Theorem pcmptdvds 12911

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 1574	. . . . . . . . . 10 |- F/ m A e. NN0
3		nfcsb1v 3158	. . . . . . . . . . 11 |- F/_ n [_ m / n ]_ A
4	3	nfel1 2383	. . . . . . . . . 10 |- F/ n [_ m / n ]_ A e. NN0
5		csbeq1a 3134	. . . . . . . . . . 11 |- ( n = m -> A = [_ m / n ]_ A )
6	5	eleq1d 2298	. . . . . . . . . 10 |- ( n = m -> ( A e. NN0 <-> [_ m / n ]_ A e. NN0 ) )
7	2, 4, 6	cbvralw 2758	. . . . . . . . 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 3128	. . . . . . . . . 10 |- ( m = p -> [_ m / n ]_ A = [_ p / n ]_ A )
10	9	eleq1d 2298	. . . . . . . . 9 |- ( m = p -> ( [_ m / n ]_ A e. NN0 <-> [_ p / n ]_ A e. NN0 ) )
11	10	rspcv 2904	. . . . . . . 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 9451	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ [_ p / n ]_ A )
14		0le0 9225	. . . . . . 7 |- 0 <_ 0
15	14	a1i 9	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ 0 )
16		prmz 12676	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
17		pcmptdvds.3	. . . . . . . . . 10 |- ( ph -> M e. ( ZZ>= ` N ) )
18		eluzelz 9758	. . . . . . . . . 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 9549	. . . . . . . 8 |- ( ( p e. ZZ /\ M e. ZZ ) -> DECID p <_ M )
22	16, 20, 21	syl2an2 596	. . . . . . 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 9594	. . . . . . . . 9 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
26		zdcle 9549	. . . . . . . . 9 |- ( ( p e. ZZ /\ N e. ZZ ) -> DECID p <_ N )
27	16, 25, 26	syl2an2 596	. . . . . . . 8 |- ( ( ph /\ p e. Prime ) -> DECID p <_ N )
28		dcn 847	. . . . . . . 8 |- (DECID p <_ N -> DECID -. p <_ N )
29	27, 28	syl 14	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> DECID -. p <_ N )
30		dcan2 940	. . . . . . 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 4090	. . . . . . 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 4090	. . . . . . 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 3638	. . . . . 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 1271	. . . . 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 2372	. . . . . . . 8 |- F/_ m if ( n e. Prime , ( n ^ A ) , 1 )
38		nfv 1574	. . . . . . . . 9 |- F/ n m e. Prime
39		nfcv 2372	. . . . . . . . . 10 |- F/_ n m
40		nfcv 2372	. . . . . . . . . 10 |- F/_ n ^
41	39, 40, 3	nfov 6043	. . . . . . . . 9 |- F/_ n ( m ^ [_ m / n ]_ A )
42		nfcv 2372	. . . . . . . . 9 |- F/_ n 1
43	38, 41, 42	nfif 3632	. . . . . . . 8 |- F/_ n if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 )
44		eleq1w 2290	. . . . . . . . 9 |- ( n = m -> ( n e. Prime <-> m e. Prime ) )
45		id 19	. . . . . . . . . 10 |- ( n = m -> n = m )
46	45, 5	oveq12d 6031	. . . . . . . . 9 |- ( n = m -> ( n ^ A ) = ( m ^ [_ m / n ]_ A ) )
47	44, 46	ifbieq1d 3626	. . . . . . . 8 |- ( n = m -> if ( n e. Prime , ( n ^ A ) , 1 ) = if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
48	37, 43, 47	cbvmpt 4182	. . . . . . 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 2250	. . . . . 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 12910	. . . . 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 4114	. . . 4 |- ( ( ph /\ p e. Prime ) -> 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
55	54	ralrimiva 2603	. . 3 |- ( ph -> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
56	36, 1	pcmptcl 12908	. . . . . . . 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 9827	. . . . . . . 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 5779	. . . . . 6 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. NN )
61	60	nnzd 9594	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. ZZ )
62	57, 23	ffvelcdmd 5779	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. NN )
63		znq 9851	. . . . 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 12898	. . . 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 9594	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. ZZ )
69	62	nnne0d 9181	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) =/= 0 )
70		dvdsval2 12344	. . 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 1271	. 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 839   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   [_csb 3125   ifcif 3603   class class class wbr 4086   |-> cmpt 4148   -->wf 5320   ` cfv 5324  (class class class)co 6013   0cc0 8025   1c1 8026   x. cmul 8030   <_ cle 8208   / cdiv 8845   NNcn 9136   NN0cn0 9395   ZZcz 9472   ZZ>=cuz 9748   QQcq 9846   seqcseq 10702   ^cexp 10793   || cdvds 12341   Primecprime 12672   pCnt cpc 12850

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-fin 6907  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-xnn0 9459  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-dvds 12342  df-gcd 12518  df-prm 12673  df-pc 12851

This theorem is referenced by: (None)