Theorem lgsdilem2 25058

Description: Lemma for lgsdi 25059. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypotheses

Ref	Expression
lgsdilem2.1	|- ( ph -> A e. ZZ )
lgsdilem2.2	|- ( ph -> M e. ZZ )
lgsdilem2.3	|- ( ph -> N e. ZZ )
lgsdilem2.4	|- ( ph -> M =/= 0 )
lgsdilem2.5	|- ( ph -> N =/= 0 )
lgsdilem2.6	|- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) )

Assertion

Ref	Expression
lgsdilem2	|- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) )

Distinct variable groups:   n, M   A, n   n, N

Allowed substitution hints:   ph( n)   F( n)

Proof of Theorem lgsdilem2

Dummy variables k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulid1 10037	. . 3 |- ( k e. CC -> ( k x. 1 ) = k )
2	1	adantl 482	. 2 |- ( ( ph /\ k e. CC ) -> ( k x. 1 ) = k )
3		lgsdilem2.2	. . . 4 |- ( ph -> M e. ZZ )
4		lgsdilem2.4	. . . 4 |- ( ph -> M =/= 0 )
5		nnabscl 14065	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6	3, 4, 5	syl2anc 693	. . 3 |- ( ph -> ( abs ` M ) e. NN )
7		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
8	6, 7	syl6eleq 2711	. 2 |- ( ph -> ( abs ` M ) e. ( ZZ>= ` 1 ) )
9	6	nnzd 11481	. . 3 |- ( ph -> ( abs ` M ) e. ZZ )
10		lgsdilem2.3	. . . . . 6 |- ( ph -> N e. ZZ )
11	3, 10	zmulcld 11488	. . . . 5 |- ( ph -> ( M x. N ) e. ZZ )
12	3	zcnd 11483	. . . . . 6 |- ( ph -> M e. CC )
13	10	zcnd 11483	. . . . . 6 |- ( ph -> N e. CC )
14		lgsdilem2.5	. . . . . 6 |- ( ph -> N =/= 0 )
15	12, 13, 4, 14	mulne0d 10679	. . . . 5 |- ( ph -> ( M x. N ) =/= 0 )
16		nnabscl 14065	. . . . 5 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
17	11, 15, 16	syl2anc 693	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) e. NN )
18	17	nnzd 11481	. . 3 |- ( ph -> ( abs ` ( M x. N ) ) e. ZZ )
19	12	abscld 14175	. . . . 5 |- ( ph -> ( abs ` M ) e. RR )
20	13	abscld 14175	. . . . 5 |- ( ph -> ( abs ` N ) e. RR )
21	12	absge0d 14183	. . . . 5 |- ( ph -> 0 <_ ( abs ` M ) )
22		nnabscl 14065	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
23	10, 14, 22	syl2anc 693	. . . . . 6 |- ( ph -> ( abs ` N ) e. NN )
24	23	nnge1d 11063	. . . . 5 |- ( ph -> 1 <_ ( abs ` N ) )
25	19, 20, 21, 24	lemulge11d 10961	. . . 4 |- ( ph -> ( abs ` M ) <_ ( ( abs ` M ) x. ( abs ` N ) ) )
26	12, 13	absmuld 14193	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
27	25, 26	breqtrrd 4681	. . 3 |- ( ph -> ( abs ` M ) <_ ( abs ` ( M x. N ) ) )
28		eluz2 11693	. . 3 |- ( ( abs ` ( M x. N ) ) e. ( ZZ>= ` ( abs ` M ) ) <-> ( ( abs ` M ) e. ZZ /\ ( abs ` ( M x. N ) ) e. ZZ /\ ( abs ` M ) <_ ( abs ` ( M x. N ) ) ) )
29	9, 18, 27, 28	syl3anbrc 1246	. 2 |- ( ph -> ( abs ` ( M x. N ) ) e. ( ZZ>= ` ( abs ` M ) ) )
30		lgsdilem2.1	. . . . . 6 |- ( ph -> A e. ZZ )
31		lgsdilem2.6	. . . . . . 7 |- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) )
32	31	lgsfcl3 25043	. . . . . 6 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> F : NN --> ZZ )
33	30, 3, 4, 32	syl3anc 1326	. . . . 5 |- ( ph -> F : NN --> ZZ )
34		elfznn 12370	. . . . 5 |- ( k e. ( 1 ... ( abs ` M ) ) -> k e. NN )
35		ffvelrn 6357	. . . . 5 |- ( ( F : NN --> ZZ /\ k e. NN ) -> ( F ` k ) e. ZZ )
36	33, 34, 35	syl2an 494	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( abs ` M ) ) ) -> ( F ` k ) e. ZZ )
37	36	zcnd 11483	. . 3 |- ( ( ph /\ k e. ( 1 ... ( abs ` M ) ) ) -> ( F ` k ) e. CC )
38		mulcl 10020	. . . 4 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
39	38	adantl 482	. . 3 |- ( ( ph /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
40	8, 37, 39	seqcl 12821	. 2 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. CC )
41	6	peano2nnd 11037	. . . . 5 |- ( ph -> ( ( abs ` M ) + 1 ) e. NN )
42		elfzuz 12338	. . . . 5 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) )
43		eluznn 11758	. . . . 5 |- ( ( ( ( abs ` M ) + 1 ) e. NN /\ k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) ) -> k e. NN )
44	41, 42, 43	syl2an 494	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. NN )
45		eleq1 2689	. . . . . 6 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
46		oveq2 6658	. . . . . . 7 |- ( n = k -> ( A /L n ) = ( A /L k ) )
47		oveq1 6657	. . . . . . 7 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
48	46, 47	oveq12d 6668	. . . . . 6 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
49	45, 48	ifbieq1d 4109	. . . . 5 |- ( n = k -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
50		ovex 6678	. . . . . 6 |- ( ( A /L k ) ^ ( k pCnt M ) ) e. _V
51		1ex 10035	. . . . . 6 |- 1 e. _V
