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Theorem expcllem 10987
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 9565 . 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2 oveq2 6093 . . . . . . 7 |- ( z = 1 -> ( A ^ z ) = ( A ^ 1 ) )
32eleq1d 2307 . . . . . 6 |- ( z = 1 -> ( ( A ^ z ) e. F <-> ( A ^ 1 ) e. F ) )
43imbi2d 230 . . . . 5 |- ( z = 1 -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ 1 ) e. F ) ) )
5 oveq2 6093 . . . . . . 7 |- ( z = w -> ( A ^ z ) = ( A ^ w ) )
65eleq1d 2307 . . . . . 6 |- ( z = w -> ( ( A ^ z ) e. F <-> ( A ^ w ) e. F ) )
76imbi2d 230 . . . . 5 |- ( z = w -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ w ) e. F ) ) )
8 oveq2 6093 . . . . . . 7 |- ( z = ( w + 1 ) -> ( A ^ z ) = ( A ^ ( w + 1 ) ) )
98eleq1d 2307 . . . . . 6 |- ( z = ( w + 1 ) -> ( ( A ^ z ) e. F <-> ( A ^ ( w + 1 ) ) e. F ) )
109imbi2d 230 . . . . 5 |- ( z = ( w + 1 ) -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
11 oveq2 6093 . . . . . . 7 |- ( z = B -> ( A ^ z ) = ( A ^ B ) )
1211eleq1d 2307 . . . . . 6 |- ( z = B -> ( ( A ^ z ) e. F <-> ( A ^ B ) e. F ) )
1312imbi2d 230 . . . . 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 3244 . . . . . . . 8 |- ( A e. F -> A e. CC )
16 exp1 10982 . . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
1715, 16syl 14 . . . . . . 7 |- ( A e. F -> ( A ^ 1 ) = A )
1817eleq1d 2307 . . . . . 6 |- ( A e. F -> ( ( A ^ 1 ) e. F <-> A e. F ) )
1918ibir 177 . . . . 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 268 . . . . . . . . . 10 |- ( ( A e. F /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
2322adantlr 481 . . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
24 nnnn0 9570 . . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25 expp1 10983 . . . . . . . . . . . 12 |- ( ( A e. CC /\ w e. NN0 ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
2615, 24, 25syl2an 289 . . . . . . . . . . 11 |- ( ( A e. F /\ w e. NN ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
2726eleq1d 2307 . . . . . . . . . 10 |- ( ( A e. F /\ w e. NN ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
2827adantr 276 . . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
2923, 28mpbird 167 . . . . . . . 8 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( A ^ ( w + 1 ) ) e. F )
3029exp31 364 . . . . . . 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 9320 . . . 4 |- ( B e. NN -> ( A e. F -> ( A ^ B ) e. F ) )
3433impcom 125 . . 3 |- ( ( A e. F /\ B e. NN ) -> ( A ^ B ) e. F )
35 oveq2 6093 . . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36 exp0 10980 . . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
3715, 36syl 14 . . . . 5 |- ( A e. F -> ( A ^ 0 ) = 1 )
3835, 37sylan9eqr 2293 . . . 4 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) = 1 )
39 expcllem.3 . . . 4 |- 1 e. F
4038, 39eqeltrdi 2329 . . 3 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) e. F )
4134, 40jaodan 809 . 2 |- ( ( A e. F /\ ( B e. NN \/ B = 0 ) ) -> ( A ^ B ) e. F )
421, 41sylan2b 287 1 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   C_ wss 3220  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NNcn 9304   NN0cn0 9563   ^cexp 10975
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-exp 10976
This theorem is used by:  expcl2lemap  10988  nnexpcl  10989  nn0expcl  10990  zexpcl  10991  qexpcl  10992  reexpcl  10993  expcl  10994  expge0  11012  expge1  11013  lgsfcl2  16125
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