Theorem lgsdilem2 23732

Description: Lemma for lgsdi 23733. (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 9610	. . 3 |- ( k e. CC -> ( k x. 1 ) = k )
2	1	adantl 466	. 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 13170	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6	3, 4, 5	syl2anc 661	. . 3 |- ( ph -> ( abs ` M ) e. NN )
7		nnuz 11141	. . 3 |- NN = ( ZZ>= ` 1 )
8	6, 7	syl6eleq 2555	. 2 |- ( ph -> ( abs ` M ) e. ( ZZ>= ` 1 ) )
9	6	nnzd 10989	. . 3 |- ( ph -> ( abs ` M ) e. ZZ )
10		lgsdilem2.3	. . . . . 6 |- ( ph -> N e. ZZ )
11	3, 10	zmulcld 10996	. . . . 5 |- ( ph -> ( M x. N ) e. ZZ )
12	3	zcnd 10991	. . . . . 6 |- ( ph -> M e. CC )
13	10	zcnd 10991	. . . . . 6 |- ( ph -> N e. CC )
14		lgsdilem2.5	. . . . . 6 |- ( ph -> N =/= 0 )
15	12, 13, 4, 14	mulne0d 10222	. . . . 5 |- ( ph -> ( M x. N ) =/= 0 )
16		nnabscl 13170	. . . . 5 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
17	11, 15, 16	syl2anc 661	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) e. NN )
18	17	nnzd 10989	. . 3 |- ( ph -> ( abs ` ( M x. N ) ) e. ZZ )
19	12	abscld 13279	. . . . 5 |- ( ph -> ( abs ` M ) e. RR )
20	13	abscld 13279	. . . . 5 |- ( ph -> ( abs ` N ) e. RR )
21	12	absge0d 13287	. . . . 5 |- ( ph -> 0 <_ ( abs ` M ) )
22		nnabscl 13170	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
23	10, 14, 22	syl2anc 661	. . . . . 6 |- ( ph -> ( abs ` N ) e. NN )
24	23	nnge1d 10599	. . . . 5 |- ( ph -> 1 <_ ( abs ` N ) )
25	19, 20, 21, 24	lemulge11d 10503	. . . 4 |- ( ph -> ( abs ` M ) <_ ( ( abs ` M ) x. ( abs ` N ) ) )
26	12, 13	absmuld 13297	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
27	25, 26	breqtrrd 4482	. . 3 |- ( ph -> ( abs ` M ) <_ ( abs ` ( M x. N ) ) )
28		eluz2 11112	. . 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 1180	. 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 23718	. . . . . 6 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> F : NN --> ZZ )
33	30, 3, 4, 32	syl3anc 1228	. . . . 5 |- ( ph -> F : NN --> ZZ )
34		elfznn 11739	. . . . 5 |- ( k e. ( 1 ... ( abs ` M ) ) -> k e. NN )
35		ffvelrn 6030	. . . . 5 |- ( ( F : NN --> ZZ /\ k e. NN ) -> ( F ` k ) e. ZZ )
36	33, 34, 35	syl2an 477	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( abs ` M ) ) ) -> ( F ` k ) e. ZZ )
37	36	zcnd 10991	. . 3 |- ( ( ph /\ k e. ( 1 ... ( abs ` M ) ) ) -> ( F ` k ) e. CC )
38		mulcl 9593	. . . 4 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
39	38	adantl 466	. . 3 |- ( ( ph /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
40	8, 37, 39	seqcl 12130	. 2 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. CC )
41	6	peano2nnd 10573	. . . . 5 |- ( ph -> ( ( abs ` M ) + 1 ) e. NN )
42		elfzuz 11709	. . . . 5 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) )
43		eluznn 11177	. . . . 5 |- ( ( ( ( abs ` M ) + 1 ) e. NN /\ k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) ) -> k e. NN )
44	41, 42, 43	syl2an 477	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. NN )
45		eleq1 2529	. . . . . 6 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
46		oveq2 6304	. . . . . . 7 |- ( n = k -> ( A /L n ) = ( A /L k ) )
47		oveq1 6303	. . . . . . 7 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
48	46, 47	oveq12d 6314	. . . . . 6 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
49	45, 48	ifbieq1d 3967	. . . . 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 6324	. . . . . 6 |- ( ( A /L k ) ^ ( k pCnt M ) ) e. _V
51		1ex 9608	. . . . . 6 |- 1 e. _V
52	50, 51	ifex 4013	. . . . 5 |- if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) e. _V
53	49, 31, 52	fvmpt 5956	. . . 4 |- ( k e. NN -> ( F ` k ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
54	44, 53	syl 16	. . 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 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. Prime )
