Theorem pcmptdvds 15598

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 1843	. . . . . . . . . 10 |- F/ m A e. NN0
3		nfcsb1v 3549	. . . . . . . . . . 11 |- F/_ n [_ m / n ]_ A
4	3	nfel1 2779	. . . . . . . . . 10 |- F/ n [_ m / n ]_ A e. NN0
5		csbeq1a 3542	. . . . . . . . . . 11 |- ( n = m -> A = [_ m / n ]_ A )
6	5	eleq1d 2686	. . . . . . . . . 10 |- ( n = m -> ( A e. NN0 <-> [_ m / n ]_ A e. NN0 ) )
7	2, 4, 6	cbvral 3167	. . . . . . . . 9 |- ( A. n e. Prime A e. NN0 <-> A. m e. Prime [_ m / n ]_ A e. NN0 )
8	1, 7	sylib 208	. . . . . . . 8 |- ( ph -> A. m e. Prime [_ m / n ]_ A e. NN0 )
9		csbeq1 3536	. . . . . . . . . 10 |- ( m = p -> [_ m / n ]_ A = [_ p / n ]_ A )
10	9	eleq1d 2686	. . . . . . . . 9 |- ( m = p -> ( [_ m / n ]_ A e. NN0 <-> [_ p / n ]_ A e. NN0 ) )
11	10	rspcv 3305	. . . . . . . 8 |- ( p e. Prime -> ( A. m e. Prime [_ m / n ]_ A e. NN0 -> [_ p / n ]_ A e. NN0 ) )
12	8, 11	mpan9 486	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> [_ p / n ]_ A e. NN0 )
13	12	nn0ge0d 11354	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ [_ p / n ]_ A )
14		0le0 11110	. . . . . 6 |- 0 <_ 0
15		breq2 4657	. . . . . . 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 ) ) )
16		breq2 4657	. . . . . . 7 |- ( 0 = if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) -> ( 0 <_ 0 <-> 0 <_ if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) ) )
17	15, 16	ifboth 4124	. . . . . 6 |- ( ( 0 <_ [_ p / n ]_ A /\ 0 <_ 0 ) -> 0 <_ if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) )
18	13, 14, 17	sylancl 694	. . . . 5 |- ( ( ph /\ p e. Prime ) -> 0 <_ if ( ( p <_ M /\ -. p <_ N ) , [_ p / n ]_ A , 0 ) )
19		pcmpt.1	. . . . . . 7 |- F = ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) )
20		nfcv 2764	. . . . . . . 8 |- F/_ m if ( n e. Prime , ( n ^ A ) , 1 )
21		nfv 1843	. . . . . . . . 9 |- F/ n m e. Prime
22		nfcv 2764	. . . . . . . . . 10 |- F/_ n m
23		nfcv 2764	. . . . . . . . . 10 |- F/_ n ^
24	22, 23, 3	nfov 6676	. . . . . . . . 9 |- F/_ n ( m ^ [_ m / n ]_ A )
25		nfcv 2764	. . . . . . . . 9 |- F/_ n 1
26	21, 24, 25	nfif 4115	. . . . . . . 8 |- F/_ n if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 )
27		eleq1 2689	. . . . . . . . 9 |- ( n = m -> ( n e. Prime <-> m e. Prime ) )
28		id 22	. . . . . . . . . 10 |- ( n = m -> n = m )
29	28, 5	oveq12d 6668	. . . . . . . . 9 |- ( n = m -> ( n ^ A ) = ( m ^ [_ m / n ]_ A ) )
30	27, 29	ifbieq1d 4109	. . . . . . . 8 |- ( n = m -> if ( n e. Prime , ( n ^ A ) , 1 ) = if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
31	20, 26, 30	cbvmpt 4749	. . . . . . 7 |- ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) ) = ( m e. NN |-> if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
32	19, 31	eqtri 2644	. . . . . 6 |- F = ( m e. NN |-> if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
33	8	adantr 481	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> A. m e. Prime [_ m / n ]_ A e. NN0 )
34		pcmpt.3	. . . . . . 7 |- ( ph -> N e. NN )
35	34	adantr 481	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> N e. NN )
36		simpr 477	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> p e. Prime )
37		pcmptdvds.3	. . . . . . 7 |- ( ph -> M e. ( ZZ>= ` N ) )
38	37	adantr 481	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> M e. ( ZZ>= ` N ) )
39	32, 33, 35, 36, 9, 38	pcmpt2 15597	. . . . 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 ) )
40	18, 39	breqtrrd 4681	. . . 4 |- ( ( ph /\ p e. Prime ) -> 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
41	40	ralrimiva 2966	. . 3 |- ( ph -> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
42	19, 1	pcmptcl 15595	. . . . . . . 8 |- ( ph -> ( F : NN --> NN /\ seq 1 ( x. , F ) : NN --> NN ) )
43	42	simprd 479	. . . . . . 7 |- ( ph -> seq 1 ( x. , F ) : NN --> NN )
44		eluznn 11758	. . . . . . . 8 |- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN )
45	34, 37, 44	syl2anc 693	. . . . . . 7 |- ( ph -> M e. NN )
46	43, 45	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. NN )
47	46	nnzd 11481	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. ZZ )
48	43, 34	ffvelrnd 6360	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. NN )
49		znq 11792	. . . . 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 )
50	47, 48, 49	syl2anc 693	. . . 4 |- ( ph -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. QQ )
51		pcz 15585	. . . 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 ) ) ) ) )
52	50, 51	syl 17	. . 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 ) ) ) ) )
53	41, 52	mpbird 247	. 2 |- ( ph -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ )
54	48	nnzd 11481	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. ZZ )
55	48	nnne0d 11065	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) =/= 0 )
56		dvdsval2 14986	. . 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 ) )
57	54, 55, 47, 56	syl3anc 1326	. 2 |- ( ph -> ( ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) <-> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ ) )
58	53, 57	mpbird 247	1 |- ( ph -> ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   seqcseq 12801   ^cexp 12860   || cdvds 14983   Primecprime 15385   pCnt cpc 15541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542

This theorem is referenced by:  bposlem6  25014