Theorem pcmptdvds 12483

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 1539	. . . . . . . . . 10 |- F/ m A e. NN0
3		nfcsb1v 3113	. . . . . . . . . . 11 |- F/_ n [_ m / n ]_ A
4	3	nfel1 2347	. . . . . . . . . 10 |- F/ n [_ m / n ]_ A e. NN0
5		csbeq1a 3089	. . . . . . . . . . 11 |- ( n = m -> A = [_ m / n ]_ A )
6	5	eleq1d 2262	. . . . . . . . . 10 |- ( n = m -> ( A e. NN0 <-> [_ m / n ]_ A e. NN0 ) )
7	2, 4, 6	cbvralw 2720	. . . . . . . . 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 3083	. . . . . . . . . 10 |- ( m = p -> [_ m / n ]_ A = [_ p / n ]_ A )
10	9	eleq1d 2262	. . . . . . . . 9 |- ( m = p -> ( [_ m / n ]_ A e. NN0 <-> [_ p / n ]_ A e. NN0 ) )
11	10	rspcv 2860	. . . . . . . 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 9296	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ [_ p / n ]_ A )
14		0le0 9071	. . . . . . 7 |- 0 <_ 0
15	14	a1i 9	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ 0 )
16		prmz 12249	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
17		pcmptdvds.3	. . . . . . . . . 10 |- ( ph -> M e. ( ZZ>= ` N ) )
18		eluzelz 9601	. . . . . . . . . 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 9393	. . . . . . . 8 |- ( ( p e. ZZ /\ M e. ZZ ) -> DECID p <_ M )
22	16, 20, 21	syl2an2 594	. . . . . . 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 9438	. . . . . . . . 9 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
26		zdcle 9393	. . . . . . . . 9 |- ( ( p e. ZZ /\ N e. ZZ ) -> DECID p <_ N )
27	16, 25, 26	syl2an2 594	. . . . . . . 8 |- ( ( ph /\ p e. Prime ) -> DECID p <_ N )
28		dcn 843	. . . . . . . 8 |- (DECID p <_ N -> DECID -. p <_ N )
29	27, 28	syl 14	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> DECID -. p <_ N )
30		dcan2 936	. . . . . . 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 4033	. . . . . . 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 4033	. . . . . . 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 3590	. . . . . 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 1249	. . . . 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 2336	. . . . . . . 8 |- F/_ m if ( n e. Prime , ( n ^ A ) , 1 )
38		nfv 1539	. . . . . . . . 9 |- F/ n m e. Prime
39		nfcv 2336	. . . . . . . . . 10 |- F/_ n m
40		nfcv 2336	. . . . . . . . . 10 |- F/_ n ^
41	39, 40, 3	nfov 5948	. . . . . . . . 9 |- F/_ n ( m ^ [_ m / n ]_ A )
42		nfcv 2336	. . . . . . . . 9 |- F/_ n 1
43	38, 41, 42	nfif 3585	. . . . . . . 8 |- F/_ n if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 )
44		eleq1w 2254	. . . . . . . . 9 |- ( n = m -> ( n e. Prime <-> m e. Prime ) )
45		id 19	. . . . . . . . . 10 |- ( n = m -> n = m )
46	45, 5	oveq12d 5936	. . . . . . . . 9 |- ( n = m -> ( n ^ A ) = ( m ^ [_ m / n ]_ A ) )
47	44, 46	ifbieq1d 3579	. . . . . . . 8 |- ( n = m -> if ( n e. Prime , ( n ^ A ) , 1 ) = if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
48	37, 43, 47	cbvmpt 4124	. . . . . . 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 2214	. . . . . 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 12482	. . . . 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 4057	. . . 4 |- ( ( ph /\ p e. Prime ) -> 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
55	54	ralrimiva 2567	. . 3 |- ( ph -> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
56	36, 1	pcmptcl 12480	. . . . . . . 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 9665	. . . . . . . 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 5694	. . . . . 6 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. NN )
61	60	nnzd 9438	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. ZZ )
62	57, 23	ffvelcdmd 5694	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. NN )
63		znq 9689	. . . . 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 12470	. . . 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 9438	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. ZZ )
69	62	nnne0d 9027	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) =/= 0 )
70		dvdsval2 11933	. . 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 1249	. 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 835   = wceq 1364   e. wcel 2164   =/= wne 2364   A.wral 2472   [_csb 3080   ifcif 3557   class class class wbr 4029   |-> cmpt 4090   -->wf 5250   ` cfv 5254  (class class class)co 5918   0cc0 7872   1c1 7873   x. cmul 7877   <_ cle 8055   / cdiv 8691   NNcn 8982   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   QQcq 9684   seqcseq 10518   ^cexp 10609   || cdvds 11930   Primecprime 12245   pCnt cpc 12422

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-fin 6797  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-xnn0 9304  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246  df-pc 12423

This theorem is referenced by: (None)