Theorem qexpz 12521

Description: If a power of a rational number is an integer, then the number is an integer. (Contributed by Mario Carneiro, 10-Aug-2015.)

Assertion

Ref	Expression
qexpz	|- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. ZZ )

Proof of Theorem qexpz

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9337	. . . 4 |- 0 e. ZZ
2		eleq1 2259	. . . 4 |- ( A = 0 -> ( A e. ZZ <-> 0 e. ZZ ) )
3	1, 2	mpbiri 168	. . 3 |- ( A = 0 -> A e. ZZ )
4	3	adantl 277	. 2 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A = 0 ) -> A e. ZZ )
5		simpll2 1039	. . . . . . . 8 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. NN )
6	5	nncnd 9004	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. CC )
7	6	mul01d 8419	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( N x. 0 ) = 0 )
8		simpr 110	. . . . . . . . 9 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> p e. Prime )
9		simpll3 1040	. . . . . . . . 9 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) e. ZZ )
10		simpll1 1038	. . . . . . . . . . . 12 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> A e. QQ )
11		qcn 9708	. . . . . . . . . . . 12 |- ( A e. QQ -> A e. CC )
12	10, 11	syl 14	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> A e. CC )
13		simplr 528	. . . . . . . . . . . 12 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> A =/= 0 )
14		zq 9700	. . . . . . . . . . . . . 14 |- ( 0 e. ZZ -> 0 e. QQ )
15	1, 14	ax-mp 5	. . . . . . . . . . . . 13 |- 0 e. QQ
16		qapne 9713	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ 0 e. QQ ) -> ( A # 0 <-> A =/= 0 ) )
17	10, 15, 16	sylancl 413	. . . . . . . . . . . 12 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A # 0 <-> A =/= 0 ) )
18	13, 17	mpbird 167	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> A # 0 )
19	5	nnzd 9447	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. ZZ )
20	12, 18, 19	expap0d 10771	. . . . . . . . . 10 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) # 0 )
21		0zd 9338	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. ZZ )
22		zapne 9400	. . . . . . . . . . 11 |- ( ( ( A ^ N ) e. ZZ /\ 0 e. ZZ ) -> ( ( A ^ N ) # 0 <-> ( A ^ N ) =/= 0 ) )
23	9, 21, 22	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( ( A ^ N ) # 0 <-> ( A ^ N ) =/= 0 ) )
24	20, 23	mpbid 147	. . . . . . . . 9 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) =/= 0 )
25		pczcl 12467	. . . . . . . . 9 |- ( ( p e. Prime /\ ( ( A ^ N ) e. ZZ /\ ( A ^ N ) =/= 0 ) ) -> ( p pCnt ( A ^ N ) ) e. NN0 )
26	8, 9, 24, 25	syl12anc 1247	. . . . . . . 8 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt ( A ^ N ) ) e. NN0 )
27	26	nn0ge0d 9305	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 <_ ( p pCnt ( A ^ N ) ) )
28		pcexp 12478	. . . . . . . 8 |- ( ( p e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ N e. ZZ ) -> ( p pCnt ( A ^ N ) ) = ( N x. ( p pCnt A ) ) )
29	8, 10, 13, 19, 28	syl121anc 1254	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt ( A ^ N ) ) = ( N x. ( p pCnt A ) ) )
30	27, 29	breqtrd 4059	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 <_ ( N x. ( p pCnt A ) ) )
31	7, 30	eqbrtrd 4055	. . . . 5 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( N x. 0 ) <_ ( N x. ( p pCnt A ) ) )
32		0red 8027	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. RR )
33		pcqcl 12475	. . . . . . . 8 |- ( ( p e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( p pCnt A ) e. ZZ )
34	8, 10, 13, 33	syl12anc 1247	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt A ) e. ZZ )
35	34	zred 9448	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt A ) e. RR )
36	5	nnred 9003	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. RR )
37	5	nngt0d 9034	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 < N )
38		lemul2 8884	. . . . . 6 |- ( ( 0 e. RR /\ ( p pCnt A ) e. RR /\ ( N e. RR /\ 0 < N ) ) -> ( 0 <_ ( p pCnt A ) <-> ( N x. 0 ) <_ ( N x. ( p pCnt A ) ) ) )
39	32, 35, 36, 37, 38	syl112anc 1253	. . . . 5 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( 0 <_ ( p pCnt A ) <-> ( N x. 0 ) <_ ( N x. ( p pCnt A ) ) ) )
40	31, 39	mpbird 167	. . . 4 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 <_ ( p pCnt A ) )
41	40	ralrimiva 2570	. . 3 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> A. p e. Prime 0 <_ ( p pCnt A ) )
42		simpl1 1002	. . . 4 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> A e. QQ )
43		pcz 12501	. . . 4 |- ( A e. QQ -> ( A e. ZZ <-> A. p e. Prime 0 <_ ( p pCnt A ) ) )
44	42, 43	syl 14	. . 3 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> ( A e. ZZ <-> A. p e. Prime 0 <_ ( p pCnt A ) ) )
45	41, 44	mpbird 167	. 2 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> A e. ZZ )
46		simp1 999	. . . 4 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. QQ )
47		qdceq 10334	. . . 4 |- ( ( A e. QQ /\ 0 e. QQ ) -> DECID A = 0 )
48	46, 15, 47	sylancl 413	. . 3 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> DECID A = 0 )
49		dcne 2378	. . 3 |- (DECID A = 0 <-> ( A = 0 \/ A =/= 0 ) )
50	48, 49	sylib 122	. 2 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> ( A = 0 \/ A =/= 0 ) )
51	4, 45, 50	mpjaodan 799	1 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. ZZ )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   class class class wbr 4033  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   x. cmul 7884   < clt 8061   <_ cle 8062   # cap 8608   NNcn 8990   NN0cn0 9249   ZZcz 9326   QQcq 9693   ^cexp 10630   Primecprime 12275   pCnt cpc 12453

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-er 6592  df-en 6800  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-xnn0 9313  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-prm 12276  df-pc 12454

This theorem is referenced by: (None)