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Theorem pellexlem1 29193
Description: Lemma for pellex 29199. 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 10349 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
213ad2ant2 1010 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  CC )
32sqcld 12025 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A ^ 2 )  e.  CC )
4 nncn 10349 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  CC )
543ad2ant1 1009 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  D  e.  CC )
6 nncn 10349 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
763ad2ant3 1011 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  CC )
87sqcld 12025 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  e.  CC )
95, 8mulcld 9425 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( D  x.  ( B ^ 2 ) )  e.  CC )
103, 9subeq0ad 9748 . . . 4  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  0  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
11 nnne0 10373 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
12113ad2ant3 1011 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  =/=  0 )
13 sqne0 11951 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( B ^ 2 )  =/=  0  <->  B  =/=  0 ) )
147, 13syl 16 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( B ^ 2 )  =/=  0  <->  B  =/=  0 ) )
1512, 14mpbird 232 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  =/=  0 )
163, 5, 8, 15divmul3d 10160 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
17 sqdiv 11950 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
) ^ 2 )  =  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )
1817fveq2d 5714 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) ) )
192, 7, 12, 18syl3anc 1218 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) ) )
20 nnre 10348 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  RR )
21203ad2ant2 1010 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  RR )
22 nnre 10348 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  RR )
23223ad2ant3 1011 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  RR )
2421, 23, 12redivcld 10178 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  RR )
25 nnnn0 10605 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  NN0 )
2625nn0ge0d 10658 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  0  <_  A )
27263ad2ant2 1010 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  A )
28 nngt0 10370 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
29283ad2ant3 1011 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <  B )
30 divge0 10217 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
3121, 27, 23, 29, 30syl22anc 1219 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  ( A  /  B
) )
3224, 31sqrsqd 12925 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( A  /  B
) )
3319, 32eqtr3d 2477 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( A  /  B
) )
34 nnq 10985 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  QQ )
35343ad2ant2 1010 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  QQ )
36 nnq 10985 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  QQ )
37363ad2ant3 1011 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  QQ )
38 qdivcl 10993 . . . . . . . 8  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
3935, 37, 12, 38syl3anc 1218 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
4033, 39eqeltrd 2517 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  e.  QQ )
41 fveq2 5710 . . . . . . 7  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( sqr `  D
) )
4241eleq1d 2509 . . . . . 6  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  (
( sqr `  (
( A ^ 2 )  /  ( B ^ 2 ) ) )  e.  QQ  <->  ( sqr `  D )  e.  QQ ) )
4340, 42syl5ibcom 220 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  D )  e.  QQ ) )
4416, 43sylbird 235 . . . 4  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( A ^ 2 )  =  ( D  x.  ( B ^
2 ) )  -> 
( sqr `  D
)  e.  QQ ) )
4510, 44sylbid 215 . . 3  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  0  -> 
( sqr `  D
)  e.  QQ ) )
4645necon3bd 2667 . 2  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( -.  ( sqr `  D
)  e.  QQ  ->  ( ( A ^ 2 )  -  ( D  x.  ( B ^
2 ) ) )  =/=  0 ) )
4746imp 429 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   CCcc 9299   RRcr 9300   0cc0 9301    x. cmul 9306    < clt 9437    <_ cle 9438    - cmin 9614    / cdiv 10012   NNcn 10341   2c2 10390   QQcq 10972   ^cexp 11884   sqrcsqr 12741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-q 10973  df-rp 11011  df-seq 11826  df-exp 11885  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743
This theorem is referenced by:  pellexlem3  29195
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