Theorem rplpwr 11960

Description: If A and B are relatively prime, then so are A ^ N and B. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
rplpwr	|- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) )

Proof of Theorem rplpwr

Dummy variables n k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5850	. . . . . . . 8 |- ( k = 1 -> ( A ^ k ) = ( A ^ 1 ) )
2	1	oveq1d 5857	. . . . . . 7 |- ( k = 1 -> ( ( A ^ k ) gcd B ) = ( ( A ^ 1 ) gcd B ) )
3	2	eqeq1d 2174	. . . . . 6 |- ( k = 1 -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ 1 ) gcd B ) = 1 ) )
4	3	imbi2d 229	. . . . 5 |- ( k = 1 -> ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ k ) gcd B ) = 1 ) <-> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ 1 ) gcd B ) = 1 ) ) )
5		oveq2 5850	. . . . . . . 8 |- ( k = n -> ( A ^ k ) = ( A ^ n ) )
6	5	oveq1d 5857	. . . . . . 7 |- ( k = n -> ( ( A ^ k ) gcd B ) = ( ( A ^ n ) gcd B ) )
7	6	eqeq1d 2174	. . . . . 6 |- ( k = n -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ n ) gcd B ) = 1 ) )
8	7	imbi2d 229	. . . . 5 |- ( k = n -> ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ k ) gcd B ) = 1 ) <-> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ n ) gcd B ) = 1 ) ) )
9		oveq2 5850	. . . . . . . 8 |- ( k = ( n + 1 ) -> ( A ^ k ) = ( A ^ ( n + 1 ) ) )
10	9	oveq1d 5857	. . . . . . 7 |- ( k = ( n + 1 ) -> ( ( A ^ k ) gcd B ) = ( ( A ^ ( n + 1 ) ) gcd B ) )
11	10	eqeq1d 2174	. . . . . 6 |- ( k = ( n + 1 ) -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) )
12	11	imbi2d 229	. . . . 5 |- ( k = ( n + 1 ) -> ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ k ) gcd B ) = 1 ) <-> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) ) )
13		oveq2 5850	. . . . . . . 8 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
14	13	oveq1d 5857	. . . . . . 7 |- ( k = N -> ( ( A ^ k ) gcd B ) = ( ( A ^ N ) gcd B ) )
15	14	eqeq1d 2174	. . . . . 6 |- ( k = N -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ N ) gcd B ) = 1 ) )
16	15	imbi2d 229	. . . . 5 |- ( k = N -> ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ k ) gcd B ) = 1 ) <-> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ N ) gcd B ) = 1 ) ) )
17		nncn 8865	. . . . . . . . . 10 |- ( A e. NN -> A e. CC )
18	17	exp1d 10583	. . . . . . . . 9 |- ( A e. NN -> ( A ^ 1 ) = A )
19	18	oveq1d 5857	. . . . . . . 8 |- ( A e. NN -> ( ( A ^ 1 ) gcd B ) = ( A gcd B ) )
20	19	adantr 274	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( ( A ^ 1 ) gcd B ) = ( A gcd B ) )
21	20	eqeq1d 2174	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( ( ( A ^ 1 ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) )
22	21	biimpar 295	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ 1 ) gcd B ) = 1 )
23		df-3an 970	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) <-> ( ( A e. NN /\ B e. NN ) /\ n e. NN ) )
24		simpl1 990	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. NN )
25	24	nncnd 8871	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. CC )
26		simpl3 992	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN )
27	26	nnnn0d 9167	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN0 )
28	25, 27	expp1d 10589	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( ( A ^ n ) x. A ) )
29		simp1 987	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. NN )
30		nnnn0 9121	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. NN -> n e. NN0 )
31	30	3ad2ant3 1010	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> n e. NN0 )
32	29, 31	nnexpcld 10610	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A ^ n ) e. NN )
33	32	nnzd 9312	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A ^ n ) e. ZZ )
34	33	adantr 274	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. ZZ )
35	34	zcnd 9314	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. CC )
36	35, 25	mulcomd 7920	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ n ) x. A ) = ( A x. ( A ^ n ) ) )
37	28, 36	eqtrd 2198	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( A x. ( A ^ n ) ) )
38	37	oveq2d 5858	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( B gcd ( A ^ ( n + 1 ) ) ) = ( B gcd ( A x. ( A ^ n ) ) ) )
39		simpl2 991	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> B e. NN )
