Theorem pellexlem1 15957

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 9262	. . . . . . 7 |- ( A e. NN -> A e. CC )
2	1	3ad2ant2 1046	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> A e. CC )
3	2	sqcld 11058	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A ^ 2 ) e. CC )
4		nncn 9262	. . . . . . 7 |- ( D e. NN -> D e. CC )
5	4	3ad2ant1 1045	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> D e. CC )
6		nncn 9262	. . . . . . . 8 |- ( B e. NN -> B e. CC )
7	6	3ad2ant3 1047	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B e. CC )
8	7	sqcld 11058	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( B ^ 2 ) e. CC )
9	5, 8	mulcld 8310	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( D x. ( B ^ 2 ) ) e. CC )
10	3, 9	subeq0ad 8610	. . . 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 9283	. . . . . . . 8 |- ( B e. NN -> B # 0 )
12	11	3ad2ant3 1047	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B # 0 )
13		sqap0 10992	. . . . . . . 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 9116	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) = D <-> ( A ^ 2 ) = ( D x. ( B ^ 2 ) ) ) )
17		sqdivap 10989	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
18	17	fveq2d 5679	. . . . . . . . 9 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( sqr ` ( ( A / B ) ^ 2 ) ) = ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) )
19	2, 7, 12, 18	syl3anc 1274	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A / B ) ^ 2 ) ) = ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) )
20		nnre 9261	. . . . . . . . . . 11 |- ( A e. NN -> A e. RR )
21	20	3ad2ant2 1046	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> A e. RR )
22		nnre 9261	. . . . . . . . . . 11 |- ( B e. NN -> B e. RR )
23	22	3ad2ant3 1047	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B e. RR )
24	21, 23, 12	redivclapd 9126	. . . . . . . . 9 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A / B ) e. RR )
25		nnnn0 9520	. . . . . . . . . . . 12 |- ( A e. NN -> A e. NN0 )
26	25	nn0ge0d 9573	. . . . . . . . . . 11 |- ( A e. NN -> 0 <_ A )
27	26	3ad2ant2 1046	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 <_ A )
28		nngt0 9279	. . . . . . . . . . 11 |- ( B e. NN -> 0 < B )
29	28	3ad2ant3 1047	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 < B )
30		divge0 9164	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
31	21, 27, 23, 29, 30	syl22anc 1275	. . . . . . . . 9 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 <_ ( A / B ) )
32	24, 31	sqrtsqd 11875	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A / B ) ^ 2 ) ) = ( A / B ) )
33	19, 32	eqtr3d 2269	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) = ( A / B ) )
34		nnq 9983	. . . . . . . . 9 |- ( A e. NN -> A e. QQ )
35	34	3ad2ant2 1046	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> A e. QQ )
36		nnq 9983	. . . . . . . . 9 |- ( B e. NN -> B e. QQ )
37	36	3ad2ant3 1047	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B e. QQ )
38		nnne0 9282	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
39	38	3ad2ant3 1047	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B =/= 0 )
40		qdivcl 9993	. . . . . . . 8 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
41	35, 37, 39, 40	syl3anc 1274	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A / B ) e. QQ )
42	33, 41	eqeltrd 2311	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) e. QQ )
43		fveq2 5675	. . . . . . 7 |- ( ( ( A ^ 2 ) / ( B ^ 2 ) ) = D -> ( sqr ` ( ( A ^ 2 ) / ( B ^ 2 ) ) ) = ( sqr ` D ) )
44	43	eleq1d 2303	. . . . . 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 2457	. 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 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   # cap 8872   / cdiv 8963   NNcn 9254   2c2 9305   QQcq 9969   ^cexp 10924   sqrcsqrt 11706

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

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-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-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-seqfrec 10834  df-exp 10925  df-rsqrt 11708

This theorem is referenced by:  pellexlem3  15959