Theorem pcexp 12574

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 5951	. . . . 5 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
2	1	oveq2d 5959	. . . 4 |- ( x = 0 -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ 0 ) ) )
3		oveq1 5950	. . . 4 |- ( x = 0 -> ( x x. ( P pCnt A ) ) = ( 0 x. ( P pCnt A ) ) )
4	2, 3	eqeq12d 2219	. . 3 |- ( x = 0 -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) ) )
5		oveq2 5951	. . . . 5 |- ( x = y -> ( A ^ x ) = ( A ^ y ) )
6	5	oveq2d 5959	. . . 4 |- ( x = y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ y ) ) )
7		oveq1 5950	. . . 4 |- ( x = y -> ( x x. ( P pCnt A ) ) = ( y x. ( P pCnt A ) ) )
8	6, 7	eqeq12d 2219	. . 3 |- ( x = y -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ y ) ) = ( y x. ( P pCnt A ) ) ) )
9		oveq2 5951	. . . . 5 |- ( x = ( y + 1 ) -> ( A ^ x ) = ( A ^ ( y + 1 ) ) )
10	9	oveq2d 5959	. . . 4 |- ( x = ( y + 1 ) -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ ( y + 1 ) ) ) )
11		oveq1 5950	. . . 4 |- ( x = ( y + 1 ) -> ( x x. ( P pCnt A ) ) = ( ( y + 1 ) x. ( P pCnt A ) ) )
12	10, 11	eqeq12d 2219	. . 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 5951	. . . . 5 |- ( x = -u y -> ( A ^ x ) = ( A ^ -u y ) )
14	13	oveq2d 5959	. . . 4 |- ( x = -u y -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ -u y ) ) )
15		oveq1 5950	. . . 4 |- ( x = -u y -> ( x x. ( P pCnt A ) ) = ( -u y x. ( P pCnt A ) ) )
16	14, 15	eqeq12d 2219	. . 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 5951	. . . . 5 |- ( x = N -> ( A ^ x ) = ( A ^ N ) )
18	17	oveq2d 5959	. . . 4 |- ( x = N -> ( P pCnt ( A ^ x ) ) = ( P pCnt ( A ^ N ) ) )
19		oveq1 5950	. . . 4 |- ( x = N -> ( x x. ( P pCnt A ) ) = ( N x. ( P pCnt A ) ) )
20	18, 19	eqeq12d 2219	. . 3 |- ( x = N -> ( ( P pCnt ( A ^ x ) ) = ( x x. ( P pCnt A ) ) <-> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
21		pc1 12570	. . . . 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 9754	. . . . . . 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 10810	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( A ^ 0 ) = 1 )
26	25	oveq2d 5959	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( P pCnt 1 ) )
27		pcqcl 12571	. . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. ZZ )
28	27	zcnd 9495	. . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. CC )
29	28	mul02d 8463	. . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( 0 x. ( P pCnt A ) ) = 0 )
30	22, 26, 29	3eqtr4d 2247	. . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( A ^ 0 ) ) = ( 0 x. ( P pCnt A ) ) )
31		oveq1 5950	. . . . 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 10689	. . . . . . . . 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 5959	. . . . . . 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 9391	. . . . . . . . . 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 10703	. . . . . . . . 9 |- ( ( A e. QQ /\ A =/= 0 /\ y e. ZZ ) -> ( A ^ y ) e. QQ )
41	36, 37, 39, 40	syl3anc 1249	. . . . . . . 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 9382	. . . . . . . . . . . . 13 |- 0 e. ZZ
44		zq 9746	. . . . . . . . . . . . 13 |- ( 0 e. ZZ -> 0 e. QQ )
45	43, 44	ax-mp 5	. . . . . . . . . . . 12 |- 0 e. QQ
46		qapne 9759	. . . . . . . . . . . 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 10822	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN0 ) -> ( A ^ y ) # 0 )
50		qapne 9759	. . . . . . . . . 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 12568	. . . . . . . 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 1258	. . . . . . 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 2237	. . . . . 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 9304	. . . . . . . 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 8098	. . . . . 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 2219	. . . . 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 8264	. . . . 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 9301	. . . . . . . . . 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 10690	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ y e. NN0 ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
69	64, 66, 67, 68	syl3anc 1249	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( A ^ -u y ) = ( 1 / ( A ^ y ) ) )
70	69	oveq2d 5959	. . . . . . 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 12573	. . . . . . . 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 1247	. . . . . . 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 2237	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) /\ y e. NN ) -> ( P pCnt ( A ^ -u y ) ) = -u ( P pCnt ( A ^ y ) ) )
77		nncn 9043	. . . . . . 7 |- ( y e. NN -> y e. CC )
78		mulneg1 8466	. . . . . . 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 2219	. . . . 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 9490	. 2 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( N e. ZZ -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) ) )
84	83	3impia 1202	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 980   = wceq 1372   e. wcel 2175   =/= wne 2375   class class class wbr 4043  (class class class)co 5943   CCcc 7922   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   -ucneg 8243   # cap 8653   / cdiv 8744   NNcn 9035   NN0cn0 9294   ZZcz 9371   QQcq 9739   ^cexp 10681   Primecprime 12371   pCnt cpc 12549

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-1o 6501  df-2o 6502  df-er 6619  df-en 6827  df-sup 7085  df-inf 7086  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-dvds 12041  df-gcd 12217  df-prm 12372  df-pc 12550

This theorem is referenced by:  qexpz  12617  expnprm  12618