Theorem rppwr 12598

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

Assertion

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

Proof of Theorem rppwr

Step	Hyp	Ref	Expression
1		simpl1 1026	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> A e. NN )
2		simpl2 1027	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> B e. NN )
3		simpl3 1028	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> N e. NN )
4	3	nnnn0d 9454	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> N e. NN0 )
5	2, 4	nnexpcld 10956	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( B ^ N ) e. NN )
6	1, 5, 3	3jca 1203	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( A e. NN /\ ( B ^ N ) e. NN /\ N e. NN ) )
7	1	nnzd 9600	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> A e. ZZ )
8	5	nnzd 9600	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( B ^ N ) e. ZZ )
9		gcdcom 12543	. . . . 5 |- ( ( A e. ZZ /\ ( B ^ N ) e. ZZ ) -> ( A gcd ( B ^ N ) ) = ( ( B ^ N ) gcd A ) )
10	7, 8, 9	syl2anc 411	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( A gcd ( B ^ N ) ) = ( ( B ^ N ) gcd A ) )
11	2, 1, 3	3jca 1203	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( B e. NN /\ A e. NN /\ N e. NN ) )
12		nnz 9497	. . . . . . . . 9 |- ( A e. NN -> A e. ZZ )
13	12	3ad2ant1 1044	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> A e. ZZ )
14		nnz 9497	. . . . . . . . 9 |- ( B e. NN -> B e. ZZ )
15	14	3ad2ant2 1045	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> B e. ZZ )
16		gcdcom 12543	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
17	13, 15, 16	syl2anc 411	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( A gcd B ) = ( B gcd A ) )
18	17	eqeq1d 2240	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 <-> ( B gcd A ) = 1 ) )
19	18	biimpa 296	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( B gcd A ) = 1 )
20		rplpwr 12597	. . . . 5 |- ( ( B e. NN /\ A e. NN /\ N e. NN ) -> ( ( B gcd A ) = 1 -> ( ( B ^ N ) gcd A ) = 1 ) )
21	11, 19, 20	sylc 62	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( B ^ N ) gcd A ) = 1 )
22	10, 21	eqtrd 2264	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( A gcd ( B ^ N ) ) = 1 )
23		rplpwr 12597	. . 3 |- ( ( A e. NN /\ ( B ^ N ) e. NN /\ N e. NN ) -> ( ( A gcd ( B ^ N ) ) = 1 -> ( ( A ^ N ) gcd ( B ^ N ) ) = 1 ) )
24	6, 22, 23	sylc 62	. 2 |- ( ( ( A e. NN /\ B e. NN /\ N e. NN ) /\ ( A gcd B ) = 1 ) -> ( ( A ^ N ) gcd ( B ^ N ) ) = 1 )
25	24	ex 115	1 |- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd ( B ^ N ) ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202  (class class class)co 6017   1c1 8032   NNcn 9142   ZZcz 9478   ^cexp 10799   gcd cgcd 12523

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524

This theorem is referenced by:  sqgcd  12599