ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expcllem Unicode version
Theorem expcllem 10965
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 9544 . 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2 oveq2 6083 . . . . . . 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 6083 . . . . . . 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 6083 . . . . . . 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 6083 . . . . . . 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 10960 . . . . . . . 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 6234 . . . . . . . . . . 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 9549 . . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25 expp1 10961 . . . . . . . . . . . 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 9299 . . . 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 6083 . . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36 exp0 10958 . . . . . 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
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   C_ wss 3220  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   NNcn 9283   NN0cn0 9542   ^cexp 10953
This theorem was proved from 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287
This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-exp 10954
This theorem is referenced by:  expcl2lemap  10966  nnexpcl  10967  nn0expcl  10968  zexpcl  10969  qexpcl  10970  reexpcl  10971  expcl  10972  expge0  10990  expge1  10991  lgsfcl2  16039
  Copyright terms: Public domain W3C validator