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Theorem expcllem 11114
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 9967 . 2  |-  ( B  e.  NN0  <->  ( B  e.  NN  \/  B  =  0 ) )
2 oveq2 5866 . . . . . . 7  |-  ( z  =  1  ->  ( A ^ z )  =  ( A ^ 1 ) )
32eleq1d 2349 . . . . . 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 5866 . . . . . . 7  |-  ( z  =  w  ->  ( A ^ z )  =  ( A ^ w
) )
65eleq1d 2349 . . . . . 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 5866 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  ( A ^ z )  =  ( A ^ (
w  +  1 ) ) )
98eleq1d 2349 . . . . . 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 5866 . . . . . . 7  |-  ( z  =  B  ->  ( A ^ z )  =  ( A ^ B
) )
1211eleq1d 2349 . . . . . 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 3176 . . . . . . . 8  |-  ( A  e.  F  ->  A  e.  CC )
16 exp1 11109 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1715, 16syl 15 . . . . . . 7  |-  ( A  e.  F  ->  ( A ^ 1 )  =  A )
1817eleq1d 2349 . . . . . 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 6014 . . . . . . . . . . 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 9972 . . . . . . . . . . . 12  |-  ( w  e.  NN  ->  w  e.  NN0 )
25 expp1 11110 . . . . . . . . . . . 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 2349 . . . . . . . . . 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 9764 . . . 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 5866 . . . . 5  |-  ( B  =  0  ->  ( A ^ B )  =  ( A ^ 0 ) )
36 exp0 11108 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3715, 36syl 15 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 0 )  =  1 )
3835, 37sylan9eqr 2337 . . . 4  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  =  1 )
39 expcllem.3 . . . 4  |-  1  e.  F
4038, 39syl6eqel 2371 . . 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 1623    e. wcel 1684    C_ wss 3152  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  expcl2lem  11115  nnexpcl  11116  nn0expcl  11117  zexpcl  11118  qexpcl  11119  reexpcl  11120  expcl  11121  expge0  11138  expge1  11139  lgsfcl2  20541
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
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-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-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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