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Theorem trilpolemisumle 13033
Description: Lemma for trilpo 13038. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)
Hypotheses
Ref Expression
trilpolemgt1.f  |-  ( ph  ->  F : NN --> { 0 ,  1 } )
trilpolemgt1.a  |-  A  = 
sum_ i  e.  NN  ( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)
trilpolemisumle.z  |-  Z  =  ( ZZ>= `  M )
trilpolemisumle.m  |-  ( ph  ->  M  e.  NN )
Assertion
Ref Expression
trilpolemisumle  |-  ( ph  -> 
sum_ i  e.  Z  ( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  <_  sum_ i  e.  Z  ( 1  / 
( 2 ^ i
) ) )
Distinct variable groups:    i, F    i, M    i, Z    ph, i
Allowed substitution hint:    A( i)

Proof of Theorem trilpolemisumle
Dummy variables  n  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trilpolemisumle.z . 2  |-  Z  =  ( ZZ>= `  M )
2 trilpolemisumle.m . . 3  |-  ( ph  ->  M  e.  NN )
32nnzd 9123 . 2  |-  ( ph  ->  M  e.  ZZ )
41eleq2i 2182 . . . . 5  |-  ( i  e.  Z  <->  i  e.  ( ZZ>= `  M )
)
54biimpi 119 . . . 4  |-  ( i  e.  Z  ->  i  e.  ( ZZ>= `  M )
)
6 eluznn 9343 . . . 4  |-  ( ( M  e.  NN  /\  i  e.  ( ZZ>= `  M ) )  -> 
i  e.  NN )
72, 5, 6syl2an 285 . . 3  |-  ( (
ph  /\  i  e.  Z )  ->  i  e.  NN )
8 eqid 2115 . . . 4  |-  ( n  e.  NN  |->  ( ( 1  /  ( 2 ^ n ) )  x.  ( F `  n ) ) )  =  ( n  e.  NN  |->  ( ( 1  /  ( 2 ^ n ) )  x.  ( F `  n
) ) )
9 oveq2 5748 . . . . . 6  |-  ( n  =  i  ->  (
2 ^ n )  =  ( 2 ^ i ) )
109oveq2d 5756 . . . . 5  |-  ( n  =  i  ->  (
1  /  ( 2 ^ n ) )  =  ( 1  / 
( 2 ^ i
) ) )
11 fveq2 5387 . . . . 5  |-  ( n  =  i  ->  ( F `  n )  =  ( F `  i ) )
1210, 11oveq12d 5758 . . . 4  |-  ( n  =  i  ->  (
( 1  /  (
2 ^ n ) )  x.  ( F `
 n ) )  =  ( ( 1  /  ( 2 ^ i ) )  x.  ( F `  i
) ) )
13 simpr 109 . . . 4  |-  ( (
ph  /\  i  e.  NN )  ->  i  e.  NN )
14 2rp 9395 . . . . . . . . 9  |-  2  e.  RR+
1514a1i 9 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN )  ->  2  e.  RR+ )
1613nnzd 9123 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN )  ->  i  e.  ZZ )
1715, 16rpexpcld 10388 . . . . . . 7  |-  ( (
ph  /\  i  e.  NN )  ->  ( 2 ^ i )  e.  RR+ )
1817rpreccld 9440 . . . . . 6  |-  ( (
ph  /\  i  e.  NN )  ->  ( 1  /  ( 2 ^ i ) )  e.  RR+ )
1918rpred 9429 . . . . 5  |-  ( (
ph  /\  i  e.  NN )  ->  ( 1  /  ( 2 ^ i ) )  e.  RR )
20 trilpolemgt1.f . . . . . . 7  |-  ( ph  ->  F : NN --> { 0 ,  1 } )
21 0re 7730 . . . . . . . . 9  |-  0  e.  RR
22 1re 7729 . . . . . . . . 9  |-  1  e.  RR
23 prssi 3646 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  { 0 ,  1 }  C_  RR )
2421, 22, 23mp2an 420 . . . . . . . 8  |-  { 0 ,  1 }  C_  RR
2524a1i 9 . . . . . . 7  |-  ( ph  ->  { 0 ,  1 }  C_  RR )
2620, 25fssd 5253 . . . . . 6  |-  ( ph  ->  F : NN --> RR )
2726ffvelrnda 5521 . . . . 5  |-  ( (
ph  /\  i  e.  NN )  ->  ( F `
 i )  e.  RR )
2819, 27remulcld 7760 . . . 4  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( 1  /  ( 2 ^ i ) )  x.  ( F `  i ) )  e.  RR )
