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Theorem lgsdilem2 23472
Description: Lemma for lgsdi 23473. (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 9605 . . 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 13138 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
63, 4, 5syl2anc 661 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  NN )
7 nnuz 11129 . . 3  |-  NN  =  ( ZZ>= `  1 )
86, 7syl6eleq 2565 . 2  |-  ( ph  ->  ( abs `  M
)  e.  ( ZZ>= ` 
1 ) )
96nnzd 10977 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  ZZ )
10 lgsdilem2.3 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
113, 10zmulcld 10984 . . . . 5  |-  ( ph  ->  ( M  x.  N
)  e.  ZZ )
123zcnd 10979 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1310zcnd 10979 . . . . . 6  |-  ( ph  ->  N  e.  CC )
14 lgsdilem2.5 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
1512, 13, 4, 14mulne0d 10213 . . . . 5  |-  ( ph  ->  ( M  x.  N
)  =/=  0 )
16 nnabscl 13138 . . . . 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 10977 . . 3  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  ZZ )
1912abscld 13247 . . . . 5  |-  ( ph  ->  ( abs `  M
)  e.  RR )
2013abscld 13247 . . . . 5  |-  ( ph  ->  ( abs `  N
)  e.  RR )
2112absge0d 13255 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  M ) )
22 nnabscl 13138 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
2310, 14, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  ( abs `  N
)  e.  NN )
2423nnge1d 10590 . . . . 5  |-  ( ph  ->  1  <_  ( abs `  N ) )
2519, 20, 21, 24lemulge11d 10495 . . . 4  |-  ( ph  ->  ( abs `  M
)  <_  ( ( abs `  M )  x.  ( abs `  N
) ) )
2612, 13absmuld 13265 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
2725, 26breqtrrd 4479 . . 3  |-  ( ph  ->  ( abs `  M
)  <_  ( abs `  ( M  x.  N
) ) )
28 eluz2 11100 . . 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 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 ) )
3231lgsfcl3 23458 . . . . . 6  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  F : NN --> ZZ )
3330, 3, 4, 32syl3anc 1228 . . . . 5  |-  ( ph  ->  F : NN --> ZZ )
34 elfznn 11726 . . . . 5  |-  ( k  e.  ( 1 ... ( abs `  M
) )  ->  k  e.  NN )
35 ffvelrn 6030 . . . . 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 10979 . . 3  |-  ( (
ph  /\  k  e.  ( 1 ... ( abs `  M ) ) )  ->  ( F `  k )  e.  CC )
38 mulcl 9588 . . . 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 12107 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 ( abs `  M
) )  e.  CC )
416peano2nnd 10565 . . . . 5  |-  ( ph  ->  ( ( abs `  M
)  +  1 )  e.  NN )
42 elfzuz 11696 . . . . 5  |-  ( k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N )
) )  ->  k  e.  ( ZZ>= `  ( ( abs `  M )  +  1 ) ) )
43 eluznn 11164 . . . . 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 2539 . . . . . 6  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
46 oveq2 6303 . . . . . . 7  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
47 oveq1 6302 . . . . . . 7  |-  ( n  =  k  ->  (
n  pCnt  M )  =  ( k  pCnt  M ) )
4846, 47oveq12d 6313 . . . . . 6  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) )
4945, 48ifbieq1d 3968 . . . . 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 6320 . . . . . 6  |-  ( ( A  /L k ) ^ ( k 
pCnt  M ) )  e. 
_V
51 1ex 9603 . . . . . 6  |-  1  e.  _V
5250, 51ifex 4014 . . . . 5  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
5349, 31, 52fvmpt 5957 . . . 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 11200 . . . . . . . . . 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 14274 . . . . . . . . 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 11701 . . . . . . . . . . . . . 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 11700 . . . . . . . . . . . . . 14  |-  ( k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N )
) )  ->  k  e.  ZZ )
64 zltp1le 10924 . . . . . . . . . . . . . 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 10978 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  RR )
7067, 69ltnled 9743 . . . . . . . . . . . 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 14097 . . . . . . . . . . . 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 13907 . . . . . . . . . . 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 14270 . . . . . . . . . 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 2510 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  M )  =  0 )
8483oveq2d 6311 . . . . . 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 23451 . . . . . . . . 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 10979 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  /L k )  e.  CC )
8988exp0d 12284 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ 0 )  =  1 )
9084, 89eqtrd 2508 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ ( k  pCnt  M ) )  =  1 )
9190ifeq1da 3975 . . . 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 3982 . . . 4  |-  if ( k  e.  Prime ,  1 ,  1 )  =  1
9391, 92syl6eq 2524 . . 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 2508 . 2  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( F `  k )  =  1 )
952, 8, 29, 40, 94seqid2 12133 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 1379    e. wcel 1767    =/= wne 2662   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    <_ cle 9641   NNcn 10548   ZZcz 10876   ZZ>=cuz 11094   QQcq 11194   ...cfz 11684    seqcseq 12087   ^cexp 12146   abscabs 13047    || cdivides 13864   Primecprime 14093    pCnt cpc 14236    /Lclgs 23435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094  df-phi 14172  df-pc 14237  df-lgs 23436
This theorem is referenced by:  lgsdi  23473
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