Theorem lgsdilem2 22806

Description: Lemma for lgsdi 22807. (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 9497	. . 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 12934	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6	3, 4, 5	syl2anc 661	. . 3 |- ( ph -> ( abs ` M ) e. NN )
7		nnuz 11010	. . 3 |- NN = ( ZZ>= ` 1 )
8	6, 7	syl6eleq 2552	. 2 |- ( ph -> ( abs ` M ) e. ( ZZ>= ` 1 ) )
9	6	nnzd 10860	. . 3 |- ( ph -> ( abs ` M ) e. ZZ )
10		lgsdilem2.3	. . . . . 6 |- ( ph -> N e. ZZ )
11	3, 10	zmulcld 10867	. . . . 5 |- ( ph -> ( M x. N ) e. ZZ )
12	3	zcnd 10862	. . . . . 6 |- ( ph -> M e. CC )
13	10	zcnd 10862	. . . . . 6 |- ( ph -> N e. CC )
14		lgsdilem2.5	. . . . . 6 |- ( ph -> N =/= 0 )
15	12, 13, 4, 14	mulne0d 10102	. . . . 5 |- ( ph -> ( M x. N ) =/= 0 )
16		nnabscl 12934	. . . . 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 10860	. . 3 |- ( ph -> ( abs ` ( M x. N ) ) e. ZZ )
19	12	abscld 13043	. . . . 5 |- ( ph -> ( abs ` M ) e. RR )
20	13	abscld 13043	. . . . 5 |- ( ph -> ( abs ` N ) e. RR )
21	12	absge0d 13051	. . . . 5 |- ( ph -> 0 <_ ( abs ` M ) )
22		nnabscl 12934	. . . . . . 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 10478	. . . . 5 |- ( ph -> 1 <_ ( abs ` N ) )
25	19, 20, 21, 24	lemulge11d 10384	. . . 4 |- ( ph -> ( abs ` M ) <_ ( ( abs ` M ) x. ( abs ` N ) ) )
26	12, 13	absmuld 13061	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
27	25, 26	breqtrrd 4429	. . 3 |- ( ph -> ( abs ` M ) <_ ( abs ` ( M x. N ) ) )
28		eluz2 10981	. . 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 1172	. 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 22792	. . . . . 6 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> F : NN --> ZZ )
33	30, 3, 4, 32	syl3anc 1219	. . . . 5 |- ( ph -> F : NN --> ZZ )
34		elfznn 11598	. . . . 5 |- ( k e. ( 1 ... ( abs ` M ) ) -> k e. NN )
35		ffvelrn 5953	. . . . 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 10862	. . 3 |- ( ( ph /\ k e. ( 1 ... ( abs ` M ) ) ) -> ( F ` k ) e. CC )
38		mulcl 9480	. . . 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 11946	. 2 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. CC )
41	6	peano2nnd 10453	. . . . 5 |- ( ph -> ( ( abs ` M ) + 1 ) e. NN )
42		elfzuz 11569	. . . . 5 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) )
43		eluznn 11039	. . . . 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 2526	. . . . . 6 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
46		oveq2 6211	. . . . . . 7 |- ( n = k -> ( A /L n ) = ( A /L k ) )
47		oveq1 6210	. . . . . . 7 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
48	46, 47	oveq12d 6221	. . . . . 6 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
49	45, 48	ifbieq1d 3923	. . . . 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 6228	. . . . . 6 |- ( ( A /L k ) ^ ( k pCnt M ) ) e. _V
51		1ex 9495	. . . . . 6 |- 1 e. _V
52	50, 51	ifex 3969	. . . . 5 |- if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) e. _V
53	49, 31, 52	fvmpt 5886	. . . 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 11073	. . . . . . . . . 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 14062	. . . . . . . . 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 11574	. . . . . . . . . . . . . 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 11573	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ZZ )
64		zltp1le 10808	. . . . . . . . . . . . . 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 10861	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. RR )
70	67, 69	ltnled 9635	. . . . . . . . . . . 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 13888	. . . . . . . . . . . 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 13699	. . . . . . . . . . 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 14058	. . . . . . . . . 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 2497	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) = 0 )
84	83	oveq2d 6219	. . . . . 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 22785	. . . . . . . . 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 10862	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
89	88	exp0d 12122	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ 0 ) = 1 )
90	84, 89	eqtrd 2495	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) = 1 )
91	90	ifeq1da 3930	. . . 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 3937	. . . 4 |- if ( k e. Prime , 1 , 1 ) = 1
93	91, 92	syl6eq 2511	. . 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 2495	. 2 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( F ` k ) = 1 )
95	2, 8, 29, 40, 94	seqid2 11972	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 1370   e. wcel 1758   =/= wne 2648   ifcif 3902   class class class wbr 4403   |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397   + caddc 9399   x. cmul 9401   < clt 9532   <_ cle 9533   NNcn 10436   ZZcz 10760   ZZ>=cuz 10975   QQcq 11067   ...cfz 11557   seqcseq 11926   ^cexp 11985   abscabs 12844   || cdivides 13656   Primecprime 13884   pCnt cpc 14024   /Lclgs 22769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-fz 11558  df-fzo 11669  df-fl 11762  df-mod 11829  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-dvds 13657  df-gcd 13812  df-prm 13885  df-phi 13962  df-pc 14025  df-lgs 22770

This theorem is referenced by:  lgsdi  22807