Theorem prmdvdsexp 12341

Description: A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.)

Assertion

Ref	Expression
prmdvdsexp	|- ( ( P e. Prime /\ A e. ZZ /\ N e. NN ) -> ( P || ( A ^ N ) <-> P || A ) )

Proof of Theorem prmdvdsexp

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

Step	Hyp	Ref	Expression
1		oveq2 5933	. . . . . . 7 |- ( m = 1 -> ( A ^ m ) = ( A ^ 1 ) )
2	1	breq2d 4046	. . . . . 6 |- ( m = 1 -> ( P || ( A ^ m ) <-> P || ( A ^ 1 ) ) )
3	2	bibi1d 233	. . . . 5 |- ( m = 1 -> ( ( P || ( A ^ m ) <-> P || A ) <-> ( P || ( A ^ 1 ) <-> P || A ) ) )
4	3	imbi2d 230	. . . 4 |- ( m = 1 -> ( ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ m ) <-> P || A ) ) <-> ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ 1 ) <-> P || A ) ) ) )
5		oveq2 5933	. . . . . . 7 |- ( m = k -> ( A ^ m ) = ( A ^ k ) )
6	5	breq2d 4046	. . . . . 6 |- ( m = k -> ( P || ( A ^ m ) <-> P || ( A ^ k ) ) )
7	6	bibi1d 233	. . . . 5 |- ( m = k -> ( ( P || ( A ^ m ) <-> P || A ) <-> ( P || ( A ^ k ) <-> P || A ) ) )
8	7	imbi2d 230	. . . 4 |- ( m = k -> ( ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ m ) <-> P || A ) ) <-> ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ k ) <-> P || A ) ) ) )
9		oveq2 5933	. . . . . . 7 |- ( m = ( k + 1 ) -> ( A ^ m ) = ( A ^ ( k + 1 ) ) )
10	9	breq2d 4046	. . . . . 6 |- ( m = ( k + 1 ) -> ( P || ( A ^ m ) <-> P || ( A ^ ( k + 1 ) ) ) )
11	10	bibi1d 233	. . . . 5 |- ( m = ( k + 1 ) -> ( ( P || ( A ^ m ) <-> P || A ) <-> ( P || ( A ^ ( k + 1 ) ) <-> P || A ) ) )
12	11	imbi2d 230	. . . 4 |- ( m = ( k + 1 ) -> ( ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ m ) <-> P || A ) ) <-> ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ ( k + 1 ) ) <-> P || A ) ) ) )
13		oveq2 5933	. . . . . . 7 |- ( m = N -> ( A ^ m ) = ( A ^ N ) )
14	13	breq2d 4046	. . . . . 6 |- ( m = N -> ( P || ( A ^ m ) <-> P || ( A ^ N ) ) )
15	14	bibi1d 233	. . . . 5 |- ( m = N -> ( ( P || ( A ^ m ) <-> P || A ) <-> ( P || ( A ^ N ) <-> P || A ) ) )
16	15	imbi2d 230	. . . 4 |- ( m = N -> ( ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ m ) <-> P || A ) ) <-> ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ N ) <-> P || A ) ) ) )
17		zcn 9348	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
18	17	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ ) -> A e. CC )
19	18	exp1d 10777	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( A ^ 1 ) = A )
20	19	breq2d 4046	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ 1 ) <-> P || A ) )
21		nnnn0 9273	. . . . . . . . . 10 |- ( k e. NN -> k e. NN0 )
22		expp1 10655	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
23	18, 21, 22	syl2an 289	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
24	23	breq2d 4046	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> ( P || ( A ^ ( k + 1 ) ) <-> P || ( ( A ^ k ) x. A ) ) )
25		simpll 527	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> P e. Prime )
26		simpr 110	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ ) -> A e. ZZ )
27		zexpcl 10663	. . . . . . . . . 10 |- ( ( A e. ZZ /\ k e. NN0 ) -> ( A ^ k ) e. ZZ )
28	26, 21, 27	syl2an 289	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> ( A ^ k ) e. ZZ )
29		simplr 528	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> A e. ZZ )
30		euclemma 12339	. . . . . . . . 9 |- ( ( P e. Prime /\ ( A ^ k ) e. ZZ /\ A e. ZZ ) -> ( P || ( ( A ^ k ) x. A ) <-> ( P || ( A ^ k ) \/ P || A ) ) )
31	25, 28, 29, 30	syl3anc 1249	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> ( P || ( ( A ^ k ) x. A ) <-> ( P || ( A ^ k ) \/ P || A ) ) )
32	24, 31	bitrd 188	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> ( P || ( A ^ ( k + 1 ) ) <-> ( P || ( A ^ k ) \/ P || A ) ) )
33		orbi1 793	. . . . . . . . 9 |- ( ( P || ( A ^ k ) <-> P || A ) -> ( ( P || ( A ^ k ) \/ P || A ) <-> ( P || A \/ P || A ) ) )
34		oridm 758	. . . . . . . . 9 |- ( ( P || A \/ P || A ) <-> P || A )
35	33, 34	bitrdi 196	. . . . . . . 8 |- ( ( P || ( A ^ k ) <-> P || A ) -> ( ( P || ( A ^ k ) \/ P || A ) <-> P || A ) )
36	35	bibi2d 232	. . . . . . 7 |- ( ( P || ( A ^ k ) <-> P || A ) -> ( ( P || ( A ^ ( k + 1 ) ) <-> ( P || ( A ^ k ) \/ P || A ) ) <-> ( P || ( A ^ ( k + 1 ) ) <-> P || A ) ) )
37	32, 36	syl5ibcom 155	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> ( ( P || ( A ^ k ) <-> P || A ) -> ( P || ( A ^ ( k + 1 ) ) <-> P || A ) ) )
38	37	expcom 116	. . . . 5 |- ( k e. NN -> ( ( P e. Prime /\ A e. ZZ ) -> ( ( P || ( A ^ k ) <-> P || A ) -> ( P || ( A ^ ( k + 1 ) ) <-> P || A ) ) ) )
39	38	a2d 26	. . . 4 |- ( k e. NN -> ( ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ k ) <-> P || A ) ) -> ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ ( k + 1 ) ) <-> P || A ) ) ) )
40	4, 8, 12, 16, 20, 39	nnind 9023	. . 3 |- ( N e. NN -> ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ N ) <-> P || A ) ) )
41	40	impcom 125	. 2 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ N e. NN ) -> ( P || ( A ^ N ) <-> P || A ) )
42	41	3impa 1196	1 |- ( ( P e. Prime /\ A e. ZZ /\ N e. NN ) -> ( P || ( A ^ N ) <-> P || A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   1c1 7897   + caddc 7899   x. cmul 7901   NNcn 9007   NN0cn0 9266   ZZcz 9343   ^cexp 10647   || cdvds 11969   Primecprime 12300

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-sup 7059  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146  df-prm 12301

This theorem is referenced by:  prmdvdsexpb  12342  rpexp  12346  pythagtriplem4  12462  lgslem4  15328  2sqlem3  15442