Theorem pcmptdvds 13068

Description: The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)

Hypotheses

Ref	Expression
pcmpt.1	|- F = ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) )
pcmpt.2	|- ( ph -> A. n e. Prime A e. NN0 )
pcmpt.3	|- ( ph -> N e. NN )
pcmptdvds.3	|- ( ph -> M e. ( ZZ>= ` N ) )

Assertion

Ref	Expression
pcmptdvds	|- ( ph -> ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) )

Proof of Theorem pcmptdvds

Dummy variables m p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcmpt.2	. . . . . . . . 9 |- ( ph -> A. n e. Prime A e. NN0 )
2		nfv 1577	. . . . . . . . . 10 |- F/ m A e. NN0
3		nfcsb1v 3174	. . . . . . . . . . 11 |- F/_ n [_ m / n ]_ A
4	3	nfel1 2397	. . . . . . . . . 10 |- F/ n [_ m / n ]_ A e. NN0
5		csbeq1a 3150	. . . . . . . . . . 11 |- ( n = m -> A = [_ m / n ]_ A )
6	5	eleq1d 2303	. . . . . . . . . 10 |- ( n = m -> ( A e. NN0 <-> [_ m / n ]_ A e. NN0 ) )
7	2, 4, 6	cbvralw 2773	. . . . . . . . 9 |- ( A. n e. Prime A e. NN0 <-> A. m e. Prime [_ m / n ]_ A e. NN0 )
8	1, 7	sylib 122	. . . . . . . 8 |- ( ph -> A. m e. Prime [_ m / n ]_ A e. NN0 )
9		csbeq1 3144	. . . . . . . . . 10 |- ( m = p -> [_ m / n ]_ A = [_ p / n ]_ A )
10	9	eleq1d 2303	. . . . . . . . 9 |- ( m = p -> ( [_ m / n ]_ A e. NN0 <-> [_ p / n ]_ A e. NN0 ) )
11	10	rspcv 2919	. . . . . . . 8 |- ( p e. Prime -> ( A. m e. Prime [_ m / n ]_ A e. NN0 -> [_ p / n ]_ A e. NN0 ) )
12	8, 11	mpan9 281	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> [_ p / n ]_ A e. NN0 )
13	12	nn0ge0d 9573	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ [_ p / n ]_ A )
14		0le0 9343	. . . . . . 7 |- 0 <_ 0
15	14	a1i 9	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ 0 )
16		prmz 12833	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
17		pcmptdvds.3	. . . . . . . . . 10 |- ( ph -> M e. ( ZZ>= ` N ) )
18		eluzelz 9881	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` N ) -> M e. ZZ )
19	17, 18	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
20	19	adantr 276	. . . . . . . 8 |- ( ( ph /\ p e. Prime ) -> M e. ZZ )
21		zdcle 9671	. . . . . . . 8 |- ( ( p e. ZZ /\ M e. ZZ ) -> DECID p <_ M )
22	16, 20, 21	syl2an2 598	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> DECID p <_ M )
23		pcmpt.3	. . . . . . . . . . 11 |- ( ph -> N e. NN )
24	23	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ p e. Prime ) -> N e. NN )
25	24	nnzd 9717	. . . . . . . . 9 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
26		zdcle 9671	. . . . . . . . 9 |- ( ( p e. ZZ /\ N e. ZZ ) -> DECID p <_ N )
27	16, 25, 26	syl2an2 598	. . . . . . . 8 |- ( ( ph /\ p e. Prime ) -> DECID p <_ N )
28		dcn 850	. . . . . . . 8 |- (DECID p <_ N -> DECID -. p <_ N )
29	27, 28	syl 14	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> DECID -. p <_ N )
30		dcan2 943	. . . . . . 7 |- (DECID p <_ M -> (DECID -. p <_ N -> DECID ( p <_ M /\ -. p <_ N ) ) )
31	22, 29, 30	sylc 62	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> DECID ( p <_ M /\ -. p <_ N ) )
32		breq2 4118	. . . . . . 7 |- ( [_ p / n ]_ A = if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) -> ( 0 <_ [_ p / n ]_ A <-> 0 <_ if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) ) )
33		breq2 4118	. . . . . . 7 |- ( 0 = if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) -> ( 0 <_ 0 <-> 0 <_ if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) ) )
34	32, 33	ifbothdc 3661	. . . . . 6 |- ( ( 0 <_ [_ p / n ]_ A /\ 0 <_ 0 /\ DECID ( p <_ M /\ -. p <_ N ) ) -> 0 <_ if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) )
35	13, 15, 31, 34	syl3anc 1274	. . . . 5 |- ( ( ph /\ p e. Prime ) -> 0 <_ if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) )
36		pcmpt.1	. . . . . . 7 |- F = ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) )
37		nfcv 2386	. . . . . . . 8 |- F/_ m if ( n e. Prime , ( n ^ A ) , 1 )
38		nfv 1577	. . . . . . . . 9 |- F/ n m e. Prime
39		nfcv 2386	. . . . . . . . . 10 |- F/_ n m
40		nfcv 2386	. . . . . . . . . 10 |- F/_ n ^
