Theorem qexpz 12646

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 9382	. . . 4 |- 0 e. ZZ
2		eleq1 2267	. . . 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 9049	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. CC )
7	6	mul01d 8464	. . . . . 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 9754	. . . . . . . . . . . 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 9746	. . . . . . . . . . . . . 14 |- ( 0 e. ZZ -> 0 e. QQ )
15	1, 14	ax-mp 5	. . . . . . . . . . . . 13 |- 0 e. QQ
16		qapne 9759	. . . . . . . . . . . . 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 9493	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. ZZ )
20	12, 18, 19	expap0d 10822	. . . . . . . . . 10 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( A ^ N ) # 0 )
21		0zd 9383	. . . . . . . . . . 11 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. ZZ )
22		zapne 9446	. . . . . . . . . . 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 12592	. . . . . . . . 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 9350	. . . . . . 7 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 <_ ( p pCnt ( A ^ N ) ) )
28		pcexp 12603	. . . . . . . 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 4069	. . . . . 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 4065	. . . . 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 8072	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 e. RR )
33		pcqcl 12600	. . . . . . . 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 9494	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> ( p pCnt A ) e. RR )
36	5	nnred 9048	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> N e. RR )
37	5	nngt0d 9079	. . . . . 6 |- ( ( ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) /\ A =/= 0 ) /\ p e. Prime ) -> 0 < N )
38		lemul2 8929	. . . . . 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 2578	. . 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 12626	. . . 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 10385	. . . 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 2386	. . 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 1372   e. wcel 2175   =/= wne 2375   A.wral 2483   class class class wbr 4043  (class class class)co 5943   CCcc 7922   RRcr 7923   0cc0 7924   x. cmul 7929   < clt 8106   <_ cle 8107   # cap 8653   NNcn 9035   NN0cn0 9294   ZZcz 9371   QQcq 9739   ^cexp 10681   Primecprime 12400   pCnt cpc 12578

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-1o 6501  df-2o 6502  df-er 6619  df-en 6827  df-sup 7085  df-inf 7086  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-xnn0 9358  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-dvds 12070  df-gcd 12246  df-prm 12401  df-pc 12579

This theorem is referenced by: (None)