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Theorem pell14qrexpclnn0 26951
Description: Lemma for pell14qrexpcl 26952. (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 5866 . . . . . 6  |-  ( a  =  0  ->  ( A ^ a )  =  ( A ^ 0 ) )
21eleq1d 2349 . . . . 5  |-  ( a  =  0  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^
0 )  e.  (Pell14QR `  D ) ) )
32imbi2d 307 . . . 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 5866 . . . . . 6  |-  ( a  =  b  ->  ( A ^ a )  =  ( A ^ b
) )
54eleq1d 2349 . . . . 5  |-  ( a  =  b  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^
b )  e.  (Pell14QR `  D ) ) )
65imbi2d 307 . . . 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 5866 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( A ^ a )  =  ( A ^ (
b  +  1 ) ) )
87eleq1d 2349 . . . . 5  |-  ( a  =  ( b  +  1 )  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^
( b  +  1 ) )  e.  (Pell14QR `  D ) ) )
98imbi2d 307 . . . 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 5866 . . . . . 6  |-  ( a  =  B  ->  ( A ^ a )  =  ( A ^ B
) )
1110eleq1d 2349 . . . . 5  |-  ( a  =  B  ->  (
( A ^ a
)  e.  (Pell14QR `  D
)  <->  ( A ^ B )  e.  (Pell14QR `  D ) ) )
1211imbi2d 307 . . . 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 26942 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
1413recnd 8861 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
1514exp0d 11239 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  =  1 )
16 pell14qrne0 26943 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
1714, 16dividd 9534 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  /  A
)  =  1 )
1815, 17eqtr4d 2318 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  =  ( A  /  A ) )
19 pell14qrdivcl 26950 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  A  e.  (Pell14QR `  D )
)  ->  ( A  /  A )  e.  (Pell14QR `  D ) )
20193anidm23 1241 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  /  A
)  e.  (Pell14QR `  D
) )
2118, 20eqeltrd 2357 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ 0 )  e.  (Pell14QR `  D
) )
22143ad2ant2 977 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  A  e.  CC )
23 simp1 955 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  b  e.  NN0 )
2422, 23expp1d 11246 . . . . . . 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 981 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  D  e.  ( NN  \NN ) )
26 simp3 957 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  ( A ^ b )  e.  (Pell14QR `  D )
)
27 simp2r 982 . . . . . . . 8  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  A  e.  (Pell14QR `  D )
)
28 pell14qrmulcl 26948 . . . . . . . 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 1182 . . . . . . 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 2357 . . . . . 6  |-  ( ( b  e.  NN0  /\  ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  /\  ( A ^ b )  e.  (Pell14QR `  D
) )  ->  ( A ^ ( b  +  1 ) )  e.  (Pell14QR `  D )
)
31303exp 1150 . . . . 5  |-  ( b  e.  NN0  ->  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( A ^
b )  e.  (Pell14QR `  D )  ->  ( A ^ ( b  +  1 ) )  e.  (Pell14QR `  D )
) ) )
3231a2d 23 . . . 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 10108 . . 3  |-  ( B  e.  NN0  ->  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A ^ B
)  e.  (Pell14QR `  D
) ) )
3433exp3acom3r 1360 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  (Pell14QR `  D )
) ) )
35343imp 1145 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    / cdiv 9423   NNcn 9746   NN0cn0 9965   ^cexp 11104  ◻NNcsquarenn 26921  Pell14QRcpell14qr 26924
This theorem is referenced by:  pell14qrexpcl  26952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-pell14qr 26928  df-pell1234qr 26929
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