Theorem prmdvdsexp 12870

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 6066	. . . . . . 7 |- ( m = 1 -> ( A ^ m ) = ( A ^ 1 ) )
2	1	breq2d 4126	. . . . . 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 6066	. . . . . . 7 |- ( m = k -> ( A ^ m ) = ( A ^ k ) )
6	5	breq2d 4126	. . . . . 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 6066	. . . . . . 7 |- ( m = ( k + 1 ) -> ( A ^ m ) = ( A ^ ( k + 1 ) ) )
10	9	breq2d 4126	. . . . . 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 6066	. . . . . . 7 |- ( m = N -> ( A ^ m ) = ( A ^ N ) )
14	13	breq2d 4126	. . . . . 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 9599	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
18	17	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ ) -> A e. CC )
19	18	exp1d 11055	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( A ^ 1 ) = A )
20	19	breq2d 4126	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ 1 ) <-> P || A ) )
21		nnnn0 9520	. . . . . . . . . 10 |- ( k e. NN -> k e. NN0 )
22		expp1 10932	. . . . . . . . . 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 4126	. . . . . . . 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 10940	. . . . . . . . . 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 529	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ k e. NN ) -> A e. ZZ )
30		euclemma 12868	. . . . . . . . 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 1274	. . . . . . . 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 800	. . . . . . . . 9 |- ( ( P || ( A ^ k ) <-> P || A ) -> ( ( P || ( A ^ k ) \/ P || A ) <-> ( P || A \/ P || A ) ) )
34		oridm 765	. . . . . . . . 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 9270	. . 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 1221	1 |- ( ( P e. Prime /\ A e. ZZ /\ N e. NN ) -> ( P || ( A ^ N ) <-> P || A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   CCcc 8141   1c1 8144   + caddc 8146   x. cmul 8148   NNcn 9254   NN0cn0 9513   ZZcz 9594   ^cexp 10924   || cdvds 12498   Primecprime 12829

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-prm 12830

This theorem is referenced by:  prmdvdsexpb  12871  rpexp  12875  pythagtriplem4  12991  lgslem4  16002  2sqlem3  16116