Theorem rplpwr 12319

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 5951	. . . . . . . 8 |- ( k = 1 -> ( A ^ k ) = ( A ^ 1 ) )
2	1	oveq1d 5958	. . . . . . 7 |- ( k = 1 -> ( ( A ^ k ) gcd B ) = ( ( A ^ 1 ) gcd B ) )
3	2	eqeq1d 2213	. . . . . 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 5951	. . . . . . . 8 |- ( k = n -> ( A ^ k ) = ( A ^ n ) )
6	5	oveq1d 5958	. . . . . . 7 |- ( k = n -> ( ( A ^ k ) gcd B ) = ( ( A ^ n ) gcd B ) )
7	6	eqeq1d 2213	. . . . . 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 5951	. . . . . . . 8 |- ( k = ( n + 1 ) -> ( A ^ k ) = ( A ^ ( n + 1 ) ) )
10	9	oveq1d 5958	. . . . . . 7 |- ( k = ( n + 1 ) -> ( ( A ^ k ) gcd B ) = ( ( A ^ ( n + 1 ) ) gcd B ) )
11	10	eqeq1d 2213	. . . . . 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 5951	. . . . . . . 8 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
14	13	oveq1d 5958	. . . . . . 7 |- ( k = N -> ( ( A ^ k ) gcd B ) = ( ( A ^ N ) gcd B ) )
15	14	eqeq1d 2213	. . . . . 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 9043	. . . . . . . . . 10 |- ( A e. NN -> A e. CC )
18	17	exp1d 10811	. . . . . . . . 9 |- ( A e. NN -> ( A ^ 1 ) = A )
19	18	oveq1d 5958	. . . . . . . 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 2213	. . . . . 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 982	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) <-> ( ( A e. NN /\ B e. NN ) /\ n e. NN ) )
24		simpl1 1002	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. NN )
25	24	nncnd 9049	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> A e. CC )
26		simpl3 1004	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN )
27	26	nnnn0d 9347	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> n e. NN0 )
28	25, 27	expp1d 10817	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( ( A ^ n ) x. A ) )
29		simp1 999	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. NN )
30		nnnn0 9301	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. NN -> n e. NN0 )
31	30	3ad2ant3 1022	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> n e. NN0 )
32	29, 31	nnexpcld 10838	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> ( A ^ n ) e. NN )
33	32	nnzd 9493	. . . . . . . . . . . . . . . . . 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 9495	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ n ) e. CC )
36	35, 25	mulcomd 8093	. . . . . . . . . . . . . . 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 2237	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) = ( A x. ( A ^ n ) ) )
38	37	oveq2d 5959	. . . . . . . . . . . . 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 1003	. . . . . . . . . . . . . 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 9390	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN -> A e. ZZ )
42	41	3ad2ant1 1020	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> A e. ZZ )
43		nnz 9390	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN -> B e. ZZ )
44	43	3ad2ant2 1021	. . . . . . . . . . . . . . . . 17 |- ( ( A e. NN /\ B e. NN /\ n e. NN ) -> B e. ZZ )
45		gcdcom 12265	. . . . . . . . . . . . . . . . 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 2213	. . . . . . . . . . . . . . 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 12318	. . . . . . . . . . . . . 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 1252	. . . . . . . . . . . . 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 2237	. . . . . . . . . . . 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 9047	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> ( n + 1 ) e. NN )
53	52	3ad2ant3 1022	. . . . . . . . . . . . . . . . 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 9347	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( n + 1 ) e. NN0 )
56	24, 55	nnexpcld 10838	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ n e. NN ) /\ ( A gcd B ) = 1 ) -> ( A ^ ( n + 1 ) ) e. NN )
57	56	nnzd 9493	. . . . . . . . . . . . 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 12265	. . . . . . . . . . . . 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 12265	. . . . . . . . . . . . 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 2247	. . . . . . . . . . 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 2213	. . . . . . . . . 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 9051	. . . 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 1202	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 980   = wceq 1372   e. wcel 2175  (class class class)co 5943   1c1 7925   + caddc 7927   x. cmul 7929   NNcn 9035   NN0cn0 9294   ZZcz 9371   ^cexp 10681   gcd cgcd 12245

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-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-sup 7085  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-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

This theorem is referenced by:  rppwr  12320  logbgcd1irr  15410  logbgcd1irraplemexp  15411  lgsne0  15486  2sqlem8  15571