Theorem pcexp 12881

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 6025	. . . . 5 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
2	1	oveq2d 6033	. . . 4 |- ( x = 0 -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ 0 ) ) )
3		oveq1 6024	. . . 4 |- ( x = 0 -> ( x x. ( P pCnt A ) ) = ( 0 x. ( P pCnt A ) ) )
4	2, 3	eqeq12d 2246	. . 3 |- ( x = 0 -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) ) )
5		oveq2 6025	. . . . 5 |- ( x = y -> ( A ^ x ) = ( A ^ y ) )
6	5	oveq2d 6033	. . . 4 |- ( x = y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ y ) ) )
7		oveq1 6024	. . . 4 |- ( x = y -> ( x x. ( P pCnt A ) ) = ( y x. ( P pCnt A ) ) )
8	6, 7	eqeq12d 2246	. . 3 |- ( x = y -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) ) )
9		oveq2 6025	. . . . 5 |- ( x = ( y + 1 ) -> ( A ^ x ) = ( A ^ ( y + 1 ) ) )
10	9	oveq2d 6033	. . . 4 |- ( x = ( y + 1 ) -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ ( y + 1 ) ) ) )
11		oveq1 6024	. . . 4 |- ( x = ( y + 1 ) -> ( x x. ( P pCnt A ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) )
12	10, 11	eqeq12d 2246	. . 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 6025	. . . . 5 |- ( x = -u y -> ( A ^ x ) = ( A ^ -u y ) )
14	13	oveq2d 6033	. . . 4 |- ( x = -u y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ -u y ) ) )
15		oveq1 6024	. . . 4 |- ( x = -u y -> ( x x. ( P pCnt A ) ) = ( -u y x. ( P pCnt A ) ) )
16	14, 15	eqeq12d 2246	. . 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 6025	. . . . 5 |- ( x = N -> ( A ^ x ) = ( A ^ N ) )
18	17	oveq2d 6033	. . . 4 |- ( x = N -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ N ) ) )
19		oveq1 6024	. . . 4 |- ( x = N -> ( x x. ( P pCnt A ) ) = ( N x. ( P pCnt A ) ) )
20	18, 19	eqeq12d 2246	. . 3 |- ( x = N -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
21		pc1 12877	. . . . 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 9867	. . . . . . 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 10928	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( A ^ 0 ) = 1 )
26	25	oveq2d 6033	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( P pCnt 1 ) )
27		pcqcl 12878	. . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. ZZ )
28	27	zcnd 9602	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. CC )
29	28	mul02d 8570	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( 0 x. ( P pCnt A ) ) = 0 )
30	22, 26, 29	3eqtr4d 2274	. . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) )
31		oveq1 6024	. . . . 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 10807	. . . . . . . . 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 6033	. . . . . . 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 537	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A e. QQ )
37		simplrr 538	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> A =/= 0 )
38		nn0z 9498	. . . . . . . . . 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 10821	. . . . . . . . 9 |- ( ( A e. QQ /\ A =/= 0 /\ y e. ZZ ) -> ( A ^ y ) e. QQ )
41	36, 37, 39, 40	syl3anc 1273	. . . . . . . 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 9489	. . . . . . . . . . . . 13 |- 0 e. ZZ
44		zq 9859	. . . . . . . . . . . . 13 |- ( 0 e. ZZ -> 0 e. QQ )
45	43, 44	ax-mp 5	. . . . . . . . . . . 12 |- 0 e. QQ
46		qapne 9872	. . . . . . . . . . . 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 10940	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) # 0 )
50		qapne 9872	. . . . . . . . . 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 12875	. . . . . . . 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 1282	. . . . . . 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 2264	. . . . . 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 9411	. . . . . . . 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 8205	. . . . . 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 2246	. . . . 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	imbitrrid 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 8371	. . . . 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 9408	. . . . . . . . . 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 10808	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ y e. NN0 ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
69	64, 66, 67, 68	syl3anc 1273	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
70	69	oveq2d 6033	. . . . . . 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 12880	. . . . . . . 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 1271	. . . . . . 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 2264	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( P pCnt ( A ^ -u y ) ) = -u ( P pCnt ( A ^ y ) ) )
77		nncn 9150	. . . . . . 7 |- ( y e. NN -> y e. CC )
78		mulneg1 8573	. . . . . . 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 2246	. . . . 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	imbitrrid 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 9597	. 2 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( N e. ZZ -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
84	83	3impia 1226	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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   -ucneg 8350   # cap 8760   / cdiv 8851   NNcn 9142   NN0cn0 9401   ZZcz 9478   QQcq 9852   ^cexp 10799   Primecprime 12678   pCnt cpc 12856

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-prm 12679  df-pc 12857

This theorem is referenced by:  qexpz  12924  expnprm  12925