Theorem expcllem 10727

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.

Step	Hyp	Ref	Expression
1		elnn0 9327	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2		oveq2 5970	. . . . . . 7 |- ( z = 1 -> ( A ^ z ) = ( A ^ 1 ) )
3	2	eleq1d 2275	. . . . . 6 |- ( z = 1 -> ( ( A ^ z ) e. F <-> ( A ^ 1 ) e. F ) )
4	3	imbi2d 230	. . . . 5 |- ( z = 1 -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ 1 ) e. F ) ) )
5		oveq2 5970	. . . . . . 7 |- ( z = w -> ( A ^ z ) = ( A ^ w ) )
6	5	eleq1d 2275	. . . . . 6 |- ( z = w -> ( ( A ^ z ) e. F <-> ( A ^ w ) e. F ) )
7	6	imbi2d 230	. . . . 5 |- ( z = w -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ w ) e. F ) ) )
8		oveq2 5970	. . . . . . 7 |- ( z = ( w + 1 ) -> ( A ^ z ) = ( A ^ ( w + 1 ) ) )
9	8	eleq1d 2275	. . . . . 6 |- ( z = ( w + 1 ) -> ( ( A ^ z ) e. F <-> ( A ^ ( w + 1 ) ) e. F ) )
10	9	imbi2d 230	. . . . 5 |- ( z = ( w + 1 ) -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
11		oveq2 5970	. . . . . . 7 |- ( z = B -> ( A ^ z ) = ( A ^ B ) )
12	11	eleq1d 2275	. . . . . 6 |- ( z = B -> ( ( A ^ z ) e. F <-> ( A ^ B ) e. F ) )
13	12	imbi2d 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
15	14	sseli 3193	. . . . . . . 8 |- ( A e. F -> A e. CC )
16		exp1 10722	. . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
17	15, 16	syl 14	. . . . . . 7 |- ( A e. F -> ( A ^ 1 ) = A )
18	17	eleq1d 2275	. . . . . 6 |- ( A e. F -> ( ( A ^ 1 ) e. F <-> A e. F ) )
19	18	ibir 177	. . . . 5 |- ( A e. F -> ( A ^ 1 ) e. F )
20		expcllem.2	. . . . . . . . . . . 12 |- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )
21	20	caovcl 6119	. . . . . . . . . . 11 |- ( ( ( A ^ w ) e. F /\ A e. F ) -> ( ( A ^ w ) x. A ) e. F )
22	21	ancoms 268	. . . . . . . . . 10 |- ( ( A e. F /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
23	22	adantlr 477	. . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
24		nnnn0 9332	. . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25		expp1 10723	. . . . . . . . . . . 12 |- ( ( A e. CC /\ w e. NN0 ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
26	15, 24, 25	syl2an 289	. . . . . . . . . . 11 |- ( ( A e. F /\ w e. NN ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
27	26	eleq1d 2275	. . . . . . . . . 10 |- ( ( A e. F /\ w e. NN ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
28	27	adantr 276	. . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
29	23, 28	mpbird 167	. . . . . . . 8 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( A ^ ( w + 1 ) ) e. F )
30	29	exp31 364	. . . . . . 7 |- ( A e. F -> ( w e. NN -> ( ( A ^ w ) e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
31	30	com12 30	. . . . . 6 |- ( w e. NN -> ( A e. F -> ( ( A ^ w ) e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
32	31	a2d 26	. . . . 5 |- ( w e. NN -> ( ( A e. F -> ( A ^ w ) e. F ) -> ( A e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
33	4, 7, 10, 13, 19, 32	nnind 9082	. . . 4 |- ( B e. NN -> ( A e. F -> ( A ^ B ) e. F ) )
34	33	impcom 125	. . 3 |- ( ( A e. F /\ B e. NN ) -> ( A ^ B ) e. F )
35		oveq2 5970	. . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36		exp0 10720	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
37	15, 36	syl 14	. . . . 5 |- ( A e. F -> ( A ^ 0 ) = 1 )
38	35, 37	sylan9eqr 2261	. . . 4 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) = 1 )
39		expcllem.3	. . . 4 |- 1 e. F
40	38, 39	eqeltrdi 2297	. . 3 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) e. F )
41	34, 40	jaodan 799	. 2 |- ( ( A e. F /\ ( B e. NN \/ B = 0 ) ) -> ( A ^ B ) e. F )
42	1, 41	sylan2b 287	1 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2177   C_ wss 3170  (class class class)co 5962   CCcc 7953   0cc0 7955   1c1 7956   + caddc 7958   x. cmul 7960   NNcn 9066   NN0cn0 9325   ^cexp 10715

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-seqfrec 10625  df-exp 10716

This theorem is referenced by:  expcl2lemap  10728  nnexpcl  10729  nn0expcl  10730  zexpcl  10731  qexpcl  10732  reexpcl  10733  expcl  10734  expge0  10752  expge1  10753  lgsfcl2  15568