Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pell14qrexpcl Structured version   Unicode version

Theorem pell14qrexpcl 26931
Description: Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell14qrexpcl  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)

Proof of Theorem pell14qrexpcl
StepHypRef Expression
1 elznn0 10297 . . 3  |-  ( B  e.  ZZ  <->  ( B  e.  RR  /\  ( B  e.  NN0  \/  -u B  e.  NN0 ) ) )
2 simplll 736 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  D  e.  ( NN  \NN ) )
3 simpllr 737 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  A  e.  (Pell14QR `  D )
)
4 simpr 449 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  B  e.  NN0 )
5 pell14qrexpclnn0 26930 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
62, 3, 4, 5syl3anc 1185 . . . . 5  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
7 pell14qrre 26921 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
87recnd 9115 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
98ad2antrr 708 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  A  e.  CC )
10 simplr 733 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  B  e.  RR )
1110recnd 9115 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  B  e.  CC )
12 simpr 449 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  -u B  e.  NN0 )
13 expneg2 11391 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
149, 11, 12, 13syl3anc 1185 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
15 simplll 736 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  D  e.  ( NN  \NN ) )
16 simpllr 737 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  A  e.  (Pell14QR `  D )
)
17 pell14qrexpclnn0 26930 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  (Pell14QR `  D
) )
1815, 16, 12, 17syl3anc 1185 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  (Pell14QR `  D
) )
19 pell14qrreccl 26928 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A ^ -u B )  e.  (Pell14QR `  D ) )  -> 
( 1  /  ( A ^ -u B ) )  e.  (Pell14QR `  D
) )
2015, 18, 19syl2anc 644 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  (
1  /  ( A ^ -u B ) )  e.  (Pell14QR `  D
) )
2114, 20eqeltrd 2511 . . . . 5  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
226, 21jaodan 762 . . . 4  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  ( B  e.  NN0  \/  -u B  e.  NN0 ) )  -> 
( A ^ B
)  e.  (Pell14QR `  D
) )
2322expl 603 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( B  e.  RR  /\  ( B  e.  NN0  \/  -u B  e.  NN0 ) )  -> 
( A ^ B
)  e.  (Pell14QR `  D
) ) )
241, 23syl5bi 210 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( B  e.  ZZ  ->  ( A ^ B
)  e.  (Pell14QR `  D
) ) )
25243impia 1151 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3318   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   1c1 8992   -ucneg 9293    / cdiv 9678   NNcn 10001   NN0cn0 10222   ZZcz 10283   ^cexp 11383  ◻NNcsquarenn 26900  Pell14QRcpell14qr 26903
This theorem is referenced by:  pellfund14  26962  pellfund14b  26963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-pell14qr 26907  df-pell1234qr 26908
  Copyright terms: Public domain W3C validator