Theorem pcexp 12292

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 5877	. . . . 5 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
2	1	oveq2d 5885	. . . 4 |- ( x = 0 -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ 0 ) ) )
3		oveq1 5876	. . . 4 |- ( x = 0 -> ( x x. ( P pCnt A ) ) = ( 0 x. ( P pCnt A ) ) )
4	2, 3	eqeq12d 2192	. . 3 |- ( x = 0 -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) ) )
5		oveq2 5877	. . . . 5 |- ( x = y -> ( A ^ x ) = ( A ^ y ) )
6	5	oveq2d 5885	. . . 4 |- ( x = y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ y ) ) )
7		oveq1 5876	. . . 4 |- ( x = y -> ( x x. ( P pCnt A ) ) = ( y x. ( P pCnt A ) ) )
8	6, 7	eqeq12d 2192	. . 3 |- ( x = y -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) ) )
9		oveq2 5877	. . . . 5 |- ( x = ( y + 1 ) -> ( A ^ x ) = ( A ^ ( y + 1 ) ) )
10	9	oveq2d 5885	. . . 4 |- ( x = ( y + 1 ) -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ ( y + 1 ) ) ) )
11		oveq1 5876	. . . 4 |- ( x = ( y + 1 ) -> ( x x. ( P pCnt A ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) )
12	10, 11	eqeq12d 2192	. . 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 5877	. . . . 5 |- ( x = -u y -> ( A ^ x ) = ( A ^ -u y ) )
14	13	oveq2d 5885	. . . 4 |- ( x = -u y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ -u y ) ) )
15		oveq1 5876	. . . 4 |- ( x = -u y -> ( x x. ( P pCnt A ) ) = ( -u y x. ( P pCnt A ) ) )
16	14, 15	eqeq12d 2192	. . 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 5877	. . . . 5 |- ( x = N -> ( A ^ x ) = ( A ^ N ) )
18	17	oveq2d 5885	. . . 4 |- ( x = N -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ N ) ) )
19		oveq1 5876	. . . 4 |- ( x = N -> ( x x. ( P pCnt A ) ) = ( N x. ( P pCnt A ) ) )
20	18, 19	eqeq12d 2192	. . 3 |- ( x = N -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
21		pc1 12288	. . . . 5 |- ( P e. Prime -> ( P pCnt 1 ) = 0 )
22	21	adantr 276	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt 1 ) = 0 )
23		qcn 9623	. . . . . . 7 |- ( A e. QQ -> A e. CC )
24	23	ad2antrl 490	. . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> A e. CC )
25	24	exp0d 10633	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( A ^ 0 ) = 1 )
26	25	oveq2d 5885	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( P pCnt 1 ) )
27		pcqcl 12289	. . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. ZZ )
28	27	zcnd 9365	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. CC )
29	28	mul02d 8339	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( 0 x. ( P pCnt A ) ) = 0 )
30	22, 26, 29	3eqtr4d 2220	. . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) )
31		oveq1 5876	. . . . 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 10513	. . . . . . . . 9 |- ( ( A e. CC /\ y e. NN0 ) -> ( A ^ ( y + 1 ) ) = ( ( A ^ y ) x. A ) )
33	24, 32	sylan 283	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ ( y + 1 ) ) = ( ( A ^ y ) x. A ) )
34	33	oveq2d 5885	. . . . . . 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 527	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> P e. Prime )
36		simplrl 535	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A e. QQ )
37		simplrr 536	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A =/= 0 )
38		nn0z 9262	. . . . . . . . . 10 |- ( y e. NN0 -> y e. ZZ )
39	38	adantl 277	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> y e. ZZ )
40		qexpclz 10527	. . . . . . . . 9 |- ( ( A e. QQ /\ A =/= 0 /\ y e. ZZ ) -> ( A ^ y ) e. QQ )
41	36, 37, 39, 40	syl3anc 1238	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) e. QQ )
42	24	adantr 276	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A e. CC )
43		0z 9253	. . . . . . . . . . . . 13 |- 0 e. ZZ
44		zq 9615	. . . . . . . . . . . . 13 |- ( 0 e. ZZ -> 0 e. QQ )
45	43, 44	ax-mp 5	. . . . . . . . . . . 12 |- 0 e. QQ
46		qapne 9628	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ 0 e. QQ ) -> ( A # 0 <-> A =/= 0 ) )
47	36, 45, 46	sylancl 413	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A # 0 <-> A =/= 0 ) )
48	37, 47	mpbird 167	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A # 0 )
49	42, 48, 39	expap0d 10645	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) # 0 )
50		qapne 9628	. . . . . . . . . 10 |- ( ( ( A ^ y ) e. QQ /\ 0 e. QQ ) -> ( ( A ^ y ) # 0 <-> ( A ^ y ) =/= 0 ) )
51	41, 45, 50	sylancl 413	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( ( A ^ y ) # 0 <-> ( A ^ y ) =/= 0 ) )
52	49, 51	mpbid 147	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) =/= 0 )
53		pcqmul 12286	. . . . . . . 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 1247	. . . . . . 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 2210	. . . . . 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 9175	. . . . . . . 8 |- ( y e. NN0 -> y e. CC )
57	56	adantl 277	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> y e. CC )
58	28	adantr 276	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( P pCnt A ) e. CC )
59	57, 58	adddirp1d 7974	. . . . . 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 2192	. . . . 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 156	. . . 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 115	. . 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 8140	. . . . 5 |- ( ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) -> -u ( P pCnt ( A ^ y ) ) = -u ( y x. ( P pCnt A ) ) )
64	24	adantr 276	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> A e. CC )
65		nnnn0 9172	. . . . . . . . . 10 |- ( y e. NN -> y e. NN0 )
66	65, 48	sylan2 286	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> A # 0 )
67	65	adantl 277	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> y e. NN0 )
68		expnegap0 10514	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ y e. NN0 ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
69	64, 66, 67, 68	syl3anc 1238	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
70	69	oveq2d 5885	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( P pCnt ( A ^ -u y ) ) = ( P pCnt ( 1 / ( A ^ y ) ) ) )
71		simpll 527	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> P e. Prime )
72	65, 41	sylan2 286	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ y ) e. QQ )
73	65, 52	sylan2 286	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ y ) =/= 0 )
74		pcrec 12291	. . . . . . . 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 1236	. . . . . . 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 2210	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( P pCnt ( A ^ -u y ) ) = -u ( P pCnt ( A ^ y ) ) )
77		nncn 8916	. . . . . . 7 |- ( y e. NN -> y e. CC )
78		mulneg1 8342	. . . . . . 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 290	. . . . . 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 2192	. . . . 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 156	. . . 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 115	. . 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 9360	. 2 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( N e. ZZ -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
84	83	3impia 1200	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 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4000  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   -ucneg 8119   # cap 8528   / cdiv 8618   NNcn 8908   NN0cn0 9165   ZZcz 9242   QQcq 9608   ^cexp 10505   Primecprime 12090   pCnt cpc 12267

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091  df-pc 12268

This theorem is referenced by:  qexpz  12333  expnprm  12334