Theorem pcexp 12263

Description: Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)

Assertion

Ref	Expression
pcexp	|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ N e. ZZ ) -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) )

Proof of Theorem pcexp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5861	. . . . 5 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
2	1	oveq2d 5869	. . . 4 |- ( x = 0 -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ 0 ) ) )
3		oveq1 5860	. . . 4 |- ( x = 0 -> ( x x. ( P pCnt A ) ) = ( 0 x. ( P pCnt A ) ) )
4	2, 3	eqeq12d 2185	. . 3 |- ( x = 0 -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) ) )
5		oveq2 5861	. . . . 5 |- ( x = y -> ( A ^ x ) = ( A ^ y ) )
6	5	oveq2d 5869	. . . 4 |- ( x = y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ y ) ) )
7		oveq1 5860	. . . 4 |- ( x = y -> ( x x. ( P pCnt A ) ) = ( y x. ( P pCnt A ) ) )
8	6, 7	eqeq12d 2185	. . 3 |- ( x = y -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) ) )
9		oveq2 5861	. . . . 5 |- ( x = ( y + 1 ) -> ( A ^ x ) = ( A ^ ( y + 1 ) ) )
10	9	oveq2d 5869	. . . 4 |- ( x = ( y + 1 ) -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ ( y + 1 ) ) ) )
11		oveq1 5860	. . . 4 |- ( x = ( y + 1 ) -> ( x x. ( P pCnt A ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) )
12	10, 11	eqeq12d 2185	. . 3 |- ( x = ( y + 1 ) -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ ( y + 1 ) ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) ) )
13		oveq2 5861	. . . . 5 |- ( x = -u y -> ( A ^ x ) = ( A ^ -u y ) )
14	13	oveq2d 5869	. . . 4 |- ( x = -u y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ -u y ) ) )
15		oveq1 5860	. . . 4 |- ( x = -u y -> ( x x. ( P pCnt A ) ) = ( -u y x. ( P pCnt A ) ) )
16	14, 15	eqeq12d 2185	. . 3 |- ( x = -u y -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ -u y ) ) = ( -u y x. ( P pCnt A ) ) ) )
17		oveq2 5861	. . . . 5 |- ( x = N -> ( A ^ x ) = ( A ^ N ) )
18	17	oveq2d 5869	. . . 4 |- ( x = N -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ N ) ) )
19		oveq1 5860	. . . 4 |- ( x = N -> ( x x. ( P pCnt A ) ) = ( N x. ( P pCnt A ) ) )
20	18, 19	eqeq12d 2185	. . 3 |- ( x = N -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
21		pc1 12259	. . . . 5 |- ( P e. Prime -> ( P pCnt 1 ) = 0 )
22	21	adantr 274	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt 1 ) = 0 )
23		qcn 9593	. . . . . . 7 |- ( A e. QQ -> A e. CC )
24	23	ad2antrl 487	. . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> A e. CC )
25	24	exp0d 10603	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( A ^ 0 ) = 1 )
26	25	oveq2d 5869	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( P pCnt 1 ) )
27		pcqcl 12260	. . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. ZZ )
28	27	zcnd 9335	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. CC )
29	28	mul02d 8311	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( 0 x. ( P pCnt A ) ) = 0 )
30	22, 26, 29	3eqtr4d 2213	. . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) )
31		oveq1 5860	. . . . 5 |- ( ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) -> ( ( P pCnt ( A ^ y ) ) + ( P pCnt A ) ) = ( ( y x. ( P pCnt A ) ) + ( P pCnt A ) ) )
32		expp1 10483	. . . . . . . . 9 |- ( ( A e. CC /\ y e. NN0 ) -> ( A ^ ( y + 1 ) ) = ( ( A ^ y ) x. A ) )
33	24, 32	sylan 281	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ ( y + 1 ) ) = ( ( A ^ y ) x. A ) )
34	33	oveq2d 5869	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( P pCnt ( A ^ ( y + 1 ) ) ) = ( P pCnt ( ( A ^ y ) x. A ) ) )
35		simpll 524	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> P e. Prime )
36		simplrl 530	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A e. QQ )
37		simplrr 531	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A =/= 0 )
38		nn0z 9232	. . . . . . . . . 10 |- ( y e. NN0 -> y e. ZZ )
39	38	adantl 275	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> y e. ZZ )
40		qexpclz 10497	. . . . . . . . 9 |- ( ( A e. QQ /\ A =/= 0 /\ y e. ZZ ) -> ( A ^ y ) e. QQ )
41	36, 37, 39, 40	syl3anc 1233	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) e. QQ )
42	24	adantr 274	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A e. CC )
43		0z 9223	. . . . . . . . . . . . 13 |- 0 e. ZZ
44		zq 9585	. . . . . . . . . . . . 13 |- ( 0 e. ZZ -> 0 e. QQ )
45	43, 44	ax-mp 5	. . . . . . . . . . . 12 |- 0 e. QQ
46		qapne 9598	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ 0 e. QQ ) -> ( A # 0 <-> A =/= 0 ) )
47	36, 45, 46	sylancl 411	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A # 0 <-> A =/= 0 ) )
48	37, 47	mpbird 166	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A # 0 )
49	42, 48, 39	expap0d 10615	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) # 0 )
