Theorem rplpwr 12543

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 6008	. . . . . . . 8 |- ( k = 1 -> ( A ^ k ) = ( A ^ 1 ) )
2	1	oveq1d 6015	. . . . . . 7 |- ( k = 1 -> ( ( A ^ k ) gcd B ) = ( ( A ^ 1 ) gcd B ) )
3	2	eqeq1d 2238	. . . . . 6 |- ( k = 1 -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ 1 ) gcd B ) = 1 ) )
4	3	imbi2d 230	. . . . 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 6008	. . . . . . . 8 |- ( k = n -> ( A ^ k ) = ( A ^ n ) )
6	5	oveq1d 6015	. . . . . . 7 |- ( k = n -> ( ( A ^ k ) gcd B ) = ( ( A ^ n ) gcd B ) )
7	6	eqeq1d 2238	. . . . . 6 |- ( k = n -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ n ) gcd B ) = 1 ) )
8	7	imbi2d 230	. . . . 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 6008	. . . . . . . 8 |- ( k = ( n + 1 ) -> ( A ^ k ) = ( A ^ ( n + 1 ) ) )
10	9	oveq1d 6015	. . . . . . 7 |- ( k = ( n + 1 ) -> ( ( A ^ k ) gcd B ) = ( ( A ^ ( n + 1 ) ) gcd B ) )
11	10	eqeq1d 2238	. . . . . 6 |- ( k = ( n + 1 ) -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ ( n + 1 ) ) gcd B ) = 1 ) )
12	11	imbi2d 230	. . . . 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 6008	. . . . . . . 8 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
14	13	oveq1d 6015	. . . . . . 7 |- ( k = N -> ( ( A ^ k ) gcd B ) = ( ( A ^ N ) gcd B ) )
15	14	eqeq1d 2238	. . . . . 6 |- ( k = N -> ( ( ( A ^ k ) gcd B ) = 1 <-> ( ( A ^ N ) gcd B ) = 1 ) )
16	15	imbi2d 230	. . . . 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 9114	. . . . . . . . . 10 |- ( A e. NN -> A e. CC )
18	17	exp1d 10885	. . . . . . . . 9 |- ( A e. NN -> ( A ^ 1 ) = A )
19	18	oveq1d 6015	. . . . . . . 8 |- ( A e. NN -> ( ( A ^ 1 ) gcd B ) = ( A gcd B ) )
20	19	adantr 276	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( ( A ^ 1 ) gcd B ) = ( A gcd B ) )
21	20	eqeq1d 2238	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( ( ( A ^ 1 ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) )
22	21	biimpar 297	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ 1 ) gcd B ) = 1 )
23		df-3an 1004	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) <-> ( ( A e. NN /\ B e. NN ) /\ n e. NN ) )
24		simpl1 1024	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. NN )
25	24	nncnd 9120	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. CC )
26		simpl3 1026	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN )
27	26	nnnn0d 9418	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN0 )
28	25, 27	expp1d 10891	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( ( A ^ n ) x. A ) )
29		simp1 1021	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. NN )
30		nnnn0 9372	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. NN -> n e. NN0 )
31	30	3ad2ant3 1044	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> n e. NN0 )
32	29, 31	nnexpcld 10912	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A ^ n ) e. NN )
33	32	nnzd 9564	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A ^ n ) e. ZZ )
34	33	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. ZZ )
35	34	zcnd 9566	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. CC )
36	35, 25	mulcomd 8164	. . . . . . . . . . . . . . 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 2262	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( A x. ( A ^ n ) ) )
38	37	oveq2d 6016	. . . . . . . . . . . . 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 1025	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> B e. NN )
40	32	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. NN )
41		nnz 9461	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN -> A e. ZZ )
42	41	3ad2ant1 1042	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. ZZ )
43		nnz 9461	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN -> B e. ZZ )
44	43	3ad2ant2 1043	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> B e. ZZ )
45		gcdcom 12489	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
46	42, 44, 45	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A gcd B ) = ( B gcd A ) )
47	46	eqeq1d 2238	. . . . . . . . . . . . . . 15 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( ( A gcd B ) = 1 <-> ( B gcd A ) = 1 ) )
48	47	biimpa 296	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( B gcd A ) = 1 )
49		rpmulgcd 12542	. . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . 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 2262	. . . . . . . . . . . 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 9118	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> ( n + 1 ) e. NN )
53	52	3ad2ant3 1044	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( n + 1 ) e. NN )
54	53	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( n + 1 ) e. NN )
55	54	nnnn0d 9418	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( n + 1 ) e. NN0 )
56	24, 55	nnexpcld 10912	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) e. NN )
57	56	nnzd 9564	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) e. ZZ )
58	44	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> B e. ZZ )
59		gcdcom 12489	. . . . . . . . . . . . 13 |- ( ( ( A ^ ( n + 1 ) ) e. ZZ /\ B e. ZZ ) -> ( ( A ^ ( n + 1 ) ) gcd B ) = ( B gcd ( A ^ ( n + 1 ) ) ) )
60	57, 58, 59	syl2anc 411	. . . . . . . . . . . 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 12489	. . . . . . . . . . . . 13 |- ( ( ( A ^ n ) e. ZZ /\ B e. ZZ ) -> ( ( A ^ n ) gcd B ) = ( B gcd ( A ^ n ) ) )
62	34, 58, 61	syl2anc 411	. . . . . . . . . . . 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 2272	. . . . . . . . . . 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 2238	. . . . . . . . . 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 158	. . . . . . . . 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 285	. . . . . . . 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 568	. . . . . . 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 116	. . . . . 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 9122	. . . 4 |- ( N e. NN -> ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ N ) gcd B ) = 1 ) )
71	70	expd 258	. . 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 1224	1 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6000   1c1 7996   + caddc 7998   x. cmul 8000   NNcn 9106   NN0cn0 9365   ZZcz 9442   ^cexp 10755   gcd cgcd 12469

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-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  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-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

This theorem is referenced by:  rppwr  12544  logbgcd1irr  15635  logbgcd1irraplemexp  15636  lgsne0  15711  2sqlem8  15796