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Theorem lgsdilem2 21117
Description: Lemma for lgsdi 21118. (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 9090 . . 3  |-  ( k  e.  CC  ->  (
k  x.  1 )  =  k )
21adantl 454 . 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 12131 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
63, 4, 5syl2anc 644 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  NN )
7 nnuz 10523 . . 3  |-  NN  =  ( ZZ>= `  1 )
86, 7syl6eleq 2528 . 2  |-  ( ph  ->  ( abs `  M
)  e.  ( ZZ>= ` 
1 ) )
96nnzd 10376 . . 3  |-  ( ph  ->  ( abs `  M
)  e.  ZZ )
10 lgsdilem2.3 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
113, 10zmulcld 10383 . . . . 5  |-  ( ph  ->  ( M  x.  N
)  e.  ZZ )
123zcnd 10378 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1310zcnd 10378 . . . . . 6  |-  ( ph  ->  N  e.  CC )
14 lgsdilem2.5 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
1512, 13, 4, 14mulne0d 9676 . . . . 5  |-  ( ph  ->  ( M  x.  N
)  =/=  0 )
16 nnabscl 12131 . . . . 5  |-  ( ( ( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
1711, 15, 16syl2anc 644 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  NN )
1817nnzd 10376 . . 3  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  e.  ZZ )
1912abscld 12240 . . . . 5  |-  ( ph  ->  ( abs `  M
)  e.  RR )
2013abscld 12240 . . . . 5  |-  ( ph  ->  ( abs `  N
)  e.  RR )
2112absge0d 12248 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  M ) )
22 nnabscl 12131 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
2310, 14, 22syl2anc 644 . . . . . 6  |-  ( ph  ->  ( abs `  N
)  e.  NN )
2423nnge1d 10044 . . . . 5  |-  ( ph  ->  1  <_  ( abs `  N ) )
2519, 20, 21, 24lemulge11d 9950 . . . 4  |-  ( ph  ->  ( abs `  M
)  <_  ( ( abs `  M )  x.  ( abs `  N
) ) )
2612, 13absmuld 12258 . . . 4  |-  ( ph  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
2725, 26breqtrrd 4240 . . 3  |-  ( ph  ->  ( abs `  M
)  <_  ( abs `  ( M  x.  N
) ) )
28 eluz2 10496 . . 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 1139 . 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 21103 . . . . . 6  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  F : NN --> ZZ )
3330, 3, 4, 32syl3anc 1185 . . . . 5  |-  ( ph  ->  F : NN --> ZZ )
34 elfznn 11082 . . . . 5  |-  ( k  e.  ( 1 ... ( abs `  M
) )  ->  k  e.  NN )
35 ffvelrn 5870 . . . . 5  |-  ( ( F : NN --> ZZ  /\  k  e.  NN )  ->  ( F `  k
)  e.  ZZ )
3633, 34, 35syl2an 465 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... ( abs `  M ) ) )  ->  ( F `  k )  e.  ZZ )
3736zcnd 10378 . . 3  |-  ( (
ph  /\  k  e.  ( 1 ... ( abs `  M ) ) )  ->  ( F `  k )  e.  CC )
38 mulcl 9076 . . . 4  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
3938adantl 454 . . 3  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
408, 37, 39seqcl 11345 . 2  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 ( abs `  M
) )  e.  CC )
416peano2nnd 10019 . . . . 5  |-  ( ph  ->  ( ( abs `  M
)  +  1 )  e.  NN )
42 elfzuz 11057 . . . . 5  |-  ( k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N )
) )  ->  k  e.  ( ZZ>= `  ( ( abs `  M )  +  1 ) ) )
437uztrn2 10505 . . . . 5  |-  ( ( ( ( abs `  M
)  +  1 )  e.  NN  /\  k  e.  ( ZZ>= `  ( ( abs `  M )  +  1 ) ) )  ->  k  e.  NN )
4441, 42, 43syl2an 465 . . . 4  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  NN )
45 eleq1 2498 . . . . . 6  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
46 oveq2 6091 . . . . . . 7  |-  ( n  =  k  ->  ( A  / L n )  =  ( A  / L k ) )
47 oveq1 6090 . . . . . . 7  |-  ( n  =  k  ->  (
n  pCnt  M )  =  ( k  pCnt  M ) )
4846, 47oveq12d 6101 . . . . . 6  |-  ( n  =  k  ->  (
( A  / L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  / L k ) ^ ( k 
pCnt  M ) ) )
49 eqidd 2439 . . . . . 6  |-  ( n  =  k  ->  1  =  1 )
5045, 48, 49ifbieq12d 3763 . . . . 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 ) )
51 ovex 6108 . . . . . 6  |-  ( ( A  / L k ) ^ ( k 
pCnt  M ) )  e. 
_V
52 1ex 9088 . . . . . 6  |-  1  e.  _V
5351, 52ifex 3799 . . . . 5  |-  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
5450, 31, 53fvmpt 5808 . . . 4  |-  ( k  e.  NN  ->  ( F `  k )  =  if ( k  e. 
