Theorem prmdvdsexp 12585

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 5975	. . . . . . 7 |- ( m = 1 -> ( A ^ m ) = ( A ^ 1 ) )
2	1	breq2d 4071	. . . . . 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 5975	. . . . . . 7 |- ( m = k -> ( A ^ m ) = ( A ^ k ) )
6	5	breq2d 4071	. . . . . 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 5975	. . . . . . 7 |- ( m = ( k + 1 ) -> ( A ^ m ) = ( A ^ ( k + 1 ) ) )
10	9	breq2d 4071	. . . . . 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 5975	. . . . . . 7 |- ( m = N -> ( A ^ m ) = ( A ^ N ) )
14	13	breq2d 4071	. . . . . 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 9412	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
18	17	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ ) -> A e. CC )
19	18	exp1d 10850	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( A ^ 1 ) = A )
20	19	breq2d 4071	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A ^ 1 ) <-> P || A ) )
21		nnnn0 9337	. . . . . . . . . 10 |- ( k e. NN -> k e. NN0 )
22		expp1 10728	. . . . . . . . . 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 4071	. . . . . . . 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 10736	. . . . . . . . . 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 12583	. . . . . . . . 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 1250	. . . . . . . 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 794	. . . . . . . . 9 |- ( ( P || ( A ^ k ) <-> P || A ) -> ( ( P || ( A ^ k ) \/ P || A ) <-> ( P || A \/ P || A ) ) )
34		oridm 759	. . . . . . . . 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 9087	. . 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 1197	1 |- ( ( P e. Prime /\ A e. ZZ /\ N e. NN ) -> ( P || ( A ^ N ) <-> P || A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   CCcc 7958   1c1 7961   + caddc 7963   x. cmul 7965   NNcn 9071   NN0cn0 9330   ZZcz 9407   ^cexp 10720   || cdvds 12213   Primecprime 12544

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545

This theorem is referenced by:  prmdvdsexpb  12586  rpexp  12590  pythagtriplem4  12706  lgslem4  15595  2sqlem3  15709