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Theorem expcllem 11397
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 10228 . 2  |-  ( B  e.  NN0  <->  ( B  e.  NN  \/  B  =  0 ) )
2 oveq2 6092 . . . . . . 7  |-  ( z  =  1  ->  ( A ^ z )  =  ( A ^ 1 ) )
32eleq1d 2504 . . . . . 6  |-  ( z  =  1  ->  (
( A ^ z
)  e.  F  <->  ( A ^ 1 )  e.  F ) )
43imbi2d 309 . . . . 5  |-  ( z  =  1  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ 1 )  e.  F ) ) )
5 oveq2 6092 . . . . . . 7  |-  ( z  =  w  ->  ( A ^ z )  =  ( A ^ w
) )
65eleq1d 2504 . . . . . 6  |-  ( z  =  w  ->  (
( A ^ z
)  e.  F  <->  ( A ^ w )  e.  F ) )
76imbi2d 309 . . . . 5  |-  ( z  =  w  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ w
)  e.  F ) ) )
8 oveq2 6092 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  ( A ^ z )  =  ( A ^ (
w  +  1 ) ) )
98eleq1d 2504 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
( A ^ z
)  e.  F  <->  ( A ^ ( w  + 
1 ) )  e.  F ) )
109imbi2d 309 . . . . 5  |-  ( z  =  ( w  + 
1 )  ->  (
( A  e.  F  ->  ( A ^ z
)  e.  F )  <-> 
( A  e.  F  ->  ( A ^ (
w  +  1 ) )  e.  F ) ) )
11 oveq2 6092 . . . . . . 7  |-  ( z  =  B  ->  ( A ^ z )  =  ( A ^ B
) )
1211eleq1d 2504 . . . . . 6  |-  ( z  =  B  ->  (
( A ^ z
)  e.  F  <->  ( A ^ B )  e.  F
) )
1312imbi2d 309 . . . . 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 3346 . . . . . . . 8  |-  ( A  e.  F  ->  A  e.  CC )
16 exp1 11392 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1715, 16syl 16 . . . . . . 7  |-  ( A  e.  F  ->  ( A ^ 1 )  =  A )
1817eleq1d 2504 . . . . . 6  |-  ( A  e.  F  ->  (
( A ^ 1 )  e.  F  <->  A  e.  F ) )
1918ibir 235 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 1 )  e.  F )
20 expcllem.2 . . . . . . . . . . . 12  |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y
)  e.  F )
2120caovcl 6244 . . . . . . . . . . 11  |-  ( ( ( A ^ w
)  e.  F  /\  A  e.  F )  ->  ( ( A ^
w )  x.  A
)  e.  F )
2221ancoms 441 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  ( A ^ w )  e.  F )  -> 
( ( A ^
w )  x.  A
)  e.  F )
2322adantlr 697 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ w )  x.  A )  e.  F
)
24 nnnn0 10233 . . . . . . . . . . . 12  |-  ( w  e.  NN  ->  w  e.  NN0 )
25 expp1 11393 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  w  e.  NN0 )  -> 
( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2615, 24, 25syl2an 465 . . . . . . . . . . 11  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( A ^ (
w  +  1 ) )  =  ( ( A ^ w )  x.  A ) )
2726eleq1d 2504 . . . . . . . . . 10  |-  ( ( A  e.  F  /\  w  e.  NN )  ->  ( ( A ^
( w  +  1 ) )  e.  F  <->  ( ( A ^ w
)  x.  A )  e.  F ) )
2827adantr 453 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( ( A ^ ( w  + 
1 ) )  e.  F  <->  ( ( A ^ w )  x.  A )  e.  F
) )
2923, 28mpbird 225 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  w  e.  NN )  /\  ( A ^
w )  e.  F
)  ->  ( A ^ ( w  + 
1 ) )  e.  F )
3029exp31 589 . . . . . . 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 25 . . . . 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 10023 . . . 4  |-  ( B  e.  NN  ->  ( A  e.  F  ->  ( A ^ B )  e.  F ) )
3433impcom 421 . . 3  |-  ( ( A  e.  F  /\  B  e.  NN )  ->  ( A ^ B
)  e.  F )
35 oveq2 6092 . . . . 5  |-  ( B  =  0  ->  ( A ^ B )  =  ( A ^ 0 ) )
36 exp0 11391 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3715, 36syl 16 . . . . 5  |-  ( A  e.  F  ->  ( A ^ 0 )  =  1 )
3835, 37sylan9eqr 2492 . . . 4  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  =  1 )
39 expcllem.3 . . . 4  |-  1  e.  F
4038, 39syl6eqel 2526 . . 3  |-  ( ( A  e.  F  /\  B  =  0 )  ->  ( A ^ B )  e.  F
)
4134, 40jaodan 762 . 2  |-  ( ( A  e.  F  /\  ( B  e.  NN  \/  B  =  0
) )  ->  ( A ^ B )  e.  F )
421, 41sylan2b 463 1  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000   NNcn 10005   NN0cn0 10226   ^cexp 11387
This theorem is referenced by:  expcl2lem  11398  nnexpcl  11399  nn0expcl  11400  zexpcl  11401  qexpcl  11402  reexpcl  11403  expcl  11404  expge0  11421  expge1  11422  lgsfcl2  21091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329  df-exp 11388
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