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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.
StepHypRef Expression
1 mulid1 9610 . . 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 13170 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
63, 4, 5syl2anc 661 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  NN )
7 nnuz 11141 . . 3  |-  NN  =  ( ZZ>= `  1 )
86, 7syl6eleq 2555 . 2  |-  ( ph  ->  ( abs `  M
)  e.  ( ZZ>= ` 
1 ) )
96nnzd 10989 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  ZZ )
10 lgsdilem2.3 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
113, 10zmulcld 10996 . . . . 5  |-  ( ph  ->  ( M  x.  N
)  e.  ZZ )
123zcnd 10991 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1310zcnd 10991 . . . . . 6  |-  ( ph  ->  N  e.  CC )
14 lgsdilem2.5 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
1512, 13, 4, 14mulne0d 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 )
1711, 15, 16syl2anc 661 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  NN )
1817nnzd 10989 . . 3  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  ZZ )
1912abscld 13279 . . . . 5  |-  ( ph  ->  ( abs `  M
)  e.  RR )
2013abscld 13279 . . . . 5  |-  ( ph  ->  ( abs `  N
)  e.  RR )
2112absge0d 13287 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  M ) )
22 nnabscl 13170 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
2310, 14, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  ( abs `  N
)  e.  NN )
2423nnge1d 10599 . . . . 5  |-  ( ph  ->  1  <_  ( abs `  N ) )
2519, 20, 21, 24lemulge11d 10503 . . . 4  |-  ( ph  ->  ( abs `  M
)  <_  ( ( abs `  M )  x.  ( abs `  N
) ) )
2612, 13absmuld 13297 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
2725, 26breqtrrd 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 )
) ) )
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 23718 . . . . . 6  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  F : NN --> ZZ )
3330, 3, 4, 32syl3anc 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 )
3633, 34, 35syl2an 477 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... ( abs `  M ) ) )  ->  ( F `  k )  e.  ZZ )
3736zcnd 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 )
3938adantl 466 . . 3  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
408, 37, 39seqcl 12130 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 ( abs `  M
) )  e.  CC )
416peano2nnd 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 )
4441, 42, 43syl2an 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 ) )
4846, 47oveq12d 6314 . . . . . 6  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) )
4945, 48ifbieq1d 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
5250, 51ifex 4013 . . . . 5  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
5349, 31, 52fvmpt 5956 . . . 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 11213 . . . . . . . . . 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 14410 . . . . . . . . 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 11714 . . . . . . . . . . . . . 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 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
) )
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 10990 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  RR )
7067, 69ltnled 9749 . . . . . . . . . . . 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 14233 . . . . . . . . . . . 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 14043 . . . . . . . . . . 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 14406 . . . . . . . . . 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 2500 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  M )  =  0 )
8483oveq2d 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 ) )
8530ad2antrr 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 )
8785, 74, 86syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  /L k )  e.  ZZ )
8887zcnd 10991 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  /L k )  e.  CC )
8988exp0d 12307 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ 0 )  =  1 )
9084, 89eqtrd 2498 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ ( k  pCnt  M ) )  =  1 )
9190ifeq1da 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
9391, 92syl6eq 2514 . . 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 2498 . 2  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( F `  k )  =  1 )
952, 8, 29, 40, 94seqid2 12156 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 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
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