Theorem prmdvdsexp 12686

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 6015	. . . . . . 7 |- ( m = 1 -> ( A ^ m ) = ( A ^ 1 ) )
2	1	breq2d 4095	. . . . . 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 6015	. . . . . . 7 |- ( m = k -> ( A ^ m ) = ( A ^ k ) )
6	5	breq2d 4095	. . . . . 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 6015	. . . . . . 7 |- ( m = ( k + 1 ) -> ( A ^ m ) = ( A ^ ( k + 1 ) ) )
10	9	breq2d 4095	. . . . . 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 6015	. . . . . . 7 |- ( m = N -> ( A ^ m ) = ( A ^ N ) )
14	13	breq2d 4095	. . . . . 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 9462	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
18	17	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ ) -> A e. CC )
19	18	exp1d 10902	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( A ^ 1 ) = A )
20	19	breq2d 4095	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ 1 ) <-> P || A ) )
21		nnnn0 9387	. . . . . . . . . 10 |- ( k e. NN -> k e. NN0 )
22		expp1 10780	. . . . . . . . . 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 4095	. . . . . . . 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 10788	. . . . . . . . . 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 12684	. . . . . . . . 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 1271	. . . . . . . 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 797	. . . . . . . . 9 |- ( ( P || ( A ^ k ) <-> P || A ) -> ( ( P || ( A ^ k ) \/ P || A ) <-> ( P || A \/ P || A ) ) )
34		oridm 762	. . . . . . . . 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 9137	. . 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 1218	1 |- ( ( P e. Prime /\ A e. ZZ /\ N e. NN ) -> ( P || ( A ^ N ) <-> P || A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   CCcc 8008   1c1 8011   + caddc 8013   x. cmul 8015   NNcn 9121   NN0cn0 9380   ZZcz 9457   ^cexp 10772   || cdvds 12314   Primecprime 12645

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7162  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-dvds 12315  df-gcd 12491  df-prm 12646

This theorem is referenced by:  prmdvdsexpb  12687  rpexp  12691  pythagtriplem4  12807  lgslem4  15698  2sqlem3  15812