Theorem pellexlem1 16074

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 9295	. . . . . . 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 11092	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A ^ 2 ) e. CC )
4		nncn 9295	. . . . . . 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 9295	. . . . . . . 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 11092	. . . . . 6 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( B ^ 2 ) e. CC )
9	5, 8	mulcld 8340	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( D x. ( B ^ 2 ) ) e. CC )
10	3, 9	subeq0ad 8641	. . . 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 9316	. . . . . . . 8 |- ( B e. NN -> B # 0 )
12	11	3ad2ant3 1051	. . . . . . 7 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B # 0 )
13		sqap0 11026	. . . . . . . 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 9149	. . . . 5 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) = D <-> ( A ^ 2 ) = ( D x. ( B ^ 2 ) ) ) )
17		sqdivap 11023	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
18	17	fveq2d 5697	. . . . . . . . 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 9294	. . . . . . . . . . 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 9294	. . . . . . . . . . 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 9159	. . . . . . . . 9 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> ( A / B ) e. RR )
25		nnnn0 9553	. . . . . . . . . . . 12 |- ( A e. NN -> A e. NN0 )
26	25	nn0ge0d 9606	. . . . . . . . . . 11 |- ( A e. NN -> 0 <_ A )
27	26	3ad2ant2 1050	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 <_ A )
28		nngt0 9312	. . . . . . . . . . 11 |- ( B e. NN -> 0 < B )
29	28	3ad2ant3 1051	. . . . . . . . . 10 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> 0 < B )
30		divge0 9197	. . . . . . . . . 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 11914	. . . . . . . 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 10016	. . . . . . . . 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 10016	. . . . . . . . 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 9315	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
39	38	3ad2ant3 1051	. . . . . . . 8 |- ( ( D e. NN /\ A e. NN /\ B e. NN ) -> B =/= 0 )
40		qdivcl 10026	. . . . . . . 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 5693	. . . . . . 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 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   RRcr 8172   0cc0 8173   x. cmul 8178   < clt 8354   <_ cle 8355   - cmin 8491   # cap 8903   / cdiv 8996   NNcn 9287   2c2 9338   QQcq 10002   ^cexp 10958   sqrcsqrt 11745

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-seqfrec 10868  df-exp 10959  df-rsqrt 11747

This theorem is referenced by:  pellexlem3  16076