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Theorem expcllem 11384
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 10215 . 2  |-  ( B  e.  NN0  <->  ( B  e.  NN  \/  B  =  0 ) )
2 oveq2 6081 . . . . . . 7  |-  ( z  =  1  ->  ( A ^ z )  =  ( A ^ 1 ) )
32eleq1d 2501 . . . . . 6  |-  ( z  =  1  ->  (
( A ^ z
)  e.  F  <->  ( A ^ 1 )  e.  F ) )
43imbi2d 308 . . . . 5  |-  ( z  =  1  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ 1 )  e.  F ) ) )
5 oveq2 6081 . . . . . . 7  |-  ( z  =  w  ->  ( A ^ z )  =  ( A ^ w
) )
65eleq1d 2501 . . . . . 6  |-  ( z  =  w  ->  (
( A ^ z
)  e.  F  <->  ( A ^ w )  e.  F ) )
76imbi2d 308 . . . . 5  |-  ( z  =  w  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ w
)  e.  F ) ) )
8 oveq2 6081 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  ( A ^ z )  =  ( A ^ (
w  +  1 ) ) )
98eleq1d 2501 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
( A ^ z
)  e.  F  <->  ( A ^ ( w  + 
1 ) )  e.  F ) )
109imbi2d 308 . . . . 5  |-  ( z  =  ( w  + 
1 )  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ (
w  +  1 ) )  e.  F ) ) )
11 oveq2 6081 . . . . . . 7  |-  ( z  =  B  ->  ( A ^ z )  =  ( A ^ B
) )
1211eleq1d 2501 . . . . . 6  |-  ( z  =  B  ->  (
( A ^ z
)  e.  F  <->  ( A ^ B )  e.  F
) )
1312imbi2d 308 . . . . 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 3336 . . . . . . . 8  |-  ( A  e.  F  ->  A  e.  CC )
16 exp1 11379 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1715, 16syl 16 . . . . . . 7  |-  ( A  e.  F  ->  ( A ^ 1 )  =  A )
1817eleq1d 2501 . . . . . 6  |-  ( A  e.  F  ->  (
( A ^ 1 )  e.  F  <->  A  e.  F ) )
1918ibir 234 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 1 )  e.  F )
20 expcllem.2 . . . . . . . . . . . 12  |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y
)  e.  F )
2120caovcl 6233 . . . . . . . . . . 11  |-  ( ( ( A ^ w
)  e.  F  /\  A  e.  F )  ->  ( ( A ^
w )  x.  A
)  e.  F )
2221ancoms 440 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  ( A ^ w )  e.  F )  -> 
( ( A ^
w )  x.  A
)  e.  F )
2322adantlr 696 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ w )  x.  A )  e.  F
)
24 nnnn0 10220 . . . . . . . . . . . 12  |-  ( w  e.  NN  ->  w  e.  NN0 )
25 expp1 11380 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  w  e.  NN0 )  -> 
( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2615, 24, 25syl2an 464 . . . . . . . . . . 11  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2726eleq1d 2501 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( ( A ^
( w  +  1 ) )  e.  F  <->  ( ( A ^ w
)  x.  A )  e.  F ) )
2827adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ ( w  + 
1 ) )  e.  F  <->  ( ( A ^ w )  x.  A )  e.  F
) )
2923, 28mpbird 224 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( A ^ ( w  + 
1 ) )  e.  F )
3029exp31 588 . . . . . . 7  |-  ( A  e.  F  ->  (
w  e.  NN  ->  ( ( A ^ w
)  e.  F  -> 
( A ^ (
w  +  1 ) )  e.  F ) ) )
3130com12 29 . . . . . 6  |-  ( w  e.  NN  ->  ( A  e.  F  ->  ( ( A ^ w
)  e.  F  -> 
( A ^ (
w  +  1 ) )  e.  F ) ) )
3231a2d 24 . . . . 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 10010 . . . 4  |-  ( B  e.  NN  ->  ( A  e.  F  ->  ( A ^ B )  e.  F ) )
3433impcom 420 . . 3  |-  ( ( A  e.  F  /\  B  e.  NN )  ->  ( A ^ B
)  e.  F )
35 oveq2 6081 . . . . 5  |-  ( B  =  0  ->  ( A ^ B )  =  ( A ^ 0 ) )
36 exp0 11378 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3715, 36syl 16 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 0 )  =  1 )
3835, 37sylan9eqr 2489 . . . 4  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  =  1 )
39 expcllem.3 . . . 4  |-  1  e.  F
4038, 39syl6eqel 2523 . . 3  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  e.  F
)
4134, 40jaodan 761 . 2  |-  ( ( A  e.  F  /\  ( B  e.  NN  \/  B  =  0
) )  ->  ( A ^ B )  e.  F )
421, 41sylan2b 462 1  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987   NNcn 9992   NN0cn0 10213   ^cexp 11374
This theorem is referenced by:  expcl2lem  11385  nnexpcl  11386  nn0expcl  11387  zexpcl  11388  qexpcl  11389  reexpcl  11390  expcl  11391  expge0  11408  expge1  11409  lgsfcl2  21078
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
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-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-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-exp 11375
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