40	32	adantr 274	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. NN )
41		nnz 9210	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN -> A e. ZZ )
42	41	3ad2ant1 1008	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. ZZ )
43		nnz 9210	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN -> B e. ZZ )
44	43	3ad2ant2 1009	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> B e. ZZ )
45		gcdcom 11906	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
46	42, 44, 45	syl2anc 409	. . . . . . . . . . . . . . . 16 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A gcd B ) = ( B gcd A ) )
47	46	eqeq1d 2174	. . . . . . . . . . . . . . 15 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( ( A gcd B ) = 1 <-> ( B gcd A ) = 1 ) )
48	47	biimpa 294	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( B gcd A ) = 1 )
49		rpmulgcd 11959	. . . . . . . . . . . . . 14 |- ( ( ( B e. NN /\ A e. NN /\ ( A ^ n ) e. NN ) /\ ( B gcd A ) = 1 ) -> ( B gcd ( A x. ( A ^ n ) ) ) = ( B gcd ( A ^ n ) ) )
50	39, 24, 40, 48, 49	syl31anc 1231	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( B gcd ( A x. ( A ^ n ) ) ) = ( B gcd ( A ^ n ) ) )
51	38, 50	eqtrd 2198	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( B gcd ( A ^ ( n + 1 ) ) ) = ( B gcd ( A ^ n ) ) )
52		peano2nn 8869	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> ( n + 1 ) e. NN )
53	52	3ad2ant3 1010	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( n + 1 ) e. NN )
54	53	adantr 274	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( n + 1 ) e. NN )
55	54	nnnn0d 9167	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( n + 1 ) e. NN0 )
56	24, 55	nnexpcld 10610	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) e. NN )
57	56	nnzd 9312	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) e. ZZ )
58	44	adantr 274	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> B e. ZZ )
59		gcdcom 11906	. . . . . . . . . . . . 13 |- ( ( ( A ^ ( n + 1 ) ) e. ZZ /\ B e. ZZ ) -> ( ( A ^ ( n + 1 ) ) gcd B ) = ( B gcd ( A ^ ( n + 1 ) ) ) )
60	57, 58, 59	syl2anc 409	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ ( n + 1 ) ) gcd B ) = ( B gcd ( A ^ ( n + 1 ) ) ) )
61		gcdcom 11906	. . . . . . . . . . . . 13 |- ( ( ( A ^ n ) e. ZZ /\ B e. ZZ ) -> ( ( A ^ n ) gcd B ) = ( B gcd ( A ^ n ) ) )
62	34, 58, 61	syl2anc 409	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ n ) gcd B ) = ( B gcd ( A ^ n ) ) )
63	51, 60, 62	3eqtr4d 2208	. . . . . . . . . . 11 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ ( n + 1 ) ) gcd B ) = ( ( A ^ n ) gcd B ) )
64	63	eqeq1d 2174	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( ( A ^ ( n + 1 ) ) gcd B ) = 1 <-> ( ( A ^ n ) gcd B ) = 1 ) )
65	64	biimprd 157	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( ( A ^ n ) gcd B ) = 1 -> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) )
66	23, 65	sylanbr 283	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( ( A ^ n ) gcd B ) = 1 -> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) )
67	66	an32s 558	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ n e. NN ) -> ( ( ( A ^ n ) gcd B ) = 1 -> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) )
68	67	expcom 115	. . . . . 6 |- ( n e. NN -> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( ( A ^ n ) gcd B ) = 1 -> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) ) )
69	68	a2d 26	. . . . 5 |- ( n e. NN -> ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ n ) gcd B ) = 1 ) -> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) ) )
70	4, 8, 12, 16, 22, 69	nnind 8873	. . . 4 |- ( N e. NN -> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ N ) gcd B ) = 1 ) )
71	70	expd 256	. . 3 |- ( N e. NN -> ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) ) )
72	71	com12 30	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( N e. NN -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) ) )
73	72	3impia 1190	1 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842   1c1 7754   + caddc 7756   x. cmul 7758   NNcn 8857   NN0cn0 9114   ZZcz 9191   ^cexp 10454   gcd cgcd 11875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  rppwr  11961  logbgcd1irr  13525  logbgcd1irraplemexp  13526  lgsne0  13579  2sqlem8  13599