298, 12, 13, 28fvmptd3 5480 . . 3  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( 1  /  (
2 ^ n ) )  x.  ( F `
 n ) ) ) `  i )  =  ( ( 1  /  ( 2 ^ i ) )  x.  ( F `  i
) ) )
307, 29syldan 278 . 2  |-  ( (
ph  /\  i  e.  Z )  ->  (
( n  e.  NN  |->  ( ( 1  / 
( 2 ^ n
) )  x.  ( F `  n )
) ) `  i
)  =  ( ( 1  /  ( 2 ^ i ) )  x.  ( F `  i ) ) )
317, 28syldan 278 . 2  |-  ( (
ph  /\  i  e.  Z )  ->  (
( 1  /  (
2 ^ i ) )  x.  ( F `
 i ) )  e.  RR )
32 eqid 2115 . . . 4  |-  ( n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) )  =  ( n  e.  NN  |->  ( 1  / 
( 2 ^ n
) ) )
3332, 10, 13, 18fvmptd3 5480 . . 3  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) `  i )  =  ( 1  / 
( 2 ^ i
) ) )
347, 33syldan 278 . 2  |-  ( (
ph  /\  i  e.  Z )  ->  (
( n  e.  NN  |->  ( 1  /  (
2 ^ n ) ) ) `  i
)  =  ( 1  /  ( 2 ^ i ) ) )
357, 19syldan 278 . 2  |-  ( (
ph  /\  i  e.  Z )  ->  (
1  /  ( 2 ^ i ) )  e.  RR )
36 simpr 109 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
( F `  i
)  =  0 )
3736oveq2d 5756 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  =  ( ( 1  /  ( 2 ^ i ) )  x.  0 ) )
3818rpcnd 9431 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN )  ->  ( 1  /  ( 2 ^ i ) )  e.  CC )
3938adantr 272 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
( 1  /  (
2 ^ i ) )  e.  CC )
4039mul01d 8119 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  0 )  =  0 )
4137, 40eqtrd 2148 . . . . 5  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  =  0 )
4218adantr 272 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
( 1  /  (
2 ^ i ) )  e.  RR+ )
4342rpge0d 9433 . . . . 5  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
0  <_  ( 1  /  ( 2 ^ i ) ) )
4441, 43eqbrtrd 3918 . . . 4  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  0 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  <_  ( 1  /  ( 2 ^ i ) ) )
45 simpr 109 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( F `  i
)  =  1 )
4645oveq2d 5756 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  =  ( ( 1  /  ( 2 ^ i ) )  x.  1 ) )
4738adantr 272 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( 1  /  (
2 ^ i ) )  e.  CC )
4847mulid1d 7747 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  1 )  =  ( 1  /  ( 2 ^ i ) ) )
4946, 48eqtrd 2148 . . . . 5  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  =  ( 1  /  ( 2 ^ i ) ) )
5019adantr 272 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( 1  /  (
2 ^ i ) )  e.  RR )
5150leidd 8240 . . . . 5  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( 1  /  (
2 ^ i ) )  <_  ( 1  /  ( 2 ^ i ) ) )
5249, 51eqbrtrd 3918 . . . 4  |-  ( ( ( ph  /\  i  e.  NN )  /\  ( F `  i )  =  1 )  -> 
( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  <_  ( 1  /  ( 2 ^ i ) ) )
5320ffvelrnda 5521 . . . . 5  |-  ( (
ph  /\  i  e.  NN )  ->  ( F `
 i )  e. 