41	39, 40, 3	nfov 6088	. . . . . . . . 9 |- F/_ n ( m ^ [_ m / n ]_ A )
42		nfcv 2386	. . . . . . . . 9 |- F/_ n 1
43	38, 41, 42	nfif 3655	. . . . . . . 8 |- F/_ n if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 )
44		eleq1w 2295	. . . . . . . . 9 |- ( n = m -> ( n e. Prime <-> m e. Prime ) )
45		id 19	. . . . . . . . . 10 |- ( n = m -> n = m )
46	45, 5	oveq12d 6076	. . . . . . . . 9 |- ( n = m -> ( n ^ A ) = ( m ^ [_ m / n ]_ A ) )
47	44, 46	ifbieq1d 3649	. . . . . . . 8 |- ( n = m -> if ( n e. Prime , ( n ^ A ) , 1 ) = if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
48	37, 43, 47	cbvmpt 4210	. . . . . . 7 |- ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) ) = ( m e. NN |-> if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
49	36, 48	eqtri 2255	. . . . . 6 |- F = ( m e. NN |-> if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
50	8	adantr 276	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> A. m e. Prime [_ m / n ]_ A e. NN0 )
51		simpr 110	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> p e. Prime )
52	17	adantr 276	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> M e. ( ZZ>= ` N ) )
53	49, 50, 24, 51, 9, 52	pcmpt2 13067	. . . . 5 |- ( ( ph /\ p e. Prime ) -> ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) = if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) )
54	35, 53	breqtrrd 4142	. . . 4 |- ( ( ph /\ p e. Prime ) -> 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
55	54	ralrimiva 2617	. . 3 |- ( ph -> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
56	36, 1	pcmptcl 13065	. . . . . . . 8 |- ( ph -> ( F : NN --> NN /\ seq 1 ( x. , F ) : NN --> NN ) )
57	56	simprd 114	. . . . . . 7 |- ( ph -> seq 1 ( x. , F ) : NN --> NN )
58		eluznn 9950	. . . . . . . 8 |- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN )
59	23, 17, 58	syl2anc 411	. . . . . . 7 |- ( ph -> M e. NN )
60	57, 59	ffvelcdmd 5818	. . . . . 6 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. NN )
61	60	nnzd 9717	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. ZZ )
62	57, 23	ffvelcdmd 5818	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. NN )
63		znq 9974	. . . . 5 |- ( ( ( seq 1 ( x. , F ) ` M ) e. ZZ /\ ( seq 1 ( x. , F ) ` N ) e. NN ) -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. QQ )
64	61, 62, 63	syl2anc 411	. . . 4 |- ( ph -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. QQ )
65		pcz 13055	. . . 4 |- ( ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. QQ -> ( ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ <-> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) ) )
66	64, 65	syl 14	. . 3 |- ( ph -> ( ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ <-> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) ) )
67	55, 66	mpbird 167	. 2 |- ( ph -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ )
68	62	nnzd 9717	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. ZZ )
69	62	nnne0d 9299	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) =/= 0 )
70		dvdsval2 12501	. . 3 |- ( ( ( seq 1 ( x. , F ) ` N ) e. ZZ /\ ( seq 1 ( x. , F ) ` N ) =/= 0 /\ ( seq 1 ( x. , F ) ` M ) e. ZZ ) -> ( ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) <-> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ ) )
71	68, 69, 61, 70	syl3anc 1274	. 2 |- ( ph -> ( ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) <-> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ ) )
72	67, 71	mpbird 167	1 |- ( ph -> ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   [_csb 3141   ifcif 3624   class class class wbr 4114   |-> cmpt 4176   -->wf 5353   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   x. cmul 8148   <_ cle 8325   / cdiv 8963   NNcn 9254   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   QQcq 9969   seqcseq 10833   ^cexp 10924   || cdvds 12498   Primecprime 12829   pCnt cpc 13007

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-xnn0 9581  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-prm 12830  df-pc 13008

This theorem is referenced by: (None)