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Theorem expcllem 11161
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 10014 . 2  |-  ( B  e.  NN0  <->  ( B  e.  NN  \/  B  =  0 ) )
2 oveq2 5908 . . . . . . 7  |-  ( z  =  1  ->  ( A ^ z )  =  ( A ^ 1 ) )
32eleq1d 2382 . . . . . 6  |-  ( z  =  1  ->  (
( A ^ z
)  e.  F  <->  ( A ^ 1 )  e.  F ) )
43imbi2d 307 . . . . 5  |-  ( z  =  1  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ 1 )  e.  F ) ) )
5 oveq2 5908 . . . . . . 7  |-  ( z  =  w  ->  ( A ^ z )  =  ( A ^ w
) )
65eleq1d 2382 . . . . . 6  |-  ( z  =  w  ->  (
( A ^ z
)  e.  F  <->  ( A ^ w )  e.  F ) )
76imbi2d 307 . . . . 5  |-  ( z  =  w  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ w
)  e.  F ) ) )
8 oveq2 5908 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  ( A ^ z )  =  ( A ^ (
w  +  1 ) ) )
98eleq1d 2382 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
( A ^ z
)  e.  F  <->  ( A ^ ( w  + 
1 ) )  e.  F ) )
109imbi2d 307 . . . . 5  |-  ( z  =  ( w  + 
1 )  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ (
w  +  1 ) )  e.  F ) ) )
11 oveq2 5908 . . . . . . 7  |-  ( z  =  B  ->  ( A ^ z )  =  ( A ^ B
) )
1211eleq1d 2382 . . . . . 6  |-  ( z  =  B  ->  (
( A ^ z
)  e.  F  <->  ( A ^ B )  e.  F
) )
1312imbi2d 307 . . . . 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 3210 . . . . . . . 8  |-  ( A  e.  F  ->  A  e.  CC )
16 exp1 11156 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1715, 16syl 15 . . . . . . 7  |-  ( A  e.  F  ->  ( A ^ 1 )  =  A )
1817eleq1d 2382 . . . . . 6  |-  ( A  e.  F  ->  (
( A ^ 1 )  e.  F  <->  A  e.  F ) )
1918ibir 233 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 1 )  e.  F )
20 expcllem.2 . . . . . . . . . . . 12  |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y
)  e.  F )
2120caovcl 6056 . . . . . . . . . . 11  |-  ( ( ( A ^ w
)  e.  F  /\  A  e.  F )  ->  ( ( A ^
w )  x.  A
)  e.  F )
2221ancoms 439 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  ( A ^ w )  e.  F )  -> 
( ( A ^
w )  x.  A
)  e.  F )
2322adantlr 695 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ w )  x.  A )  e.  F
)
24 nnnn0 10019 . . . . . . . . . . . 12  |-  ( w  e.  NN  ->  w  e.  NN0 )
25 expp1 11157 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  w  e.  NN0 )  -> 
( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2615, 24, 25syl2an 463 . . . . . . . . . . 11  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2726eleq1d 2382 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( ( A ^
( w  +  1 ) )  e.  F  <->  ( ( A ^ w
)  x.  A )  e.  F ) )
2827adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ ( w  + 
1 ) )  e.  F  <->  ( ( A ^ w )  x.  A )  e.  F
) )
2923, 28mpbird 223 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( A ^ ( w  + 
1 ) )  e.  F )
3029exp31 587 . . . . . . 7  |-  ( A  e.  F  ->  (
w  e.  NN  ->  ( ( A ^ w
)  e.  F  -> 
( A ^ (
w  +  1 ) )  e.  F ) ) )
3130com12 27 . . . . . 6  |-  ( w  e.  NN  ->  ( A  e.  F  ->  ( ( A ^ w
)  e.  F  -> 
( A ^ (
w  +  1 ) )  e.  F ) ) )
3231a2d 23 . . . . 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 9809 . . . 4  |-  ( B  e.  NN  ->  ( A  e.  F  ->  ( A ^ B )  e.  F ) )
3433impcom 419 . . 3  |-  ( ( A  e.  F  /\  B  e.  NN )  ->  ( A ^ B
)  e.  F )
35 oveq2 5908 . . . . 5  |-  ( B  =  0  ->  ( A ^ B )  =  ( A ^ 0 ) )
36 exp0 11155 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3715, 36syl 15 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 0 )  =  1 )
3835, 37sylan9eqr 2370 . . . 4  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  =  1 )
39 expcllem.3 . . . 4  |-  1  e.  F
4038, 39syl6eqel 2404 . . 3  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  e.  F
)
4134, 40jaodan 760 . 2  |-  ( ( A  e.  F  /\  ( B  e.  NN  \/  B  =  0
) )  ->  ( A ^ B )  e.  F )
421, 41sylan2b 461 1  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701    C_ wss 3186  (class class class)co 5900   CCcc 8780   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787   NNcn 9791   NN0cn0 10012   ^cexp 11151
This theorem is referenced by:  expcl2lem  11162  nnexpcl  11163  nn0expcl  11164  zexpcl  11165  qexpcl  11166  reexpcl  11167  expcl  11168  expge0  11185  expge1  11186  lgsfcl2  20594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-seq 11094  df-exp 11152
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