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Theorem pellexlem1 26246
Description: Lemma for pellex 26252. 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 9687 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
213ad2ant2 982 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  CC )
32sqcld 11174 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A ^ 2 )  e.  CC )
4 nncn 9687 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  CC )
543ad2ant1 981 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  D  e.  CC )
6 nncn 9687 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
763ad2ant3 983 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  CC )
87sqcld 11174 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  e.  CC )
95, 8mulcld 8788 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( D  x.  ( B ^ 2 ) )  e.  CC )
10 subeq0 9006 . . . . 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 645 . . . 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 9711 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
13123ad2ant3 983 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  =/=  0 )
14 sqne0 11101 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( B ^ 2 )  =/=  0  <->  B  =/=  0 ) )
157, 14syl 17 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( B ^ 2 )  =/=  0  <->  B  =/=  0 ) )
1613, 15mpbird 225 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  =/=  0 )
173, 5, 8, 16divmul3d 9503 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
18 sqdiv 11100 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
) ^ 2 )  =  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )
1918fveq2d 5427 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) ) )
202, 7, 13, 19syl3anc 1187 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) ) )
21 nnre 9686 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  RR )
22213ad2ant2 982 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  RR )
23 nnre 9686 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  RR )
24233ad2ant3 983 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  RR )
2522, 24, 13redivcld 9521 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  RR )
26 nnnn0 9904 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  NN0 )
2726nn0ge0d 9953 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  0  <_  A )
28273ad2ant2 982 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  A )
29 nngt0 9708 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
30293ad2ant3 983 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <  B )
31 divge0 9558 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
3222, 28, 24, 30, 31syl22anc 1188 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  ( A  /  B
) )
3325, 32sqrsqd 11832 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( A  /  B
) )
3420, 33eqtr3d 2290 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( A  /  B
) )
35 nnq 10261 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  QQ )
36353ad2ant2 982 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  QQ )
37 nnq 10261 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  QQ )
38373ad2ant3 983 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  QQ )
39 qdivcl 10269 . . . . . . . 8  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
4036, 38, 13, 39syl3anc 1187 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
4134, 40eqeltrd 2330 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  e.  QQ )
42 fveq2 5423 . . . . . . 7  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( sqr `  D
) )
4342eleq1d 2322 . . . . . 6  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  (
( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) )  e.  QQ  <->  ( sqr `  D )  e.  QQ ) )
4441, 43syl5ibcom 213 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  D )  e.  QQ ) )
4517, 44sylbird 228 . . . 4  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( A ^ 2 )  =  ( D  x.  ( B ^
2 ) )  -> 
( sqr `  D
)  e.  QQ ) )
4611, 45sylbid 208 . . 3  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  0  -> 
( sqr `  D
)  e.  QQ ) )
4746necon3bd 2456 . 2  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( -.  ( sqr `  D
)  e.  QQ  ->  ( ( A ^ 2 )  -  ( D  x.  ( B ^
2 ) ) )  =/=  0 ) )
4847imp 420 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670    x. cmul 8675    < clt 8800    <_ cle 8801    - cmin 8970    / cdiv 9356   NNcn 9679   2c2 9728   QQcq 10248   ^cexp 11035   sqrcsqr 11648
This theorem is referenced by:  pellexlem3  26248
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-q 10249  df-rp 10287  df-seq 10978  df-exp 11036  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650
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