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Theorem expcllem 9584
Description: Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
Hypotheses
Ref Expression
expcllem.1  |-  F  C_  CC
expcllem.2  |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y
)  e.  F )
expcllem.3  |-  1  e.  F
Assertion
Ref Expression
expcllem  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
Distinct variable groups:    x, y, A   
x, B    x, F, y
Allowed substitution hint:    B( y)

Proof of Theorem expcllem
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 8357 . 2  |-  ( B  e.  NN0  <->  ( B  e.  NN  \/  B  =  0 ) )
2 oveq2 5551 . . . . . . 7  |-  ( z  =  1  ->  ( A ^ z )  =  ( A ^ 1 ) )
32eleq1d 2148 . . . . . 6  |-  ( z  =  1  ->  (
( A ^ z
)  e.  F  <->  ( A ^ 1 )  e.  F ) )
43imbi2d 228 . . . . 5  |-  ( z  =  1  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ 1 )  e.  F ) ) )
5 oveq2 5551 . . . . . . 7  |-  ( z  =  w  ->  ( A ^ z )  =  ( A ^ w
) )
65eleq1d 2148 . . . . . 6  |-  ( z  =  w  ->  (
( A ^ z
)  e.  F  <->  ( A ^ w )  e.  F ) )
76imbi2d 228 . . . . 5  |-  ( z  =  w  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ w
)  e.  F ) ) )
8 oveq2 5551 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  ( A ^ z )  =  ( A ^ (
w  +  1 ) ) )
98eleq1d 2148 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
( A ^ z
)  e.  F  <->  ( A ^ ( w  + 
1 ) )  e.  F ) )
109imbi2d 228 . . . . 5  |-  ( z  =  ( w  + 
1 )  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ (
w  +  1 ) )  e.  F ) ) )
11 oveq2 5551 . . . . . . 7  |-  ( z  =  B  ->  ( A ^ z )  =  ( A ^ B
) )
1211eleq1d 2148 . . . . . 6  |-  ( z  =  B  ->  (
( A ^ z
)  e.  F  <->  ( A ^ B )  e.  F
) )
1312imbi2d 228 . . . . 5  |-  ( z  =  B  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ B
)  e.  F ) ) )
14 expcllem.1 . . . . . . . . 9  |-  F  C_  CC
1514sseli 2996 . . . . . . . 8  |-  ( A  e.  F  ->  A  e.  CC )
16 exp1 9579 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1715, 16syl 14 . . . . . . 7  |-  ( A  e.  F  ->  ( A ^ 1 )  =  A )
1817eleq1d 2148 . . . . . 6  |-  ( A  e.  F  ->  (
( A ^ 1 )  e.  F  <->  A  e.  F ) )
1918ibir 175 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 1 )  e.  F )
20 expcllem.2 . . . . . . . . . . . 12  |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y
)  e.  F )
2120caovcl 5686 . . . . . . . . . . 11  |-  ( ( ( A ^ w
)  e.  F  /\  A  e.  F )  ->  ( ( A ^
w )  x.  A
)  e.  F )
2221ancoms 264 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  ( A ^ w )  e.  F )  -> 
( ( A ^
w )  x.  A
)  e.  F )
2322adantlr 461 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ w )  x.  A )  e.  F
)
24 nnnn0 8362 . . . . . . . . . . . 12  |-  ( w  e.  NN  ->  w  e.  NN0 )
25 expp1 9580 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  w  e.  NN0 )  -> 
( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2615, 24, 25syl2an 283 . . . . . . . . . . 11  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2726eleq1d 2148 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( ( A ^
( w  +  1 ) )  e.  F  <->  ( ( A ^ w
)  x.  A )  e.  F ) )
2827adantr 270 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ ( w  + 
1 ) )  e.  F  <->  ( ( A ^ w )  x.  A )  e.  F
) )
2923, 28mpbird 165 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( A ^ ( w  + 
1 ) )  e.  F )
3029exp31 356 . . . . . . 7  |-  ( A  e.  F  ->  (
w  e.  NN  ->  ( ( A ^ w
)  e.  F  -> 
( A ^ (
w  +  1 ) )  e.  F ) ) )
3130com12 30 . . . . . 6  |-  ( w  e.  NN  ->  ( A  e.  F  ->  ( ( A ^ w
)  e.  F  -> 
( A ^ (
w  +  1 ) )  e.  F ) ) )
3231a2d 26 . . . . 5  |-  ( w  e.  NN  ->  (
( A  e.  F  ->  ( A ^ w
)  e.  F )  ->  ( A  e.  F  ->  ( A ^ ( w  + 
1 ) )  e.  F ) ) )
334, 7, 10, 13, 19, 32nnind 8122 . . . 4  |-  ( B  e.  NN  ->  ( A  e.  F  ->  ( A ^ B )  e.  F ) )
3433impcom 123 . . 3  |-  ( ( A  e.  F  /\  B  e.  NN )  ->  ( A ^ B
)  e.  F )
35 oveq2 5551 . . . . 5  |-  ( B  =  0  ->  ( A ^ B )  =  ( A ^ 0 ) )
36 exp0 9577 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3715, 36syl 14 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 0 )  =  1 )
3835, 37sylan9eqr 2136 . . . 4  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  =  1 )
39 expcllem.3 . . . 4  |-  1  e.  F
4038, 39syl6eqel 2170 . . 3  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  e.  F
)
4134, 40jaodan 744 . 2  |-  ( ( A  e.  F  /\  ( B  e.  NN  \/  B  =  0
) )  ->  ( A ^ B )  e.  F )
421, 41sylan2b 281 1  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434    C_ wss 2974  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048   NNcn 8106   NN0cn0 8355   ^cexp 9572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522  df-iexp 9573
This theorem is referenced by:  expcl2lemap  9585  nnexpcl  9586  nn0expcl  9587  zexpcl  9588  qexpcl  9589  reexpcl  9590  expcl  9591  expge0  9609  expge1  9610
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