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Theorem pellexlem1 26080
Description: Lemma for pellex 26086. 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 9634 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
213ad2ant2 982 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  CC )
32sqcld 11121 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A ^ 2 )  e.  CC )
4 nncn 9634 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  CC )
543ad2ant1 981 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  D  e.  CC )
6 nncn 9634 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
763ad2ant3 983 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  CC )
87sqcld 11121 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  e.  CC )
95, 8mulcld 8735 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( D  x.  ( B ^ 2 ) )  e.  CC )
10 subeq0 8953 . . . . 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 9658 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
13123ad2ant3 983 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  =/=  0 )
14 sqne0 11048 . . . . . . . 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 9450 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
18 sqdiv 11047 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
) ^ 2 )  =  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )
1918fveq2d 5381 . . . . . . . . 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 9633 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  RR )
22213ad2ant2 982 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  RR )
23 nnre 9633 . . . . . . . . . . 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 9468 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  RR )
26 nnnn0 9851 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  NN0 )
2726nn0ge0d 9900 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  0  <_  A )
28273ad2ant2 982 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  A )
29 nngt0 9655 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
30293ad2ant3 983 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <  B )
31 divge0 9505 . . . . . . . . . 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 11779 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( A  /  B
) )
3420, 33eqtr3d 2287 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( A  /  B
) )
35 nnq 10208 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  QQ )
36353ad2ant2 982 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  QQ )
37 nnq 10208 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  QQ )
38373ad2ant3 983 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  QQ )
39 qdivcl 10216 . . . . . . . 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 2327 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  e.  QQ )
42 fveq2 5377 . . . . . . 7  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( sqr `  D
) )
4342eleq1d 2319 . . . . . 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 2449 . 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 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617    x. cmul 8622    < clt 8747    <_ cle 8748    - cmin 8917    / cdiv 9303   NNcn 9626   2c2 9675   QQcq 10195   ^cexp 10982   sqrcsqr 11595
This theorem is referenced by:  pellexlem3  26082
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-q 10196  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597
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