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Theorem expcllem 12871
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 11294 . 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2 oveq2 6658 . . . . . . 7 |- ( z = 1 -> ( A ^ z ) = ( A ^ 1 ) )
32eleq1d 2686 . . . . . 6 |- ( z = 1 -> ( ( A ^ z ) e. F <-> ( A ^ 1 ) e. F ) )
43imbi2d 330 . . . . 5 |- ( z = 1 -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ 1 ) e. F ) ) )
5 oveq2 6658 . . . . . . 7 |- ( z = w -> ( A ^ z ) = ( A ^ w ) )
65eleq1d 2686 . . . . . 6 |- ( z = w -> ( ( A ^ z ) e. F <-> ( A ^ w ) e. F ) )
76imbi2d 330 . . . . 5 |- ( z = w -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ w ) e. F ) ) )
8 oveq2 6658 . . . . . . 7 |- ( z = ( w + 1 ) -> ( A ^ z ) = ( A ^ ( w + 1 ) ) )
98eleq1d 2686 . . . . . 6 |- ( z = ( w + 1 ) -> ( ( A ^ z ) e. F <-> ( A ^ ( w + 1 ) ) e. F ) )
109imbi2d 330 . . . . 5 |- ( z = ( w + 1 ) -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
11 oveq2 6658 . . . . . . 7 |- ( z = B -> ( A ^ z ) = ( A ^ B ) )
1211eleq1d 2686 . . . . . 6 |- ( z = B -> ( ( A ^ z ) e. F <-> ( A ^ B ) e. F ) )
1312imbi2d 330 . . . . 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 3599 . . . . . . . 8 |- ( A e. F -> A e. CC )
16 exp1 12866 . . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
1715, 16syl 17 . . . . . . 7 |- ( A e. F -> ( A ^ 1 ) = A )
1817eleq1d 2686 . . . . . 6 |- ( A e. F -> ( ( A ^ 1 ) e. F <-> A e. F ) )
1918ibir 257 . . . . 5 |- ( A e. F -> ( A ^ 1 ) e. F )
20 expcllem.2 . . . . . . . . . . . 12 |- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )
2120caovcl 6828 . . . . . . . . . . 11 |- ( ( ( A ^ w ) e. F /\ A e. F ) -> ( ( A ^ w ) x. A ) e. F )
2221ancoms 469 . . . . . . . . . 10 |- ( ( A e. F /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
2322adantlr 751 . . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
24 nnnn0 11299 . . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25 expp1 12867 . . . . . . . . . . . 12 |- ( ( A e. CC /\ w e. NN0 ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
2615, 24, 25syl2an 494 . . . . . . . . . . 11 |- ( ( A e. F /\ w e. NN ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
2726eleq1d 2686 . . . . . . . . . 10 |- ( ( A e. F /\ w e. NN ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
2827adantr 481 . . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
2923, 28mpbird 247 . . . . . . . 8 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( A ^ ( w + 1 ) ) e. F )
3029exp31 630 . . . . . . 7 |- ( A e. F -> ( w e. NN -> ( ( A ^ w ) e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
3130com12 32 . . . . . 6 |- ( w e. NN -> ( A e. F -> ( ( A ^ w ) e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
3231a2d 29 . . . . 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 11038 . . . 4 |- ( B e. NN -> ( A e. F -> ( A ^ B ) e. F ) )
3433impcom 446 . . 3 |- ( ( A e. F /\ B e. NN ) -> ( A ^ B ) e. F )
35 oveq2 6658 . . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36 exp0 12864 . . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
3715, 36syl 17 . . . . 5 |- ( A e. F -> ( A ^ 0 ) = 1 )
3835, 37sylan9eqr 2678 . . . 4 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) = 1 )
39 expcllem.3 . . . 4 |- 1 e. F
4038, 39syl6eqel 2709 . . 3 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) e. F )
4134, 40jaodan 826 . 2 |- ( ( A e. F /\ ( B e. NN \/ B = 0 ) ) -> ( A ^ B ) e. F )
421, 41sylan2b 492 1 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   NN0cn0 11292   ^cexp 12860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861
This theorem is referenced by:  expcl2lem  12872  nnexpcl  12873  nn0expcl  12874  zexpcl  12875  qexpcl  12876  reexpcl  12877  expcl  12878  expge0  12896  expge1  12897  lgsfcl2  25028
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