Theorem pcmptdvds 13126

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 1581	. . . . . . . . . 10 |- F/ m A e. NN0
3		nfcsb1v 3180	. . . . . . . . . . 11 |- F/_ n [_ m / n ]_ A
4	3	nfel1 2403	. . . . . . . . . 10 |- F/ n [_ m / n ]_ A e. NN0
5		csbeq1a 3156	. . . . . . . . . . 11 |- ( n = m -> A = [_ m / n ]_ A )
6	5	eleq1d 2307	. . . . . . . . . 10 |- ( n = m -> ( A e. NN0 <-> [_ m / n ]_ A e. NN0 ) )
7	2, 4, 6	cbvralw 2779	. . . . . . . . 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 3150	. . . . . . . . . 10 |- ( m = p -> [_ m / n ]_ A = [_ p / n ]_ A )
10	9	eleq1d 2307	. . . . . . . . 9 |- ( m = p -> ( [_ m / n ]_ A e. NN0 <-> [_ p / n ]_ A e. NN0 ) )
11	10	rspcv 2925	. . . . . . . 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 9625	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ [_ p / n ]_ A )
14		0le0 9394	. . . . . . 7 |- 0 <_ 0
15	14	a1i 9	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> 0 <_ 0 )
16		prmz 12891	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
17		pcmptdvds.3	. . . . . . . . . 10 |- ( ph -> M e. ( ZZ>= ` N ) )
18		eluzelz 9933	. . . . . . . . . 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 9723	. . . . . . . 8 |- ( ( p e. ZZ /\ M e. ZZ ) -> DECID p <_ M )
22	16, 20, 21	syl2an2 602	. . . . . . 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 9769	. . . . . . . . 9 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
26		zdcle 9723	. . . . . . . . 9 |- ( ( p e. ZZ /\ N e. ZZ ) -> DECID p <_ N )
27	16, 25, 26	syl2an2 602	. . . . . . . 8 |- ( ( ph /\ p e. Prime ) -> DECID p <_ N )
28		dcn 854	. . . . . . . 8 |- (DECID p <_ N -> DECID -. p <_ N )
29	27, 28	syl 14	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> DECID -. p <_ N )
30		dcan2 947	. . . . . . 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 4134	. . . . . . 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 4134	. . . . . . 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 3675	. . . . . 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 1278	. . . . 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 2392	. . . . . . . 8 |- F/_ m if ( n e. Prime , ( n ^ A ) , 1 )
38		nfv 1581	. . . . . . . . 9 |- F/ n m e. Prime
39		nfcv 2392	. . . . . . . . . 10 |- F/_ n m
40		nfcv 2392	. . . . . . . . . 10 |- F/_ n ^
41	39, 40, 3	nfov 6115	. . . . . . . . 9 |- F/_ n ( m ^ [_ m / n ]_ A )
42		nfcv 2392	. . . . . . . . 9 |- F/_ n 1
43	38, 41, 42	nfif 3669	. . . . . . . 8 |- F/_ n if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 )
44		eleq1w 2299	. . . . . . . . 9 |- ( n = m -> ( n e. Prime <-> m e. Prime ) )
45		id 19	. . . . . . . . . 10 |- ( n = m -> n = m )
46	45, 5	oveq12d 6103	. . . . . . . . 9 |- ( n = m -> ( n ^ A ) = ( m ^ [_ m / n ]_ A ) )
47	44, 46	ifbieq1d 3663	. . . . . . . 8 |- ( n = m -> if ( n e. Prime , ( n ^ A ) , 1 ) = if ( m e. Prime , ( m ^ [_ m / n ]_ A ) , 1 ) )
48	37, 43, 47	cbvmpt 4226	. . . . . . 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 2259	. . . . . 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 13125	. . . . 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 4158	. . . 4 |- ( ( ph /\ p e. Prime ) -> 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
55	54	ralrimiva 2623	. . 3 |- ( ph -> A. p e. Prime 0 <_ ( p pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) )
56	36, 1	pcmptcl 13123	. . . . . . . 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 10002	. . . . . . . 8 |- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN )
59	23, 17, 58	syl2anc 415	. . . . . . 7 |- ( ph -> M e. NN )
60	57, 59	ffvelcdmd 5844	. . . . . 6 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. NN )
61	60	nnzd 9769	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` M ) e. ZZ )
62	57, 23	ffvelcdmd 5844	. . . . 5 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. NN )
63		znq 10026	. . . . 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 415	. . . 4 |- ( ph -> ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) e. QQ )
65		pcz 13113	. . . 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 9769	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) e. ZZ )
69	62	nnne0d 9350	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` N ) =/= 0 )
70		dvdsval2 12559	. . 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 1278	. 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   [_csb 3147   ifcif 3638   class class class wbr 4130   |-> cmpt 4192   -->wf 5373   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   x. cmul 8184   <_ cle 8361   / cdiv 9003   NNcn 9305   NN0cn0 9565   ZZcz 9646   ZZ>=cuz 9923   QQcq 10021   seqcseq 10886   ^cexp 10977   || cdvds 12556   Primecprime 12887   pCnt cpc 13065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-xnn0 9633  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-fl 10707  df-mod 10762  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-dvds 12557  df-gcd 12733  df-prm 12888  df-pc 13066

This theorem is used by: (None)