52	50, 51	ifex 4156	. . . . 5 |- if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) e. _V
53	49, 31, 52	fvmpt 6282	. . . 4 |- ( k e. NN -> ( F ` k ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
54	44, 53	syl 17	. . 3 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( F ` k ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
55		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. Prime )
56	3	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M e. ZZ )
57		zq 11794	. . . . . . . . . 10 |- ( M e. ZZ -> M e. QQ )
58	56, 57	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M e. QQ )
59		pcabs 15579	. . . . . . . . 9 |- ( ( k e. Prime /\ M e. QQ ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
60	55, 58, 59	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
61		elfzle1 12344	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
62	61	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
63		elfzelz 12342	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ZZ )
64		zltp1le 11427	. . . . . . . . . . . . . 14 |- ( ( ( abs ` M ) e. ZZ /\ k e. ZZ ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
65	9, 63, 64	syl2an 494	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
66	62, 65	mpbird 247	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( abs ` M ) < k )
67	19	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( abs ` M ) e. RR )
68	63	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. ZZ )
69	68	zred 11482	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. RR )
70	67, 69	ltnled 10184	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> -. k <_ ( abs ` M ) ) )
71	66, 70	mpbid 222	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> -. k <_ ( abs ` M ) )
72	71	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k <_ ( abs ` M ) )
73		prmz 15389	. . . . . . . . . . . 12 |- ( k e. Prime -> k e. ZZ )
74	73	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. ZZ )
75	4	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M =/= 0 )
76	56, 75, 5	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( abs ` M ) e. NN )
77		dvdsle 15032	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ ( abs ` M ) e. NN ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
78	74, 76, 77	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
79	72, 78	mtod 189	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k || ( abs ` M ) )
80		pceq0 15575	. . . . . . . . . 10 |- ( ( k e. Prime /\ ( abs ` M ) e. NN ) -> ( ( k pCnt ( abs ` M ) ) = 0 <-> -. k || ( abs ` M ) ) )
81	55, 76, 80	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( k pCnt ( abs ` M ) ) = 0 <-> -. k || ( abs ` M ) ) )
82	79, 81	mpbird 247	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = 0 )
83	60, 82	eqtr3d 2658	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) = 0 )
84	83	oveq2d 6666	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) = ( ( A /L k ) ^ 0 ) )
85	30	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> A e. ZZ )
86		lgscl 25036	. . . . . . . . 9 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
87	85, 74, 86	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
88	87	zcnd 11483	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
89	88	exp0d 13002	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ 0 ) = 1 )
90	84, 89	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) = 1 )
91	90	ifeq1da 4116	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = if ( k e. Prime , 1 , 1 ) )
92		ifid 4125	. . . 4 |- if ( k e. Prime , 1 , 1 ) = 1
93	91, 92	syl6eq 2672	. . 3 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = 1 )
94	54, 93	eqtrd 2656	. 2 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( F ` k ) = 1 )
95	2, 8, 29, 40, 94	seqid2 12847	1 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   ...cfz 12326   seqcseq 12801   ^cexp 12860   abscabs 13974   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   /Lclgs 25019

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-rep 4771  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-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-int 4476  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-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  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-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  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-phi 15471  df-pc 15542  df-lgs 25020

This theorem is referenced by:  lgsdi  25059