Theorem lgsdilem2 24252

Description: Lemma for lgsdi 24253. (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 9637	. . 3 |- ( k e. CC -> ( k x. 1 ) = k )
2	1	adantl 468	. 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 13381	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6	3, 4, 5	syl2anc 666	. . 3 |- ( ph -> ( abs ` M ) e. NN )
7		nnuz 11191	. . 3 |- NN = ( ZZ>= ` 1 )
8	6, 7	syl6eleq 2538	. 2 |- ( ph -> ( abs ` M ) e. ( ZZ>= ` 1 ) )
9	6	nnzd 11036	. . 3 |- ( ph -> ( abs ` M ) e. ZZ )
10		lgsdilem2.3	. . . . . 6 |- ( ph -> N e. ZZ )
11	3, 10	zmulcld 11043	. . . . 5 |- ( ph -> ( M x. N ) e. ZZ )
12	3	zcnd 11038	. . . . . 6 |- ( ph -> M e. CC )
13	10	zcnd 11038	. . . . . 6 |- ( ph -> N e. CC )
14		lgsdilem2.5	. . . . . 6 |- ( ph -> N =/= 0 )
15	12, 13, 4, 14	mulne0d 10261	. . . . 5 |- ( ph -> ( M x. N ) =/= 0 )
16		nnabscl 13381	. . . . 5 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
17	11, 15, 16	syl2anc 666	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) e. NN )
18	17	nnzd 11036	. . 3 |- ( ph -> ( abs ` ( M x. N ) ) e. ZZ )
19	12	abscld 13491	. . . . 5 |- ( ph -> ( abs ` M ) e. RR )
20	13	abscld 13491	. . . . 5 |- ( ph -> ( abs ` N ) e. RR )
21	12	absge0d 13499	. . . . 5 |- ( ph -> 0 <_ ( abs ` M ) )
22		nnabscl 13381	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
23	10, 14, 22	syl2anc 666	. . . . . 6 |- ( ph -> ( abs ` N ) e. NN )
24	23	nnge1d 10649	. . . . 5 |- ( ph -> 1 <_ ( abs ` N ) )
25	19, 20, 21, 24	lemulge11d 10541	. . . 4 |- ( ph -> ( abs ` M ) <_ ( ( abs ` M ) x. ( abs ` N ) ) )
26	12, 13	absmuld 13509	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
27	25, 26	breqtrrd 4428	. . 3 |- ( ph -> ( abs ` M ) <_ ( abs ` ( M x. N ) ) )
28		eluz2 11162	. . 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 1191	. 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 24238	. . . . . 6 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> F : NN --> ZZ )
33	30, 3, 4, 32	syl3anc 1267	. . . . 5 |- ( ph -> F : NN --> ZZ )
34		elfznn 11825	. . . . 5 |- ( k e. ( 1 ... ( abs ` M ) ) -> k e. NN )
35		ffvelrn 6018	. . . . 5 |- ( ( F : NN --> ZZ /\ k e. NN ) -> ( F ` k ) e. ZZ )
36	33, 34, 35	syl2an 480	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( abs ` M ) ) ) -> ( F ` k ) e. ZZ )
37	36	zcnd 11038	. . 3 |- ( ( ph /\ k e. ( 1 ... ( abs ` M ) ) ) -> ( F ` k ) e. CC )
38		mulcl 9620	. . . 4 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
39	38	adantl 468	. . 3 |- ( ( ph /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
40	8, 37, 39	seqcl 12230	. 2 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. CC )
41	6	peano2nnd 10623	. . . . 5 |- ( ph -> ( ( abs ` M ) + 1 ) e. NN )
42		elfzuz 11793	. . . . 5 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) )
43		eluznn 11226	. . . . 5 |- ( ( ( ( abs ` M ) + 1 ) e. NN /\ k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) ) -> k e. NN )
44	41, 42, 43	syl2an 480	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. NN )
45		eleq1 2516	. . . . . 6 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
46		oveq2 6296	. . . . . . 7 |- ( n = k -> ( A /L n ) = ( A /L k ) )
47		oveq1 6295	. . . . . . 7 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
48	46, 47	oveq12d 6306	. . . . . 6 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
49	45, 48	ifbieq1d 3903	. . . . 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 6316	. . . . . 6 |- ( ( A /L k ) ^ ( k pCnt M ) ) e. _V
51		1ex 9635	. . . . . 6 |- 1 e. _V
52	50, 51	ifex 3948	. . . . 5 |- if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) e. _V
53	49, 31, 52	fvmpt 5946	. . . 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 463	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. Prime )
56	3	ad2antrr 731	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M e. ZZ )
