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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.
StepHypRef Expression
1 mulid1 9497 . . 3  |-  ( k  e.  CC  ->  (
k  x.  1 )  =  k )
21adantl 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 )
63, 4, 5syl2anc 661 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  NN )
7 nnuz 11010 . . 3  |-  NN  =  ( ZZ>= `  1 )
86, 7syl6eleq 2552 . 2  |-  ( ph  ->  ( abs `  M
)  e.  ( ZZ>= ` 
1 ) )
96nnzd 10860 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  ZZ )
10 lgsdilem2.3 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
113, 10zmulcld 10867 . . . . 5  |-  ( ph  ->  ( M  x.  N
)  e.  ZZ )
123zcnd 10862 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1310zcnd 10862 . . . . . 6  |-  ( ph  ->  N  e.  CC )
14 lgsdilem2.5 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
1512, 13, 4, 14mulne0d 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 )
1711, 15, 16syl2anc 661 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  NN )
1817nnzd 10860 . . 3  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  ZZ )
1912abscld 13043 . . . . 5  |-  ( ph  ->  ( abs `  M
)  e.  RR )
2013abscld 13043 . . . . 5  |-  ( ph  ->  ( abs `  N
)  e.  RR )
2112absge0d 13051 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  M ) )
22 nnabscl 12934 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
2310, 14, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  ( abs `  N
)  e.  NN )
2423nnge1d 10478 . . . . 5  |-  ( ph  ->  1  <_  ( abs `  N ) )
2519, 20, 21, 24lemulge11d 10384 . . . 4  |-  ( ph  ->  ( abs `  M
)  <_  ( ( abs `  M )  x.  ( abs `  N
) ) )
2612, 13absmuld 13061 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
2725, 26breqtrrd 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 )
) ) )
299, 18, 27, 28syl3anbrc 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 ) )
3231lgsfcl3 22792 . . . . . 6  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  F : NN --> ZZ )
3330, 3, 4, 32syl3anc 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 )
3633, 34, 35syl2an 477 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... ( abs `  M ) ) )  ->  ( F `  k )  e.  ZZ )
3736zcnd 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 )
3938adantl 466 . . 3  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
408, 37, 39seqcl 11946 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 ( abs `  M
) )  e.  CC )
416peano2nnd 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 )
4441, 42, 43syl2an 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 ) )
4846, 47oveq12d 6221 . . . . . 6  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) )
4945, 48ifbieq1d 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
5250, 51ifex 3969 . . . . 5  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
5349, 31, 52fvmpt 5886 . . . 4  |-  ( k  e.  NN  ->  ( F `  k )  =  if ( k  e. 
Prime ,  ( ( A  /L k ) ^ ( k  pCnt  M ) ) ,  1 ) )
5444, 53syl 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 )
563ad2antrr 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 )
5856, 57syl 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
) )
6055, 58, 59syl2anc 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 )
6261adantl 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
) )
659, 63, 64syl2an 477 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( ( abs `  M )  <  k  <->  ( ( abs `  M
)  +  1 )  <_  k ) )
6662, 65mpbird 232 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( abs `  M
)  <  k )
6719adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( abs `  M
)  e.  RR )
6863adantl 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  ZZ )
6968zred 10861 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  RR )
7067, 69ltnled 9635 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( ( abs `  M )  <  k  <->  -.  k  <_  ( abs `  M ) ) )
7166, 70mpbid 210 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  -.  k  <_  ( abs `  M ) )
7271adantr 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 )
7473adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  k  e.  ZZ )
754ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  M  =/=  0
)
7656, 75, 5syl2anc 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 ) ) )
7874, 76, 77syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  ||  ( abs `  M )  ->  k  <_  ( abs `  M ) ) )
7972, 78mtod 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
) ) )
8155, 76, 80syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( k 
pCnt  ( abs `  M
) )  =  0  <->  -.  k  ||  ( abs `  M ) ) )
8279, 81mpbird 232 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  ( abs `  M ) )  =  0 )
8360, 82eqtr3d 2497 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  M )  =  0 )
8483oveq2d 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 ) )
8530ad2antrr 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 )
8785, 74, 86syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  /L k )  e.  ZZ )
8887zcnd 10862 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  /L k )  e.  CC )
8988exp0d 12122 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ 0 )  =  1 )
9084, 89eqtrd 2495 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ ( k  pCnt  M ) )  =  1 )
9190ifeq1da 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
9391, 92syl6eq 2511 . . 3  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) ,  1 )  =  1 )
9454, 93eqtrd 2495 . 2  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( F `  k )  =  1 )
952, 8, 29, 40, 94seqid2 11972 1  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 ( abs `  M
) )  =  (  seq 1 (  x.  ,  F ) `  ( abs `  ( M  x.  N ) ) ) )
Colors of variables: wff setvar class
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
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