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Theorem etransclem18 38117
Description: The given function is integrable . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem18.s  |-  ( ph  ->  RR  e.  { RR ,  CC } )
etransclem18.x  |-  ( ph  ->  RR  e.  ( (
TopOpen ` fld )t  RR ) )
etransclem18.p  |-  ( ph  ->  P  e.  NN )
etransclem18.m  |-  ( ph  ->  M  e.  NN0 )
etransclem18.f  |-  F  =  ( x  e.  RR  |->  ( ( x ^
( P  -  1 ) )  x.  prod_ j  e.  ( 1 ... M ) ( ( x  -  j ) ^ P ) ) )
etransclem18.a  |-  ( ph  ->  A  e.  RR )
etransclem18.b  |-  ( ph  ->  B  e.  RR )
Assertion
Ref Expression
etransclem18  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( _e  ^c  -u x )  x.  ( F `  x
) ) )  e.  L^1 )
Distinct variable groups:    x, A    x, B    j, M, x    P, j, x    ph, j, x
Allowed substitution hints:    A( j)    B( j)    F( x, j)

Proof of Theorem etransclem18
Dummy variables  k 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioossicc 11720 . . 3  |-  ( A (,) B )  C_  ( A [,] B )
21a1i 11 . 2  |-  ( ph  ->  ( A (,) B
)  C_  ( A [,] B ) )
3 ioombl 22518 . . 3  |-  ( A (,) B )  e. 
dom  vol
43a1i 11 . 2  |-  ( ph  ->  ( A (,) B
)  e.  dom  vol )
5 ere 14143 . . . . . 6  |-  _e  e.  RR
65recni 9655 . . . . 5  |-  _e  e.  CC
76a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  _e  e.  CC )
8 etransclem18.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
9 etransclem18.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
108, 9iccssred 37602 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
1110sselda 3432 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
1211recnd 9669 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
1312negcld 9973 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u x  e.  CC )
147, 13cxpcld 23653 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( _e  ^c  -u x )  e.  CC )
15 etransclem18.s . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
16 etransclem18.x . . . . . . 7  |-  ( ph  ->  RR  e.  ( (
TopOpen ` fld )t  RR ) )
1715, 16dvdmsscn 37811 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
18 etransclem18.p . . . . . 6  |-  ( ph  ->  P  e.  NN )
19 etransclem18.f . . . . . 6  |-  F  =  ( x  e.  RR  |->  ( ( x ^
( P  -  1 ) )  x.  prod_ j  e.  ( 1 ... M ) ( ( x  -  j ) ^ P ) ) )
2017, 18, 19etransclem8 38107 . . . . 5  |-  ( ph  ->  F : RR --> CC )
2120adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F : RR
--> CC )
2221, 11ffvelrnd 6023 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
2314, 22mulcld 9663 . 2  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
_e  ^c  -u x
)  x.  ( F `
 x ) )  e.  CC )
24 eqidd 2452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( y  e.  CC  |->  ( _e  ^c  y ) )  =  ( y  e.  CC  |->  ( _e  ^c  y ) ) )
25 oveq2 6298 . . . . . . . . 9  |-  ( y  =  -u x  ->  (
_e  ^c  y )  =  ( _e  ^c  -u x ) )
2625adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  y  =  -u x )  -> 
( _e  ^c 
y )  =  ( _e  ^c  -u x ) )
2710, 17sstrd 3442 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  CC )
2827sselda 3432 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
2928negcld 9973 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u x  e.  CC )
306a1i 11 . . . . . . . . . 10  |-  ( x  e.  CC  ->  _e  e.  CC )
31 negcl 9875 . . . . . . . . . 10  |-  ( x  e.  CC  ->  -u x  e.  CC )
3230, 31cxpcld 23653 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
_e  ^c  -u x
)  e.  CC )
3328, 32syl 17 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( _e  ^c  -u x )  e.  CC )
3424, 26, 29, 33fvmptd 5954 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
y  e.  CC  |->  ( _e  ^c  y ) ) `  -u x
)  =  ( _e 
^c  -u x
) )
3534eqcomd 2457 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( _e  ^c  -u x )  =  ( ( y  e.  CC  |->  ( _e  ^c  y ) ) `
 -u x ) )