Prime ,  ( ( A  / L k ) ^ ( k  pCnt  M ) ) ,  1 ) )
5544, 54syl 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 ) )
56 simpr 449 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  k  e.  Prime )
573ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  M  e.  ZZ )
58 zq 10582 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  QQ )
5957, 58syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  M  e.  QQ )
60 pcabs 13250 . . . . . . . . 9  |-  ( ( k  e.  Prime  /\  M  e.  QQ )  ->  (
k  pCnt  ( abs `  M ) )  =  ( k  pCnt  M
) )
6156, 59, 60syl2anc 644 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  ( abs `  M ) )  =  ( k 
pCnt  M ) )
62 elfzle1 11062 . . . . . . . . . . . . . 14  |-  ( k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N )
) )  ->  (
( abs `  M
)  +  1 )  <_  k )
6362adantl 454 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( ( abs `  M )  +  1 )  <_  k )
64 elfzelz 11061 . . . . . . . . . . . . . 14  |-  ( k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N )
) )  ->  k  e.  ZZ )
65 zltp1le 10327 . . . . . . . . . . . . . 14  |-  ( ( ( abs `  M
)  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( abs `  M
)  <  k  <->  ( ( abs `  M )  +  1 )  <_  k
) )
669, 64, 65syl2an 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( ( abs `  M )  <  k  <->  ( ( abs `  M
)  +  1 )  <_  k ) )
6763, 66mpbird 225 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( abs `  M
)  <  k )
6819adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( abs `  M
)  e.  RR )
6964adantl 454 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  ZZ )
7069zred 10377 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  k  e.  RR )
7168, 70ltnled 9222 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( ( abs `  M )  <  k  <->  -.  k  <_  ( abs `  M ) ) )
7267, 71mpbid 203 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  -.  k  <_  ( abs `  M ) )
7372adantr 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  -.  k  <_  ( abs `  M ) )
74 prmz 13085 . . . . . . . . . . . 12  |-  ( k  e.  Prime  ->  k  e.  ZZ )
7574adantl 454 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  k  e.  ZZ )
764ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  M  =/=  0
)
7757, 76, 5syl2anc 644 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( abs `  M
)  e.  NN )
78 dvdsle 12897 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  ( abs `  M )  e.  NN )  -> 
( k  ||  ( abs `  M )  -> 
k  <_  ( abs `  M ) ) )
7975, 77, 78syl2anc 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  ||  ( abs `  M )  ->  k  <_  ( abs `  M ) ) )
8073, 79mtod 171 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  -.  k  ||  ( abs `  M ) )
81 pceq0 13246 . . . . . . . . . 10  |-  ( ( k  e.  Prime  /\  ( abs `  M )  e.  NN )  ->  (
( k  pCnt  ( abs `  M ) )  =  0  <->  -.  k  ||  ( abs `  M
) ) )
8256, 77, 81syl2anc 644 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( k 
pCnt  ( abs `  M
) )  =  0  <->  -.  k  ||  ( abs `  M ) ) )
8380, 82mpbird 225 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  ( abs `  M ) )  =  0 )
8461, 83eqtr3d 2472 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( k  pCnt  M )  =  0 )
8584oveq2d 6099 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  / L k ) ^ ( k  pCnt  M ) )  =  ( ( A  / L
k ) ^ 0 ) )
8630ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  A  e.  ZZ )
87 lgscl 21096 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  / L
k )  e.  ZZ )
8886, 75, 87syl2anc 644 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  / L k )  e.  ZZ )
8988zcnd 10378 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( A  / L k )  e.  CC )
9089exp0d 11519 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  / L k ) ^ 0 )  =  1 )
9185, 90eqtrd 2470 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  /\  k  e.  Prime )  ->  ( ( A  / L k ) ^ ( k  pCnt  M ) )  =  1 )
9291ifeq1da 3766 . . . 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 ) )
93 ifid 3773 . . . 4  |-  if ( k  e.  Prime ,  1 ,  1 )  =  1
9492, 93syl6eq 2486 . . 3  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  if ( k  e.  Prime ,  ( ( A  / L k ) ^ ( k 
pCnt  M ) ) ,  1 )  =  1 )
9555, 94eqtrd 2470 . 2  |-  ( (
ph  /\  k  e.  ( ( ( abs `  M )  +  1 ) ... ( abs `  ( M  x.  N
) ) ) )  ->  ( F `  k )  =  1 )
962, 8, 29, 40, 95seqid2 11371 1  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 ( abs `  M
) )  =  (  seq  1 (  x.  ,  F ) `  ( abs `  ( M  x.  N ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ifcif 3741   class class class wbr 4214    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    < clt 9122    <_ cle 9123   NNcn 10002   ZZcz 10284   ZZ>=cuz 10490   QQcq 10576   ...cfz 11045    seq cseq 11325   ^cexp 11384   abscabs 12041    || cdivides 12854   Primecprime 13081    pCnt cpc 13212    / Lclgs 21080
This theorem is referenced by:  lgsdi  21118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-prm 13082  df-phi 13157  df-pc 13213  df-lgs 21081
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