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Theorem pell14qrexpcl 29231
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 10680 . . 3  |-  ( B  e.  ZZ  <->  ( B  e.  RR  /\  ( B  e.  NN0  \/  -u B  e.  NN0 ) ) )
2 simplll 757 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  D  e.  ( NN  \NN ) )
3 simpllr 758 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  A  e.  (Pell14QR `  D )
)
4 simpr 461 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  B  e.  NN0 )
5 pell14qrexpclnn0 29230 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
62, 3, 4, 5syl3anc 1218 . . . . 5  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  B  e. 
NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
7 pell14qrre 29221 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
87recnd 9431 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
98ad2antrr 725 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  A  e.  CC )
10 simplr 754 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  B  e.  RR )
1110recnd 9431 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  B  e.  CC )
12 simpr 461 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  -u B  e.  NN0 )
13 expneg2 11893 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
149, 11, 12, 13syl3anc 1218 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
15 simplll 757 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  D  e.  ( NN  \NN ) )
16 simpllr 758 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  A  e.  (Pell14QR `  D )
)
17 pell14qrexpclnn0 29230 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  (Pell14QR `  D
) )
1815, 16, 12, 17syl3anc 1218 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  (Pell14QR `  D
) )
19 pell14qrreccl 29228 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A ^ -u B )  e.  (Pell14QR `  D ) )  -> 
( 1  /  ( A ^ -u B ) )  e.  (Pell14QR `  D
) )
2015, 18, 19syl2anc 661 . . . . . 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 2517 . . . . 5  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  B  e.  RR )  /\  -u B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
226, 21jaodan 783 . . . 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 618 . . 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 217 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( B  e.  ZZ  ->  ( A ^ B
)  e.  (Pell14QR `  D
) ) )
25243impia 1184 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3344   ` cfv 5437  (class class class)co 6110   CCcc 9299   RRcr 9300   1c1 9302   -ucneg 9615    / cdiv 10012   NNcn 10341   NN0cn0 10598   ZZcz 10665   ^cexp 11884  ◻NNcsquarenn 29200  Pell14QRcpell14qr 29203
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-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-rp 11011  df-seq 11826  df-exp 11885  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-pell14qr 29207  df-pell1234qr 29208
This theorem is referenced by:  pellfund14  29262  pellfund14b  29263
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