56	3	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M e. ZZ )
57		zq 11213	. . . . . . . . . 10 |- ( M e. ZZ -> M e. QQ )
58	56, 57	syl 16	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M e. QQ )
59		pcabs 14410	. . . . . . . . 9 |- ( ( k e. Prime /\ M e. QQ ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
60	55, 58, 59	syl2anc 661	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
61		elfzle1 11714	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
62	61	adantl 466	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
63		elfzelz 11713	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ZZ )
64		zltp1le 10934	. . . . . . . . . . . . . 14 |- ( ( ( abs ` M ) e. ZZ /\ k e. ZZ ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
65	9, 63, 64	syl2an 477	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
66	62, 65	mpbird 232	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( abs ` M ) < k )
67	19	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( abs ` M ) e. RR )
68	63	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. ZZ )
69	68	zred 10990	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. RR )
70	67, 69	ltnled 9749	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> -. k <_ ( abs ` M ) ) )
71	66, 70	mpbid 210	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> -. k <_ ( abs ` M ) )
72	71	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k <_ ( abs ` M ) )
73		prmz 14233	. . . . . . . . . . . 12 |- ( k e. Prime -> k e. ZZ )
74	73	adantl 466	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. ZZ )
75	4	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M =/= 0 )
76	56, 75, 5	syl2anc 661	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( abs ` M ) e. NN )
77		dvdsle 14043	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ ( abs ` M ) e. NN ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
78	74, 76, 77	syl2anc 661	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
79	72, 78	mtod 177	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k || ( abs ` M ) )
80		pceq0 14406	. . . . . . . . . 10 |- ( ( k e. Prime /\ ( abs ` M ) e. NN ) -> ( ( k pCnt ( abs ` M ) ) = 0 <-> -. k || ( abs ` M ) ) )
81	55, 76, 80	syl2anc 661	. . . . . . . . 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 232	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = 0 )
83	60, 82	eqtr3d 2500	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) = 0 )
84	83	oveq2d 6312	. . . . . 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 725	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> A e. ZZ )
86		lgscl 23711	. . . . . . . . 9 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
87	85, 74, 86	syl2anc 661	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
88	87	zcnd 10991	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
89	88	exp0d 12307	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ 0 ) = 1 )
90	84, 89	eqtrd 2498	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) = 1 )
91	90	ifeq1da 3974	. . . 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 3981	. . . 4 |- if ( k e. Prime , 1 , 1 ) = 1
93	91, 92	syl6eq 2514	. . 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 2498	. 2 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( F ` k ) = 1 )
95	2, 8, 29, 40, 94	seqid2 12156	1 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   ifcif 3944   class class class wbr 4456   |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   < clt 9645   <_ cle 9646   NNcn 10556   ZZcz 10885   ZZ>=cuz 11106   QQcq 11207   ...cfz 11697   seqcseq 12110   ^cexp 12169   abscabs 13079   || cdvds 13998   Primecprime 14229   pCnt cpc 14372   /Lclgs 23695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-prm 14230  df-phi 14308  df-pc 14373  df-lgs 23696

This theorem is referenced by:  lgsdi  23733