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Theorem pell14qrexpcl 26820
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 10252 . . 3  |-  ( B  e.  ZZ  <->  ( B  e.  RR  /\  ( B  e.  NN0  \/  -u B  e.  NN0 ) ) )
2 simplll 735 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  D  e.  ( NN  \NN ) )
3 simpllr 736 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  A  e.  (Pell14QR `  D )
)
4 simpr 448 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  B  e.  NN0 )
5 pell14qrexpclnn0 26819 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
62, 3, 4, 5syl3anc 1184 . . . . 5  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
7 pell14qrre 26810 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
87recnd 9070 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
98ad2antrr 707 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  A  e.  CC )
10 simplr 732 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  B  e.  RR )
1110recnd 9070 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  B  e.  CC )
12 simpr 448 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  -u B  e.  NN0 )
13 expneg2 11345 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
149, 11, 12, 13syl3anc 1184 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
15 simplll 735 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  D  e.  ( NN  \NN ) )
16 simpllr 736 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  A  e.  (Pell14QR `  D )
)
17 pell14qrexpclnn0 26819 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  (Pell14QR `  D
) )
1815, 16, 12, 17syl3anc 1184 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  (Pell14QR `  D
) )
19 pell14qrreccl 26817 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A ^ -u B )  e.  (Pell14QR `  D ) )  -> 
( 1  /  ( A ^ -u B ) )  e.  (Pell14QR `  D
) )
2015, 18, 19syl2anc 643 . . . . . 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 2478 . . . . 5  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
226, 21jaodan 761 . . . 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 602 . . 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 209 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( B  e.  ZZ  ->  ( A ^ B
)  e.  (Pell14QR `  D
) ) )
25243impia 1150 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 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3277   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   1c1 8947   -ucneg 9248    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ^cexp 11337  ◻NNcsquarenn 26789  Pell14QRcpell14qr 26792
This theorem is referenced by:  pellfund14  26851  pellfund14b  26852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-pell14qr 26796  df-pell1234qr 26797
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