3635mpteq2dva 4489 . . . . 5  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( _e  ^c  -u x ) )  =  ( x  e.  ( A [,] B ) 
|->  ( ( y  e.  CC  |->  ( _e  ^c  y ) ) `
 -u x ) ) )
37 epr 14260 . . . . . . . . 9  |-  _e  e.  RR+
38 mnfxr 11414 . . . . . . . . . . 11  |- -oo  e.  RR*
3938a1i 11 . . . . . . . . . 10  |-  ( _e  e.  RR+  -> -oo  e.  RR* )
40 0red 9644 . . . . . . . . . 10  |-  ( _e  e.  RR+  ->  0  e.  RR )
41 rpxr 11309 . . . . . . . . . 10  |-  ( _e  e.  RR+  ->  _e  e.  RR* )
42 rpgt0 11313 . . . . . . . . . 10  |-  ( _e  e.  RR+  ->  0  < 
_e )
4339, 40, 41, 42gtnelioc 37587 . . . . . . . . 9  |-  ( _e  e.  RR+  ->  -.  _e  e.  ( -oo (,] 0
) )
4437, 43ax-mp 5 . . . . . . . 8  |-  -.  _e  e.  ( -oo (,] 0
)
45 eldif 3414 . . . . . . . 8  |-  ( _e  e.  ( CC  \ 
( -oo (,] 0 ) )  <->  ( _e  e.  CC  /\  -.  _e  e.  ( -oo (,] 0 ) ) )
466, 44, 45mpbir2an 931 . . . . . . 7  |-  _e  e.  ( CC  \  ( -oo (,] 0 ) )
47 cxpcncf2 37778 . . . . . . 7  |-  ( _e  e.  ( CC  \ 
( -oo (,] 0 ) )  ->  ( y  e.  CC  |->  ( _e  ^c  y ) )  e.  ( CC -cn-> CC ) )
4846, 47mp1i 13 . . . . . 6  |-  ( ph  ->  ( y  e.  CC  |->  ( _e  ^c 
y ) )  e.  ( CC -cn-> CC ) )
49 eqid 2451 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  -u x
)  =  ( x  e.  ( A [,] B )  |->  -u x
)
5049negcncf 21950 . . . . . . 7  |-  ( ( A [,] B ) 
C_  CC  ->  ( x  e.  ( A [,] B )  |->  -u x
)  e.  ( ( A [,] B )
-cn-> CC ) )
5127, 50syl 17 . . . . . 6  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  -u x )  e.  ( ( A [,] B ) -cn-> CC ) )
5248, 51cncfmpt1f 21945 . . . . 5  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( ( y  e.  CC  |->  ( _e  ^c  y ) ) `
 -u x ) )  e.  ( ( A [,] B ) -cn-> CC ) )
5336, 52eqeltrd 2529 . . . 4  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( _e  ^c  -u x ) )  e.  ( ( A [,] B ) -cn-> CC ) )
54 ax-resscn 9596 . . . . . . . 8  |-  RR  C_  CC
5554a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  RR  C_  CC )
5618adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  P  e.  NN )
57 etransclem18.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
5857adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  M  e.  NN0 )
59 etransclem6 38105 . . . . . . . 8  |-  ( x  e.  RR  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  ( 1 ... M ) ( ( x  -  j ) ^ P
) ) )  =  ( y  e.  RR  |->  ( ( y ^
( P  -  1 ) )  x.  prod_ k  e.  ( 1 ... M ) ( ( y  -  k ) ^ P ) ) )
6019, 59eqtri 2473 . . . . . . 7  |-  F  =  ( y  e.  RR  |->  ( ( y ^
( P  -  1 ) )  x.  prod_ k  e.  ( 1 ... M ) ( ( y  -  k ) ^ P ) ) )
6155, 56, 58, 60, 11etransclem13 38112 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  prod_ k  e.  ( 0 ... M ) ( ( x  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )
6261mpteq2dva 4489 . . . . 5  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  =  ( x  e.  ( A [,] B )  |->  prod_
k  e.  ( 0 ... M ) ( ( x  -  k
) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
63 fzfid 12186 . . . . . 6  |-  ( ph  ->  ( 0 ... M
)  e.  Fin )
64123adant3 1028 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  x  e.  CC )
65 elfzelz 11800 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ZZ )
6665zcnd 11041 . . . . . . . . 9  |-  ( k  e.  ( 0 ... M )  ->  k  e.  CC )
67663ad2ant3 1031 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  k  e.  CC )
6864, 67subcld 9986 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  ( x  -  k )  e.  CC )
69 nnm1nn0 10911 . . . . . . . . . 10  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
7018, 69syl 17 . . . . . . . . 9  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
7118nnnn0d 10925 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN0 )
7270, 71ifcld 3924 . . . . . . . 8  |-  ( ph  ->  if ( k  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
73723ad2ant1 1029 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  if (
k  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