50		qapne 9598	. . . . . . . . . 10 |- ( ( ( A ^ y ) e. QQ /\ 0 e. QQ ) -> ( ( A ^ y ) # 0 <-> ( A ^ y ) =/= 0 ) )
51	41, 45, 50	sylancl 411	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( ( A ^ y ) # 0 <-> ( A ^ y ) =/= 0 ) )
52	49, 51	mpbid 146	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) =/= 0 )
53		pcqmul 12257	. . . . . . . 8 |- ( ( P e. Prime /\ ( ( A ^ y ) e. QQ /\ ( A ^ y ) =/= 0 ) /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( ( A ^ y ) x. A ) ) = ( ( P pCnt ( A ^ y ) ) + ( P pCnt A ) ) )
54	35, 41, 52, 36, 37, 53	syl122anc 1242	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( P pCnt ( ( A ^ y ) x. A ) ) = ( ( P pCnt ( A ^ y ) ) + ( P pCnt A ) ) )
55	34, 54	eqtrd 2203	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( P pCnt ( A ^ ( y + 1 ) ) ) = ( ( P pCnt ( A ^ y ) ) + ( P pCnt A ) ) )
56		nn0cn 9145	. . . . . . . 8 |- ( y e. NN0 -> y e. CC )
57	56	adantl 275	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> y e. CC )
58	28	adantr 274	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( P pCnt A ) e. CC )
59	57, 58	adddirp1d 7946	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( ( y + 1 ) x. ( P pCnt A ) ) = ( ( y x. ( P pCnt A ) ) + ( P pCnt A ) ) )
60	55, 59	eqeq12d 2185	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( ( P pCnt ( A ^ ( y + 1 ) ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) <-> ( ( P pCnt ( A ^ y ) ) + ( P pCnt A ) ) = ( ( y x. ( P pCnt A ) ) + ( P pCnt A ) ) ) )
61	31, 60	syl5ibr 155	. . . 4 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) -> ( P pCnt ( A ^ ( y + 1 ) ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) ) )
62	61	ex 114	. . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( y e. NN0 -> ( ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) -> ( P pCnt ( A ^ ( y + 1 ) ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) ) ) )
63		negeq 8112	. . . . 5 |- ( ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) -> -u ( P pCnt ( A ^ y ) ) = -u ( y x. ( P pCnt A ) ) )
64	24	adantr 274	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> A e. CC )
65		nnnn0 9142	. . . . . . . . . 10 |- ( y e. NN -> y e. NN0 )
66	65, 48	sylan2 284	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> A # 0 )
67	65	adantl 275	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> y e. NN0 )
68		expnegap0 10484	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ y e. NN0 ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
69	64, 66, 67, 68	syl3anc 1233	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
70	69	oveq2d 5869	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( P pCnt ( A ^ -u y ) ) = ( P pCnt ( 1 / ( A ^ y ) ) ) )
71		simpll 524	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> P e. Prime )
72	65, 41	sylan2 284	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ y ) e. QQ )
73	65, 52	sylan2 284	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ y ) =/= 0 )
74		pcrec 12262	. . . . . . . 8 |- ( ( P e. Prime /\ ( ( A ^ y ) e. QQ /\ ( A ^ y ) =/= 0 ) ) -> ( P pCnt ( 1 / ( A ^ y ) ) ) = -u ( P pCnt ( A ^ y ) ) )
75	71, 72, 73, 74	syl12anc 1231	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( P pCnt ( 1 / ( A ^ y ) ) ) = -u ( P pCnt ( A ^ y ) ) )
76	70, 75	eqtrd 2203	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( P pCnt ( A ^ -u y ) ) = -u ( P pCnt ( A ^ y ) ) )
77		nncn 8886	. . . . . . 7 |- ( y e. NN -> y e. CC )
78		mulneg1 8314	. . . . . . 7 |- ( ( y e. CC /\ ( P pCnt A ) e. CC ) -> ( -u y x. ( P pCnt A ) ) = -u ( y x. ( P pCnt A ) ) )
79	77, 28, 78	syl2anr 288	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( -u y x. ( P pCnt A ) ) = -u ( y x. ( P pCnt A ) ) )
80	76, 79	eqeq12d 2185	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( ( P pCnt ( A ^ -u y ) ) = ( -u y x. ( P pCnt A ) ) <-> -u ( P pCnt ( A ^ y ) ) = -u ( y x. ( P pCnt A ) ) ) )
81	63, 80	syl5ibr 155	. . . 4 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) -> ( P pCnt ( A ^ -u y ) ) = ( -u y x. ( P pCnt A ) ) ) )
82	81	ex 114	. . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( y e. NN -> ( ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) -> ( P pCnt ( A ^ -u y ) ) = ( -u y x. ( P pCnt A ) ) ) ) )
83	4, 8, 12, 16, 20, 30, 62, 82	zindd 9330	. 2 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( N e. ZZ -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
84	83	3impia 1195	1 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ N e. ZZ ) -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   -ucneg 8091   # cap 8500   / cdiv 8589   NNcn 8878   NN0cn0 9135   ZZcz 9212   QQcq 9578   ^cexp 10475   Primecprime 12061   pCnt cpc 12238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062  df-pc 12239

This theorem is referenced by:  qexpz  12304  expnprm  12305