{ 0 ,  1 } )
54 elpri 3518 . . . . 5  |-  ( ( F `  i )  e.  { 0 ,  1 }  ->  (
( F `  i
)  =  0  \/  ( F `  i
)  =  1 ) )
5553, 54syl 14 . . . 4  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( F `  i )  =  0  \/  ( F `  i )  =  1 ) )
5644, 52, 55mpjaodan 770 . . 3  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( 1  /  ( 2 ^ i ) )  x.  ( F `  i ) )  <_ 
( 1  /  (
2 ^ i ) ) )
577, 56syldan 278 . 2  |-  ( (
ph  /\  i  e.  Z )  ->  (
( 1  /  (
2 ^ i ) )  x.  ( F `
 i ) )  <_  ( 1  / 
( 2 ^ i
) ) )
5820, 8trilpolemclim 13031 . . 3  |-  ( ph  ->  seq 1 (  +  ,  ( n  e.  NN  |->  ( ( 1  /  ( 2 ^ n ) )  x.  ( F `  n
) ) ) )  e.  dom  ~~>  )
59 nnuz 9310 . . . 4  |-  NN  =  ( ZZ>= `  1 )
6029, 28eqeltrd 2192 . . . . 5  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( 1  /  (
2 ^ n ) )  x.  ( F `
 n ) ) ) `  i )  e.  RR )
6160recnd 7758 . . . 4  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( 1  /  (
2 ^ n ) )  x.  ( F `
 n ) ) ) `  i )  e.  CC )
6259, 2, 61iserex 11048 . . 3  |-  ( ph  ->  (  seq 1 (  +  ,  ( n  e.  NN  |->  ( ( 1  /  ( 2 ^ n ) )  x.  ( F `  n ) ) ) )  e.  dom  ~~>  <->  seq M (  +  ,  ( n  e.  NN  |->  ( ( 1  /  ( 2 ^ n ) )  x.  ( F `  n ) ) ) )  e.  dom  ~~>  ) )
6358, 62mpbid 146 . 2  |-  ( ph  ->  seq M (  +  ,  ( n  e.  NN  |->  ( ( 1  /  ( 2 ^ n ) )  x.  ( F `  n
) ) ) )  e.  dom  ~~>  )
64 seqex 10160 . . . 4  |-  seq 1
(  +  ,  ( n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) )  e.  _V
65 rpreccl 9416 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
6614, 65ax-mp 5 . . . . . . 7  |-  ( 1  /  2 )  e.  RR+
6766a1i 9 . . . . . 6  |-  ( ph  ->  ( 1  /  2
)  e.  RR+ )
68 1zzd 9032 . . . . . 6  |-  ( ph  ->  1  e.  ZZ )
6967, 68rpexpcld 10388 . . . . 5  |-  ( ph  ->  ( ( 1  / 
2 ) ^ 1 )  e.  RR+ )
70 1mhlfehlf 8889 . . . . . . 7  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
7170, 66eqeltri 2188 . . . . . 6  |-  ( 1  -  ( 1  / 
2 ) )  e.  RR+
7271a1i 9 . . . . 5  |-  ( ph  ->  ( 1  -  (
1  /  2 ) )  e.  RR+ )
7369, 72rpdivcld 9447 . . . 4  |-  ( ph  ->  ( ( ( 1  /  2 ) ^
1 )  /  (
1  -  ( 1  /  2 ) ) )  e.  RR+ )
74 halfcn 8885 . . . . . 6  |-  ( 1  /  2 )  e.  CC
7574a1i 9 . . . . 5  |-  ( ph  ->  ( 1  /  2
)  e.  CC )
76 halfge0 8887 . . . . . . . 8  |-  0  <_  ( 1  /  2
)
77 halfre 8884 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
7877absidi 10838 . . . . . . . 8  |-  ( 0  <_  ( 1  / 
2 )  ->  ( abs `  ( 1  / 
2 ) )  =  ( 1  /  2
) )
7976, 78ax-mp 5 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
80 halflt1 8888 . . . . . . 7  |-  ( 1  /  2 )  <  1
8179, 80eqbrtri 3917 . . . . . 6  |-  ( abs `  ( 1  /  2
) )  <  1
8281a1i 9 . . . . 5  |-  ( ph  ->  ( abs `  (
1  /  2 ) )  <  1 )
83 1nn0 8944 . . . . . 6  |-  1  e.  NN0
8483a1i 9 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
85 oveq2 5748 . . . . . . . 8  |-  ( n  =  j  ->  (
2 ^ n )  =  ( 2 ^ j ) )
8685oveq2d 5756 . . . . . . 7  |-  ( n  =  j  ->  (
1  /  ( 2 ^ n ) )  =  ( 1  / 
( 2 ^ j
) ) )
87 elnnuz 9311 . . . . . . . . 9  |-  ( j  e.  NN  <->  j  e.  ( ZZ>= `  1 )
)
8887biimpri 132 . . . . . . . 8  |-  ( j  e.  ( ZZ>= `  1
)  ->  j  e.  NN )