57		zq 11267	. . . . . . . . . 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 14817	. . . . . . . . 9 |- ( ( k e. Prime /\ M e. QQ ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
60	55, 58, 59	syl2anc 666	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
61		elfzle1 11799	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
62	61	adantl 468	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
63		elfzelz 11797	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ZZ )
64		zltp1le 10983	. . . . . . . . . . . . . 14 |- ( ( ( abs ` M ) e. ZZ /\ k e. ZZ ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
65	9, 63, 64	syl2an 480	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
66	62, 65	mpbird 236	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( abs ` M ) < k )
67	19	adantr 467	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( abs ` M ) e. RR )
68	63	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. ZZ )
69	68	zred 11037	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. RR )
70	67, 69	ltnled 9779	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> -. k <_ ( abs ` M ) ) )
71	66, 70	mpbid 214	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> -. k <_ ( abs ` M ) )
72	71	adantr 467	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k <_ ( abs ` M ) )
73		prmz 14619	. . . . . . . . . . . 12 |- ( k e. Prime -> k e. ZZ )
74	73	adantl 468	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. ZZ )
75	4	ad2antrr 731	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M =/= 0 )
76	56, 75, 5	syl2anc 666	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( abs ` M ) e. NN )
77		dvdsle 14343	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ ( abs ` M ) e. NN ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
78	74, 76, 77	syl2anc 666	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
79	72, 78	mtod 181	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k || ( abs ` M ) )
80		pceq0 14813	. . . . . . . . . 10 |- ( ( k e. Prime /\ ( abs ` M ) e. NN ) -> ( ( k pCnt ( abs ` M ) ) = 0 <-> -. k || ( abs ` M ) ) )
81	55, 76, 80	syl2anc 666	. . . . . . . . 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 236	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = 0 )
83	60, 82	eqtr3d 2486	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) = 0 )
84	83	oveq2d 6304	. . . . . 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 731	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> A e. ZZ )
86		lgscl 24231	. . . . . . . . 9 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
87	85, 74, 86	syl2anc 666	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
88	87	zcnd 11038	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
89	88	exp0d 12407	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ 0 ) = 1 )
90	84, 89	eqtrd 2484	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) = 1 )
91	90	ifeq1da 3910	. . . 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 3917	. . . 4 |- if ( k e. Prime , 1 , 1 ) = 1
93	91, 92	syl6eq 2500	. . 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 2484	. 2 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( F ` k ) = 1 )
95	2, 8, 29, 40, 94	seqid2 12256	1 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   ifcif 3880   class class class wbr 4401   |-> cmpt 4460   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   x. cmul 9541   < clt 9672   <_ cle 9673   NNcn 10606   ZZcz 10934   ZZ>=cuz 11156   QQcq 11261   ...cfz 11781   seqcseq 12210   ^cexp 12269   abscabs 13290   || cdvds 14298   Primecprime 14615   pCnt cpc 14779   /Lclgs 24215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-dvds 14299  df-gcd 14462  df-prm 14616  df-phi 14707  df-pc 14780  df-lgs 24216

This theorem is referenced by:  lgsdi  24253