Theorem rplpwr 12829

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 6096	. . . . . . . 8 |- ( k = 1 -> ( A ^ k ) = ( A ^ 1 ) )
2	1	oveq1d 6103	. . . . . . 7 |- ( k = 1 -> ( ( A ^ k ) gcd B ) = ( ( A ^ 1 ) gcd B ) )
3	2	eqeq1d 2247	. . . . . 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 6096	. . . . . . . 8 |- ( k = n -> ( A ^ k ) = ( A ^ n ) )
6	5	oveq1d 6103	. . . . . . 7 |- ( k = n -> ( ( A ^ k ) gcd B ) = ( ( A ^ n ) gcd B ) )
7	6	eqeq1d 2247	. . . . . 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 6096	. . . . . . . 8 |- ( k = ( n + 1 ) -> ( A ^ k ) = ( A ^ ( n + 1 ) ) )
10	9	oveq1d 6103	. . . . . . 7 |- ( k = ( n + 1 ) -> ( ( A ^ k ) gcd B ) = ( ( A ^ ( n + 1 ) ) gcd B ) )
11	10	eqeq1d 2247	. . . . . 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 6096	. . . . . . . 8 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
14	13	oveq1d 6103	. . . . . . 7 |- ( k = N -> ( ( A ^ k ) gcd B ) = ( ( A ^ N ) gcd B ) )
15	14	eqeq1d 2247	. . . . . 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 9321	. . . . . . . . . 10 |- ( A e. NN -> A e. CC )
18	17	exp1d 11127	. . . . . . . . 9 |- ( A e. NN -> ( A ^ 1 ) = A )
19	18	oveq1d 6103	. . . . . . . 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 2247	. . . . . 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 1011	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) <-> ( ( A e. NN /\ B e. NN ) /\ n e. NN ) )
24		simpl1 1031	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. NN )
25	24	nncnd 9327	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. CC )
26		simpl3 1033	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN )
27	26	nnnn0d 9631	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN0 )
28	25, 27	expp1d 11133	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( ( A ^ n ) x. A ) )
29		simp1 1028	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. NN )
30		nnnn0 9581	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. NN -> n e. NN0 )
31	30	3ad2ant3 1051	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> n e. NN0 )
32	29, 31	nnexpcld 11154	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A ^ n ) e. NN )
33	32	nnzd 9778	. . . . . . . . . . . . . . . . . 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 9780	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. CC )
36	35, 25	mulcomd 8353	. . . . . . . . . . . . . . 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 2271	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( A x. ( A ^ n ) ) )
38	37	oveq2d 6104	. . . . . . . . . . . . 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 1032	. . . . . . . . . . . . . 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 9674	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN -> A e. ZZ )
42	41	3ad2ant1 1049	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. ZZ )
43		nnz 9674	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN -> B e. ZZ )
44	43	3ad2ant2 1050	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> B e. ZZ )
45		gcdcom 12775	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
46	42, 44, 45	syl2anc 415	. . . . . . . . . . . . . . . 16 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A gcd B ) = ( B gcd A ) )
47	46	eqeq1d 2247	. . . . . . . . . . . . . . 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 12828	. . . . . . . . . . . . . 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 1281	. . . . . . . . . . . . 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 2271	. . . . . . . . . . . 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 9325	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> ( n + 1 ) e. NN )
53	52	3ad2ant3 1051	. . . . . . . . . . . . . . . . 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 9631	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( n + 1 ) e. NN0 )
56	24, 55	nnexpcld 11154	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) e. NN )
57	56	nnzd 9778	. . . . . . . . . . . . 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 12775	. . . . . . . . . . . . 13 |- ( ( ( A ^ ( n + 1 ) ) e. ZZ /\ B e. ZZ ) -> ( ( A ^ ( n + 1 ) ) gcd B ) = ( B gcd ( A ^ ( n + 1 ) ) ) )
60	57, 58, 59	syl2anc 415	. . . . . . . . . . . 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 12775	. . . . . . . . . . . . 13 |- ( ( ( A ^ n ) e. ZZ /\ B e. ZZ ) -> ( ( A ^ n ) gcd B ) = ( B gcd ( A ^ n ) ) )
62	34, 58, 61	syl2anc 415	. . . . . . . . . . . 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 2281	. . . . . . . . . . 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 2247	. . . . . . . . . 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 574	. . . . . . 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 9329	. . . 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 1231	1 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209  (class class class)co 6088   1c1 8186   + caddc 8188   x. cmul 8190   NNcn 9313   NN0cn0 9574   ZZcz 9655   ^cexp 10996   gcd cgcd 12755

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-gcd 12756

This theorem is used by:  rppwr  12830  logbgcd1irr  16203  logbgcd1irraplemexp  16204  lgsne0  16362  2sqlem8  16447