Theorem qexpz 13131

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 9655	. . . 4 |- 0 e. ZZ
2		eleq1 2301	. . . 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 1068	. . . . . . . 8 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. NN )
6	5	nncnd 9318	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. CC )
7	6	mul01d 8720	. . . . . 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 1069	. . . . . . . . 9 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) e. ZZ )
10		simpll1 1067	. . . . . . . . . . . 12 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> A e. QQ )
11		qcn 10034	. . . . . . . . . . . 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 533	. . . . . . . . . . . 12 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> A =/= 0 )
14		zq 10026	. . . . . . . . . . . . . 14 |- ( 0 e. ZZ -> 0 e. QQ )
15	1, 14	ax-mp 5	. . . . . . . . . . . . 13 |- 0 e. QQ
16		qapne 10039	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ 0 e. QQ ) -> ( A # 0 <-> A =/= 0 ) )
17	10, 15, 16	sylancl 417	. . . . . . . . . . . 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 9767	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. ZZ )
20	12, 18, 19	expap0d 11117	. . . . . . . . . 10 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) # 0 )
21		0zd 9656	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. ZZ )
22		zapne 9719	. . . . . . . . . . 11 |- ( ( ( A ^ N ) e. ZZ /\ 0 e. ZZ ) -> ( ( A ^ N ) # 0 <-> ( A ^ N ) =/= 0 ) )
23	9, 21, 22	syl2anc 415	. . . . . . . . . 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 13077	. . . . . . . . 9 |- ( ( p e. Prime /\ ( ( A ^ N ) e. ZZ /\ ( A ^ N ) =/= 0 ) ) -> ( p pCnt ( A ^ N ) ) e. NN0 )
26	8, 9, 24, 25	syl12anc 1276	. . . . . . . 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 9623	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 <_ ( p pCnt ( A ^ N ) ) )
28		pcexp 13088	. . . . . . . 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 1283	. . . . . . 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 4156	. . . . . 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 4152	. . . . 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 8327	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. RR )
33		pcqcl 13085	. . . . . . . 8 |- ( ( p e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( p pCnt A ) e. ZZ )
34	8, 10, 13, 33	syl12anc 1276	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt A ) e. ZZ )
35	34	zred 9768	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt A ) e. RR )
36	5	nnred 9317	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. RR )
37	5	nngt0d 9348	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 < N )
38		lemul2 9187	. . . . . 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 1282	. . . . 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 2623	. . 3 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> A. p e. Prime 0 <_ ( p pCnt A ) )
42		simpl1 1031	. . . 4 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> A e. QQ )
43		pcz 13111	. . . 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 1028	. . . 4 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. QQ )
47		qdceq 10679	. . . 4 |- ( ( A e. QQ /\ 0 e. QQ ) -> DECID A = 0 )
48	46, 15, 47	sylancl 417	. . 3 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> DECID A = 0 )
49		dcne 2431	. . 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 810	1 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. ZZ )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   x. cmul 8184   < clt 8360   <_ cle 8361   # cap 8909   NNcn 9304   NN0cn0 9563   ZZcz 9644   QQcq 10019   ^cexp 10975   Primecprime 12885   pCnt cpc 13063

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  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  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-isom 5386  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-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-inf 7325  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-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-xnn0 9631  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886  df-pc 13064

This theorem is used by: (None)