Theorem pcmptdvds 12297

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 1521	. . . . . . . . . 10 |- F/ m A e. NN0
3		nfcsb1v 3082	. . . . . . . . . . 11 |- F/_ n [_ m / n ]_ A
4	3	nfel1 2323	. . . . . . . . . 10 |- F/ n [_ m / n ]_ A e. NN0
5		csbeq1a 3058	. . . . . . . . . . 11 |- ( n = m -> A = [_ m / n ]_ A )
6	5	eleq1d 2239	. . . . . . . . . 10 |- ( n = m -> ( A e. NN0 <-> [_ m / n ]_ A e. NN0 ) )
7	2, 4, 6	cbvralw 2691	. . . . . . . . 9 |- ( A. n e. Prime A e. NN0 <-> A. m e. Prime [_ m / n ]_ A e. NN0 )
8	1, 7	sylib 121	. . . . . . . 8 |- ( ph -> A. m e. Prime [_ m / n ]_ A e. NN0 )
9		csbeq1 3052	. . . . . . . . . 10 |- ( m = p -> [_ m / n ]_ A = [_ p / n ]_ A )
10	9	eleq1d 2239	. . . . . . . . 9 |- ( m = p -> ( [_ m / n ]_ A e. NN0 <-> [_ p / n ]_ A e. NN0 ) )
11	10	rspcv 2830	. . . . . . . 8 |- ( p e. Prime -> ( A. m e. Prime [_ m / n ]_ A e. NN0 -> [_ p / n ]_ A e. NN0 ) )
12	8, 11	mpan9 279	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> [_ p / n ]_ A e. NN0 )
13	12	nn0ge0d 9191	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ [_ p / n ]_ A )
14		0le0 8967	. . . . . . 7 |- 0 <_ 0
15	14	a1i 9	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ 0 )
16		prmz 12065	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
17		pcmptdvds.3	. . . . . . . . . 10 |- ( ph -> M e. ( ZZ>= ` N ) )
18		eluzelz 9496	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` N ) -> M e. ZZ )
19	17, 18	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
20	19	adantr 274	. . . . . . . 8 |- ( ( ph /\ p e. Prime ) -> M e. ZZ )
21		zdcle 9288	. . . . . . . 8 |- ( ( p e. ZZ /\ M e. ZZ ) -> DECID p <_ M )
22	16, 20, 21	syl2an2 589	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> DECID p <_ M )
23		pcmpt.3	. . . . . . . . . . 11 |- ( ph -> N e. NN )
24	23	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ p e. Prime ) -> N e. NN )
25	24	nnzd 9333	. . . . . . . . 9 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
26		zdcle 9288	. . . . . . . . 9 |- ( ( p e. ZZ /\ N e. ZZ ) -> DECID p <_ N )
27	16, 25, 26	syl2an2 589	. . . . . . . 8 |- ( ( ph /\ p e. Prime ) -> DECID p <_ N )
28		dcn 837	. . . . . . . 8 |- (DECID p <_ N -> DECID -. p <_ N )
29	27, 28	syl 14	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> DECID -. p <_ N )
30		dcan2 929	. . . . . . 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 3993	. . . . . . 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 3993	. . . . . . 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 3558	. . . . . 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 1233	. . . . 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 2312	. . . . . . . 8 |- F/_ m if ( n e. Prime , ( n ^ A ) , 1 )
38		nfv 1521	. . . . . . . . 9 |- F/ n m e. Prime
39		nfcv 2312	. . . . . . . . . 10 |- F/_ n m
40		nfcv 2312	. . . . . . . . . 10 |- F/_ n ^
41	39, 40, 3	nfov 5883	. . . . . . . . 9 |- F/_ n ( m ^ [_ m / n ]_ A )
42		nfcv 2312	. . . . . . . . 9 |- F/_ n 1
43	38, 41, 42	nfif 3554	. . . . . . . 8 |- F/_ n if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 )
44		eleq1w 2231	. . . . . . . . 9 |- ( n = m -> ( n e. Prime <-> m e. Prime ) )
45		id 19	. . . . . . . . . 10 |- ( n = m -> n = m )
46	45, 5	oveq12d 5871	. . . . . . . . 9 |- ( n = m -> ( n ^ A ) = ( m ^ [_ m / n ]_ A ) )
47	44, 46	ifbieq1d 3548	. . . . . . . 8 |- ( n = m -> if ( n e. Prime , ( n ^ A ) , 1 ) = if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
48	37, 43, 47	cbvmpt 4084	. . . . . . 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 2191	. . . . . 6 |- F = ( m e. NN |-> if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
50	8	adantr 274	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> A. m e. Prime [_ m / n ]_ A e. NN0 )
51		simpr 109	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> p e. Prime )
52	17	adantr 274	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> M e. ( ZZ>= ` N ) )
53	49, 50, 24, 51, 9, 52	pcmpt2 12296	. . . . 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 4017	. . . 4 |- ( ( ph /\ p e. Prime ) -> 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
55	54	ralrimiva 2543	. . 3 |- ( ph -> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
56	36, 1	pcmptcl 12294	. . . . . . . 8 |- ( ph -> ( F : NN --> NN /\ seq 1 ( x. , F ) : NN --> NN ) )
57	56	simprd 113	. . . . . . 7 |- ( ph -> seq 1 ( x. , F ) : NN --> NN )
58		eluznn 9559	. . . . . . . 8 |- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN )
59	23, 17, 58	syl2anc 409	. . . . . . 7 |- ( ph -> M e. NN )
60	57, 59	ffvelrnd 5632	. . . . . 6 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. NN )
61	60	nnzd 9333	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. ZZ )
62	57, 23	ffvelrnd 5632	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. NN )
63		znq 9583	. . . . 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 409	. . . 4 |- ( ph -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. QQ )
65		pcz 12285	. . . 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 166	. 2 |- ( ph -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. ZZ )
68	62	nnzd 9333	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. ZZ )
69	62	nnne0d 8923	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) =/= 0 )
70		dvdsval2 11752	. . 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 1233	. 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 166	1 |- ( ph -> ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   [_csb 3049   ifcif 3526   class class class wbr 3989   |-> cmpt 4050   -->wf 5194   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   x. cmul 7779   <_ cle 7955   / cdiv 8589   NNcn 8878   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   QQcq 9578   seqcseq 10401   ^cexp 10475   || cdvds 11749   Primecprime 12061   pCnt cpc 12238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-xnn0 9199  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062  df-pc 12239

This theorem is referenced by: (None)