8988adantl 273 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  j  e.  NN )
9014a1i 9 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  2  e.  RR+ )
9189nnzd 9123 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  j  e.  ZZ )
9290, 91rpexpcld 10388 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  ( 2 ^ j )  e.  RR+ )
9392rpreccld 9440 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  ( 1  /  ( 2 ^ j ) )  e.  RR+ )
9432, 86, 89, 93fvmptd3 5480 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  ( (
n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) `  j )  =  ( 1  / 
( 2 ^ j
) ) )
95 2cnd 8750 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  2  e.  CC )
9690rpap0d 9435 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  2 #  0
)
9795, 96, 91exprecapd 10372 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  ( (
1  /  2 ) ^ j )  =  ( 1  /  (
2 ^ j ) ) )
9894, 97eqtr4d 2151 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  1 )
)  ->  ( (
n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) `  j )  =  ( ( 1  /  2 ) ^
j ) )
9975, 82, 84, 98geolim2 11221 . . . 4  |-  ( ph  ->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( 2 ^ n
) ) ) )  ~~>  ( ( ( 1  /  2 ) ^
1 )  /  (
1  -  ( 1  /  2 ) ) ) )
100 breldmg 4713 . . . 4  |-  ( (  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( 2 ^ n
) ) ) )  e.  _V  /\  (
( ( 1  / 
2 ) ^ 1 )  /  ( 1  -  ( 1  / 
2 ) ) )  e.  RR+  /\  seq 1
(  +  ,  ( n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) )  ~~>  ( ( ( 1  /  2
) ^ 1 )  /  ( 1  -  ( 1  /  2
) ) ) )  ->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( 2 ^ n
) ) ) )  e.  dom  ~~>  )
10164, 73, 99, 100mp3an2i 1303 . . 3  |-  ( ph  ->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( 2 ^ n
) ) ) )  e.  dom  ~~>  )
10233, 38eqeltrd 2192 . . . 4  |-  ( (
ph  /\  i  e.  NN )  ->  ( ( n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) `  i )  e.  CC )
10359, 2, 102iserex 11048 . . 3  |-  ( ph  ->  (  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) )  e.  dom  ~~>  <->  seq M (  +  ,  ( n  e.  NN  |->  ( 1  /  ( 2 ^ n ) ) ) )  e.  dom  ~~>  ) )
104101, 103mpbid 146 . 2  |-  ( ph  ->  seq M (  +  ,  ( n  e.  NN  |->  ( 1  / 
( 2 ^ n
) ) ) )  e.  dom  ~~>  )
1051, 3, 30, 31, 34, 35, 57, 63, 104isumle 11204 1  |-  ( ph  -> 
sum_ i  e.  Z  ( ( 1  / 
( 2 ^ i
) )  x.  ( F `  i )
)  <_  sum_ i  e.  Z  ( 1  / 
( 2 ^ i
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314    e. wcel 1463   _Vcvv 2658    C_ wss 3039   {cpr 3496   class class class wbr 3897    |-> cmpt 3957   dom cdm 4507   -->wf 5087   ` cfv 5091  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589    < clt 7764    <_ cle 7765    - cmin 7897    / cdiv 8392   NNcn 8677   2c2 8728   NN0cn0 8928   ZZ>=cuz 9275   RR+crp 9390    seqcseq 10158   ^cexp 10232   abscabs 10709    ~~> cli 10987   sum_csu 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-ico 9617  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063
This theorem is referenced by:  trilpolemgt1  13034  trilpolemeq1  13035
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