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Theorem pellexlem1 26925
Description: Lemma for pellex 26931. 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
StepHypRef Expression
1 nncn 9756 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
213ad2ant2 977 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  CC )
32sqcld 11245 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A ^ 2 )  e.  CC )
4 nncn 9756 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  CC )
543ad2ant1 976 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  D  e.  CC )
6 nncn 9756 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
763ad2ant3 978 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  CC )
87sqcld 11245 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  e.  CC )
95, 8mulcld 8857 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( D  x.  ( B ^ 2 ) )  e.  CC )
10 subeq0 9075 . . . . 5  |-  ( ( ( A ^ 2 )  e.  CC  /\  ( D  x.  ( B ^ 2 ) )  e.  CC )  -> 
( ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  0  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
113, 9, 10syl2anc 642 . . . 4  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  0  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
12 nnne0 9780 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
13123ad2ant3 978 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  =/=  0 )
14 sqne0 11172 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( B ^ 2 )  =/=  0  <->  B  =/=  0 ) )
157, 14syl 15 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( B ^ 2 )  =/=  0  <->  B  =/=  0 ) )
1613, 15mpbird 223 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  =/=  0 )
173, 5, 8, 16divmul3d 9572 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
18 sqdiv 11171 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
) ^ 2 )  =  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )
1918fveq2d 5531 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) ) )
202, 7, 13, 19syl3anc 1182 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) ) )
21 nnre 9755 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  RR )
22213ad2ant2 977 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  RR )
23 nnre 9755 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  RR )
24233ad2ant3 978 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  RR )
2522, 24, 13redivcld 9590 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  RR )
26 nnnn0 9974 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  NN0 )
2726nn0ge0d 10023 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  0  <_  A )
28273ad2ant2 977 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  A )
29 nngt0 9777 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
30293ad2ant3 978 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <  B )
31 divge0 9627 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
3222, 28, 24, 30, 31syl22anc 1183 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  ( A  /  B
) )
3325, 32sqrsqd 11904 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( A  /  B
) )
3420, 33eqtr3d 2319 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( A  /  B
) )
35 nnq 10331 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  QQ )
36353ad2ant2 977 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  QQ )
37 nnq 10331 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  QQ )
38373ad2ant3 978 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  QQ )
39 qdivcl 10339 . . . . . . . 8  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
4036, 38, 13, 39syl3anc 1182 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
4134, 40eqeltrd 2359 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  e.  QQ )
42 fveq2 5527 . . . . . . 7  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( sqr `  D
) )
4342eleq1d 2351 . . . . . 6  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  (
( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) )  e.  QQ  <->  ( sqr `  D )  e.  QQ ) )
4441, 43syl5ibcom 211 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  D )  e.  QQ ) )
4517, 44sylbird 226 . . . 4  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( A ^ 2 )  =  ( D  x.  ( B ^
2 ) )  -> 
( sqr `  D
)  e.  QQ ) )
4611, 45sylbid 206 . . 3  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  0  -> 
( sqr `  D
)  e.  QQ ) )
4746necon3bd 2485 . 2  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( -.  ( sqr `  D
)  e.  QQ  ->  ( ( A ^ 2 )  -  ( D  x.  ( B ^
2 ) ) )  =/=  0 ) )
4847imp 418 1  |-  ( ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =/=  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739    x. cmul 8744    < clt 8869    <_ cle 8870    - cmin 9039    / cdiv 9425   NNcn 9748   2c2 9797   QQcq 10318   ^cexp 11106   sqrcsqr 11720
This theorem is referenced by:  pellexlem3  26927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722
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