7468, 73expcld 12416 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  ( (
x  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) )  e.  CC )
75 nfv 1761 . . . . . . 7  |-  F/ x
( ph  /\  k  e.  ( 0 ... M
) )
76 ssid 3451 . . . . . . . . . . 11  |-  CC  C_  CC
7776a1i 11 . . . . . . . . . 10  |-  ( ph  ->  CC  C_  CC )
7827, 77idcncfg 37749 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  x )  e.  ( ( A [,] B
) -cn-> CC ) )
7978adantr 467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
x  e.  ( A [,] B )  |->  x )  e.  ( ( A [,] B )
-cn-> CC ) )
8027adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  ( A [,] B )  C_  CC )
8166adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  k  e.  CC )
8276a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  CC  C_  CC )
8380, 81, 82constcncfg 37748 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
x  e.  ( A [,] B )  |->  k )  e.  ( ( A [,] B )
-cn-> CC ) )
8479, 83subcncf 37746 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
x  e.  ( A [,] B )  |->  ( x  -  k ) )  e.  ( ( A [,] B )
-cn-> CC ) )
85 expcncf 21954 . . . . . . . . 9  |-  ( if ( k  =  0 ,  ( P  - 
1 ) ,  P
)  e.  NN0  ->  ( y  e.  CC  |->  ( y ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )  e.  ( CC -cn-> CC ) )
8672, 85syl 17 . . . . . . . 8  |-  ( ph  ->  ( y  e.  CC  |->  ( y ^ if ( k  =  0 ,  ( P  - 
1 ) ,  P
) ) )  e.  ( CC -cn-> CC ) )
8786adantr 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
y  e.  CC  |->  ( y ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )  e.  ( CC -cn-> CC ) )
88 oveq1 6297 . . . . . . 7  |-  ( y  =  ( x  -  k )  ->  (
y ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) )  =  ( ( x  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )
8975, 84, 87, 82, 88cncfcompt2 37777 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
x  e.  ( A [,] B )  |->  ( ( x  -  k
) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )  e.  ( ( A [,] B
) -cn-> CC ) )
9027, 63, 74, 89fprodcncf 37779 . . . . 5  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  prod_ k  e.  ( 0 ... M ) ( ( x  -  k ) ^ if ( k  =  0 ,  ( P  - 
1 ) ,  P
) ) )  e.  ( ( A [,] B ) -cn-> CC ) )
9162, 90eqeltrd 2529 . . . 4  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
9253, 91mulcncf 22398 . . 3  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( ( _e  ^c  -u x )  x.  ( F `  x
) ) )  e.  ( ( A [,] B ) -cn-> CC ) )
93 cniccibl 22798 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
x  e.  ( A [,] B )  |->  ( ( _e  ^c  -u x )  x.  ( F `  x )
) )  e.  ( ( A [,] B
) -cn-> CC ) )  -> 
( x  e.  ( A [,] B ) 
|->  ( ( _e  ^c  -u x )  x.  ( F `  x
) ) )  e.  L^1 )
948, 9, 92, 93syl3anc 1268 . 2  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( ( _e  ^c  -u x )  x.  ( F `  x
) ) )  e.  L^1 )
952, 4, 23, 94iblss 22762 1  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( _e  ^c  -u x )  x.  ( F `  x
) ) )  e.  L^1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    \ cdif 3401    C_ wss 3404   ifcif 3881   {cpr 3970    |-> cmpt 4461   dom cdm 4834   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544   -oocmnf 9673   RR*cxr 9674    - cmin 9860   -ucneg 9861   NNcn 10609   NN0cn0 10869   RR+crp 11302   (,)cioo 11635   (,]cioc 11636   [,]cicc 11638   ...cfz 11784   ^cexp 12272   prod_cprod 13959   _eceu 14115   ↾t crest 15319   TopOpenctopn 15320  ℂfldccnfld 18970   -cn->ccncf 21908   volcvol 22415   L^1cibl 22575    ^c ccxp 23505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-prod 13960  df-ef 14121  df-e 14122  df-sin 14123  df-cos 14124  df-tan 14125  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-ibl 22580  df-0p 22628  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507
This theorem is referenced by:  etransclem23  38122  etransclem46  38145
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