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Theorem pellexlem1 26282
Description: Lemma for pellex 26288. 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 9722 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
213ad2ant2 982 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  CC )
32sqcld 11210 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A ^ 2 )  e.  CC )
4 nncn 9722 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  CC )
543ad2ant1 981 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  D  e.  CC )
6 nncn 9722 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
763ad2ant3 983 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  CC )
87sqcld 11210 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( B ^ 2 )  e.  CC )
95, 8mulcld 8823 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( D  x.  ( B ^ 2 ) )  e.  CC )
10 subeq0 9041 . . . . 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 9746 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
13123ad2ant3 983 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  =/=  0 )
14 sqne0 11137 . . . . . . . 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 9538 . . . . 5  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  (
( ( A ^
2 )  /  ( B ^ 2 ) )  =  D  <->  ( A ^ 2 )  =  ( D  x.  ( B ^ 2 ) ) ) )
18 sqdiv 11136 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
) ^ 2 )  =  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )
1918fveq2d 5462 . . . . . . . . 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 9721 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  RR )
22213ad2ant2 982 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  RR )
23 nnre 9721 . . . . . . . . . . 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 9556 . . . . . . . . 9  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( A  /  B )  e.  RR )
26 nnnn0 9940 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  NN0 )
2726nn0ge0d 9989 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  0  <_  A )
28273ad2ant2 982 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <_  A )
29 nngt0 9743 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
30293ad2ant3 983 . . . . . . . . . 10  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  0  <  B )
31 divge0 9593 . . . . . . . . . 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 11868 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A  /  B ) ^
2 ) )  =  ( A  /  B
) )
3420, 33eqtr3d 2292 . . . . . . 7  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( A  /  B
) )
35 nnq 10297 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  QQ )
36353ad2ant2 982 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  A  e.  QQ )
37 nnq 10297 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  QQ )
38373ad2ant3 983 . . . . . . . 8  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  B  e.  QQ )
39 qdivcl 10305 . . . . . . . 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 2332 . . . . . 6  |-  ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  e.  QQ )
42 fveq2 5458 . . . . . . 7  |-  ( ( ( A ^ 2 )  /  ( B ^ 2 ) )  =  D  ->  ( sqr `  ( ( A ^ 2 )  / 
( B ^ 2 ) ) )  =  ( sqr `  D
) )
4342eleq1d 2324 . . . . . 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 2458 . 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 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705    x. cmul 8710    < clt 8835    <_ cle 8836    - cmin 9005    / cdiv 9391   NNcn 9714   2c2 9763   QQcq 10284   ^cexp 11071   sqrcsqr 11684
This theorem is referenced by:  pellexlem3  26284
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-q 10285  df-rp 10323  df-seq 11014  df-exp 11072  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686
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