Theorem qexpz 12870

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 9453	. . . 4 |- 0 e. ZZ
2		eleq1 2292	. . . 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 1061	. . . . . . . 8 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. NN )
6	5	nncnd 9120	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. CC )
7	6	mul01d 8535	. . . . . 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 1062	. . . . . . . . 9 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) e. ZZ )
10		simpll1 1060	. . . . . . . . . . . 12 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> A e. QQ )
11		qcn 9825	. . . . . . . . . . . 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 9817	. . . . . . . . . . . . . 14 |- ( 0 e. ZZ -> 0 e. QQ )
15	1, 14	ax-mp 5	. . . . . . . . . . . . 13 |- 0 e. QQ
16		qapne 9830	. . . . . . . . . . . . 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 9564	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. ZZ )
20	12, 18, 19	expap0d 10896	. . . . . . . . . 10 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) # 0 )
21		0zd 9454	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. ZZ )
22		zapne 9517	. . . . . . . . . . 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 12816	. . . . . . . . 9 |- ( ( p e. Prime /\ ( ( A ^ N ) e. ZZ /\ ( A ^ N ) =/= 0 ) ) -> ( p pCnt ( A ^ N ) ) e. NN0 )
26	8, 9, 24, 25	syl12anc 1269	. . . . . . . 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 9421	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 <_ ( p pCnt ( A ^ N ) ) )
28		pcexp 12827	. . . . . . . 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 1276	. . . . . . 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 4108	. . . . . 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 4104	. . . . 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 8143	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. RR )
33		pcqcl 12824	. . . . . . . 8 |- ( ( p e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( p pCnt A ) e. ZZ )
34	8, 10, 13, 33	syl12anc 1269	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt A ) e. ZZ )
35	34	zred 9565	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt A ) e. RR )
36	5	nnred 9119	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. RR )
37	5	nngt0d 9150	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 < N )
38		lemul2 9000	. . . . . 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 1275	. . . . 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 2603	. . 3 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> A. p e. Prime 0 <_ ( p pCnt A ) )
42		simpl1 1024	. . . 4 |- ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) -> A e. QQ )
43		pcz 12850	. . . 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 1021	. . . 4 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. QQ )
47		qdceq 10459	. . . 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 2411	. . 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 803	1 |- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. ZZ )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   class class class wbr 4082  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   x. cmul 8000   < clt 8177   <_ cle 8178   # cap 8724   NNcn 9106   NN0cn0 9365   ZZcz 9442   QQcq 9810   ^cexp 10755   Primecprime 12624   pCnt cpc 12802

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-2o 6561  df-er 6678  df-en 6886  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-xnn0 9429  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470  df-prm 12625  df-pc 12803

This theorem is referenced by: (None)