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Theorem pell14qrexpclnn0 26920
Description: Lemma for pell14qrexpcl 26921. (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 6081 . . . . . 6  |-  ( a  =  0  ->  ( A ^ a )  =  ( A ^ 0 ) )
21eleq1d 2501 . . . . 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 6081 . . . . . 6  |-  ( a  =  b  ->  ( A ^ a )  =  ( A ^ b
) )
54eleq1d 2501 . . . . 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 6081 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( A ^ a )  =  ( A ^ (
b  +  1 ) ) )
87eleq1d 2501 . . . . 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 6081 . . . . . 6  |-  ( a  =  B  ->  ( A ^ a )  =  ( A ^ B
) )
1110eleq1d 2501 . . . . 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 26911 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
1413recnd 9106 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
1514exp0d 11509 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  =  1 )
16 pell14qrne0 26912 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
1714, 16dividd 9780 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  /  A
)  =  1 )
1815, 17eqtr4d 2470 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  =  ( A  /  A ) )
19 pell14qrdivcl 26919 . . . . . 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 2509 . . . 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 11516 . . . . . . 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 26917 . . . . . . . 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 2509 . . . . . 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 10358 . . 3  |-  ( B  e.  NN0  ->  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ B
)  e.  (Pell14QR `  D
) ) )
3433exp3acom3r 1379 . 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 1652    e. wcel 1725    \ cdif 3309   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    / cdiv 9669   NNcn 9992   NN0cn0 10213   ^cexp 11374  ◻NNcsquarenn 26890  Pell14QRcpell14qr 26893
This theorem is referenced by:  pell14qrexpcl  26921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-pell14qr 26897  df-pell1234qr 26898
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