46	16, 45	sylbird 170	. . . 4 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( A ^ 2 ) = ( D x. ( B ^ 2 ) ) -> ( sqr ` D ) e. QQ ) )
47	10, 46	sylbid 150	. . 3 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( ( A ^ 2 ) - ( D x. ( B ^ 2 ) ) ) = 0 -> ( sqr ` D ) e. QQ ) )
48	47	necon3bd 2463	. 2 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( -. ( sqr ` D ) e. QQ -> ( ( A ^ 2 ) - ( D x. ( B ^ 2 ) ) ) =/= 0 ) )
49	48	imp 124	1 |- ( ( ( D e. NN /\ A e. NN /\ B e. NN ) /\ -. ( sqr ` D ) e. QQ ) -> ( ( A ^ 2 ) - ( D x. ( B ^ 2 ) ) ) =/= 0 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8497   # cap 8910   / cdiv 9003   NNcn 9305   2c2 9356   QQcq 10021   ^cexp 10977   sqrcsqrt 11764

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

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-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-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-seqfrec 10887  df-exp 10978  df-rsqrt 11766

This theorem is used by:  pellexlem3  16099