Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pell14qrexpclnn0 Unicode version

Theorem pell14qrexpclnn0 26621
Description: Lemma for pell14qrexpcl 26622. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell14qrexpclnn0  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)

Proof of Theorem pell14qrexpclnn0
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6029 . . . . . 6  |-  ( a  =  0  ->  ( A ^ a )  =  ( A ^ 0 ) )
21eleq1d 2454 . . . . 5  |-  ( a  =  0  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^
0 )  e.  (Pell14QR `  D ) ) )
32imbi2d 308 . . . 4  |-  ( a  =  0  ->  (
( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  ->  ( A ^ a )  e.  (Pell14QR `  D )
)  <->  ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  ->  ( A ^ 0 )  e.  (Pell14QR `  D )
) ) )
4 oveq2 6029 . . . . . 6  |-  ( a  =  b  ->  ( A ^ a )  =  ( A ^ b
) )
54eleq1d 2454 . . . . 5  |-  ( a  =  b  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^
b )  e.  (Pell14QR `  D ) ) )
65imbi2d 308 . . . 4  |-  ( a  =  b  ->  (
( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  ->  ( A ^ a )  e.  (Pell14QR `  D )
)  <->  ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  ->  ( A ^ b )  e.  (Pell14QR `  D )
) ) )
7 oveq2 6029 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( A ^ a )  =  ( A ^ (
b  +  1 ) ) )
87eleq1d 2454 . . . . 5  |-  ( a  =  ( b  +  1 )  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^
( b  +  1 ) )  e.  (Pell14QR `  D ) ) )
98imbi2d 308 . . . 4  |-  ( a  =  ( b  +  1 )  ->  (
( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  ->  ( A ^ a )  e.  (Pell14QR `  D )
)  <->  ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  ->  ( A ^ ( b  +  1 ) )  e.  (Pell14QR `  D )
) ) )
10 oveq2 6029 . . . . . 6  |-  ( a  =  B  ->  ( A ^ a )  =  ( A ^ B
) )
1110eleq1d 2454 . . . . 5  |-  ( a  =  B  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^ B )  e.  (Pell14QR `  D ) ) )
1211imbi2d 308 . . . 4  |-  ( a  =  B  ->  (
( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  ->  ( A ^ a )  e.  (Pell14QR `  D )
)  <->  ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  ->  ( A ^ B )  e.  (Pell14QR `  D )
) ) )
13 pell14qrre 26612 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
1413recnd 9048 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
1514exp0d 11445 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  =  1 )
16 pell14qrne0 26613 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
1714, 16dividd 9721 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  /  A
)  =  1 )
1815, 17eqtr4d 2423 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  =  ( A  /  A ) )
19 pell14qrdivcl 26620 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  A  e.  (Pell14QR `  D )
)  ->  ( A  /  A )  e.  (Pell14QR `  D ) )
20193anidm23 1243 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  /  A
)  e.  (Pell14QR `  D
) )
2118, 20eqeltrd 2462 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  e.  (Pell14QR `  D
) )
22143ad2ant2 979 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  A  e.  CC )
23 simp1 957 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  b  e.  NN0 )
2422, 23expp1d 11452 . . . . . . 7  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  ( A ^ ( b  +  1 ) )  =  ( ( A ^
b )  x.  A
) )
25 simp2l 983 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  D  e.  ( NN  \NN ) )
26 simp3 959 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  ( A ^ b )  e.  (Pell14QR `  D )
)
27 simp2r 984 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  A  e.  (Pell14QR `  D )
)
28 pell14qrmulcl 26618 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A ^
b )  e.  (Pell14QR `  D )  /\  A  e.  (Pell14QR `  D )
)  ->  ( ( A ^ b )  x.  A )  e.  (Pell14QR `  D ) )
2925, 26, 27, 28syl3anc 1184 . . . . . . 7  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  (
( A ^ b
)  x.  A )  e.  (Pell14QR `  D
) )
3024, 29eqeltrd 2462 . . . . . 6  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  ( A ^ ( b  +  1 ) )  e.  (Pell14QR `  D )
)
31303exp 1152 . . . . 5  |-  ( b  e.  NN0  ->  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( A ^
b )  e.  (Pell14QR `  D )  ->  ( A ^ ( b  +  1 ) )  e.  (Pell14QR `  D )
) ) )
3231a2d 24 . . . 4  |-  ( b  e.  NN0  ->  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  ->  ( A ^ b )  e.  (Pell14QR `  D )
)  ->  ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ (
b  +  1 ) )  e.  (Pell14QR `  D
) ) ) )
333, 6, 9, 12, 21, 32nn0ind 10299 . . 3  |-  ( B  e.  NN0  ->  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ B
)  e.  (Pell14QR `  D
) ) )
3433exp3acom3r 1376 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  (Pell14QR `  D )
) ) )
35343imp 1147 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    \ cdif 3261   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    / cdiv 9610   NNcn 9933   NN0cn0 10154   ^cexp 11310  ◻NNcsquarenn 26591  Pell14QRcpell14qr 26594
This theorem is referenced by:  pell14qrexpcl  26622
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-pell14qr 26598  df-pell1234qr 26599
  Copyright terms: Public domain W3C validator