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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.
StepHypRef Expression
1 mulid1 9637 . . 3  |-  ( k  e.  CC  ->  (
k  x.  1 )  =  k )
21adantl 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 )
63, 4, 5syl2anc 666 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  NN )
7 nnuz 11191 . . 3  |-  NN  =  ( ZZ>= `  1 )
86, 7syl6eleq 2538 . 2  |-  ( ph  ->  ( abs `  M
)  e.  ( ZZ>= ` 
1 ) )
96nnzd 11036 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  ZZ )
10 lgsdilem2.3 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
113, 10zmulcld 11043 . . . . 5  |-  ( ph  ->  ( M  x.  N
)  e.  ZZ )
123zcnd 11038 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1310zcnd 11038 . . . . . 6  |-  ( ph  ->  N  e.  CC )
14 lgsdilem2.5 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
1512, 13, 4, 14mulne0d 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 )
1711, 15, 16syl2anc 666 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  NN )
1817nnzd 11036 . . 3  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  ZZ )
1912abscld 13491 . . . . 5  |-  ( ph  ->  ( abs `  M
)  e.  RR )
2013abscld 13491 . . . . 5  |-  ( ph  ->  ( abs `  N
)  e.  RR )
2112absge0d 13499 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  M ) )
22 nnabscl 13381 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
2310, 14, 22syl2anc 666 . . . . . 6  |-  ( ph  ->  ( abs `  N
)  e.  NN )
2423nnge1d 10649 . . . . 5  |-  ( ph  ->  1  <_  ( abs `  N ) )
2519, 20, 21, 24lemulge11d 10541 . . . 4  |-  ( ph  ->  ( abs `  M
)  <_  ( ( abs `  M )  x.  ( abs `  N
) ) )
2612, 13absmuld 13509 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
2725, 26breqtrrd 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 )
) ) )
299, 18, 27, 28syl3anbrc 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 ) )
3231lgsfcl3 24238 . . . . . 6  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  F : NN --> ZZ )
3330, 3, 4, 32syl3anc 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 )
3633, 34, 35syl2an 480 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... ( abs `  M ) ) )  ->  ( F `  k )  e.  ZZ )
3736zcnd 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 )
3938adantl 468 . . 3  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
408, 37, 39seqcl 12230 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 ( abs `  M
) )  e.  CC )
416peano2nnd 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 )
4441, 42, 43syl2an 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 ) )
4846, 47oveq12d 6306 . . . . . 6  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) )
4945, 48ifbieq1d 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
5250, 51ifex 3948 . . . . 5  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
5349, 31, 52fvmpt 5946 . . . 4  |-  ( k  e.  NN  ->  ( F `  k )  =  if ( k  e. 
Prime ,  ( ( A  /L k ) ^ ( k  pCnt  M ) ) ,  1 ) )
5444, 53syl 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 )
563ad2antrr 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 )
5856, 57syl 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
) )
6055, 58, 59syl2anc 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 )
6261adantl 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
) )
659, 63, 64syl2an 480 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( ( abs `  M )  <  k  <->  ( ( abs `  M
)  +  1 )  <_  k ) )
6662, 65mpbird 236 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( abs `  M
)  <  k )
6719adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( abs `  M
)  e.  RR )
6863adantl 468 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  ZZ )
6968zred 11037 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  RR )
7067, 69ltnled 9779 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( ( abs `  M )  <  k  <->  -.  k  <_  ( abs `  M ) ) )
7166, 70mpbid 214 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  -.  k  <_  ( abs `  M ) )
7271adantr 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 )
7473adantl 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  k  e.  ZZ )
754ad2antrr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  M  =/=  0
)
7656, 75, 5syl2anc 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 ) ) )
7874, 76, 77syl2anc 666 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  ||  ( abs `  M )  ->  k  <_  ( abs `  M ) ) )
7972, 78mtod 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
) ) )
8155, 76, 80syl2anc 666 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( k 
pCnt  ( abs `  M
) )  =  0  <->  -.  k  ||  ( abs `  M ) ) )
8279, 81mpbird 236 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  ( abs `  M ) )  =  0 )
8360, 82eqtr3d 2486 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  M )  =  0 )
8483oveq2d 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 ) )
8530ad2antrr 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 )
8785, 74, 86syl2anc 666 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  /L k )  e.  ZZ )
8887zcnd 11038 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  /L k )  e.  CC )
8988exp0d 12407 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ 0 )  =  1 )
9084, 89eqtrd 2484 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  /L k ) ^ ( k  pCnt  M ) )  =  1 )
9190ifeq1da 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
9391, 92syl6eq 2500 . . 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 2484 . 2  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( F `  k )  =  1 )
952, 8, 29, 40, 94seqid2 12256 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 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
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