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Theorem etransclem18 38229
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 11745 . . 3  |-  ( A (,) B )  C_  ( A [,] B )
21a1i 11 . 2  |-  ( ph  ->  ( A (,) B
)  C_  ( A [,] B ) )
3 ioombl 22597 . . 3  |-  ( A (,) B )  e. 
dom  vol
43a1i 11 . 2  |-  ( ph  ->  ( A (,) B
)  e.  dom  vol )
5 ere 14220 . . . . . 6  |-  _e  e.  RR
65recni 9673 . . . . 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 37698 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
1110sselda 3418 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
1211recnd 9687 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
1312negcld 9992 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u x  e.  CC )
147, 13cxpcld 23732 . . 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 37908 . . . . . 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 38219 . . . . 5  |-  ( ph  ->  F : RR --> CC )
2120adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F : RR
--> CC )
2221, 11ffvelrnd 6038 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
2314, 22mulcld 9681 . 2  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
_e  ^c  -u x
)  x.  ( F `
 x ) )  e.  CC )
24 eqidd 2472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( y  e.  CC  |->  ( _e  ^c  y ) )  =  ( y  e.  CC  |->  ( _e  ^c  y ) ) )
25 oveq2 6316 . . . . . . . . 9  |-  ( y  =  -u x  ->  (
_e  ^c  y )  =  ( _e  ^c  -u x ) )
2625adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  y  =  -u x )  -> 
( _e  ^c 
y )  =  ( _e  ^c  -u x ) )
2710, 17sstrd 3428 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  CC )
2827sselda 3418 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
2928negcld 9992 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u x  e.  CC )
306a1i 11 . . . . . . . . . 10  |-  ( x  e.  CC  ->  _e  e.  CC )
31 negcl 9895 . . . . . . . . . 10  |-  ( x  e.  CC  ->  -u x  e.  CC )
3230, 31cxpcld 23732 . . . . . . . . 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 5969 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
y  e.  CC  |->  ( _e  ^c  y ) ) `  -u x
)  =  ( _e 
^c  -u x
) )
3534eqcomd 2477 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( _e  ^c  -u x )  =  ( ( y  e.  CC  |->  ( _e  ^c  y ) ) `
 -u x ) )
3635mpteq2dva 4482 . . . . 5  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( _e  ^c  -u x ) )  =  ( x  e.  ( A [,] B ) 
|->  ( ( y  e.  CC  |->  ( _e  ^c  y ) ) `
 -u x ) ) )
37 epr 14337 . . . . . . . . 9  |-  _e  e.  RR+
38 mnfxr 11437 . . . . . . . . . . 11  |- -oo  e.  RR*
3938a1i 11 . . . . . . . . . 10  |-  ( _e  e.  RR+  -> -oo  e.  RR* )
40 0red 9662 . . . . . . . . . 10  |-  ( _e  e.  RR+  ->  0  e.  RR )
41 rpxr 11332 . . . . . . . . . 10  |-  ( _e  e.  RR+  ->  _e  e.  RR* )
42 rpgt0 11336 . . . . . . . . . 10  |-  ( _e  e.  RR+  ->  0  < 
_e )
4339, 40, 41, 42gtnelioc 37683 . . . . . . . . 9  |-  ( _e  e.  RR+  ->  -.  _e  e.  ( -oo (,] 0
) )
4437, 43ax-mp 5 . . . . . . . 8  |-  -.  _e  e.  ( -oo (,] 0
)
45 eldif 3400 . . . . . . . 8  |-  ( _e  e.  ( CC  \ 
( -oo (,] 0 ) )  <->  ( _e  e.  CC  /\  -.  _e  e.  ( -oo (,] 0 ) ) )
466, 44, 45mpbir2an 934 . . . . . . 7  |-  _e  e.  ( CC  \  ( -oo (,] 0 ) )
47 cxpcncf2 37875 . . . . . . 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 2471 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  -u x
)  =  ( x  e.  ( A [,] B )  |->  -u x
)
5049negcncf 22028 . . . . . . 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 22023 . . . . 5  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( ( y  e.  CC  |->  ( _e  ^c  y ) ) `
 -u x ) )  e.  ( ( A [,] B ) -cn-> CC ) )
5336, 52eqeltrd 2549 . . . 4  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( _e  ^c  -u x ) )  e.  ( ( A [,] B ) -cn-> CC ) )
54 ax-resscn 9614 . . . . . . . 8  |-  RR  C_  CC
5554a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  RR  C_  CC )
5618adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  P  e.  NN )
57 etransclem18.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
5857adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  M  e.  NN0 )
59 etransclem6 38217 . . . . . . . 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 2493 . . . . . . 7  |-  F  =  ( y  e.  RR  |->  ( ( y ^
( P  -  1 ) )  x.  prod_ k  e.  ( 1 ... M ) ( ( y  -  k ) ^ P ) ) )
6155, 56, 58, 60, 11etransclem13 38224 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  prod_ k  e.  ( 0 ... M ) ( ( x  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )
6261mpteq2dva 4482 . . . . 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 12224 . . . . . 6  |-  ( ph  ->  ( 0 ... M
)  e.  Fin )
64123adant3 1050 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  x  e.  CC )
65 elfzelz 11826 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ZZ )
6665zcnd 11064 . . . . . . . . 9  |-  ( k  e.  ( 0 ... M )  ->  k  e.  CC )
67663ad2ant3 1053 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  k  e.  CC )
6864, 67subcld 10005 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  ( x  -  k )  e.  CC )
69 nnm1nn0 10935 . . . . . . . . . 10  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
7018, 69syl 17 . . . . . . . . 9  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
7118nnnn0d 10949 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN0 )
7270, 71ifcld 3915 . . . . . . . 8  |-  ( ph  ->  if ( k  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
73723ad2ant1 1051 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  if (
k  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
7468, 73expcld 12454 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B )  /\  k  e.  ( 0 ... M ) )  ->  ( (
x  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) )  e.  CC )
75 nfv 1769 . . . . . . 7  |-  F/ x
( ph  /\  k  e.  ( 0 ... M
) )
76 ssid 3437 . . . . . . . . . . 11  |-  CC  C_  CC
7776a1i 11 . . . . . . . . . 10  |-  ( ph  ->  CC  C_  CC )
7827, 77idcncfg 37846 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  x )  e.  ( ( A [,] B
) -cn-> CC ) )
7978adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
x  e.  ( A [,] B )  |->  x )  e.  ( ( A [,] B )
-cn-> CC ) )
8027adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  ( A [,] B )  C_  CC )
8166adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  k  e.  CC )
8276a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  CC  C_  CC )
8380, 81, 82constcncfg 37845 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
x  e.  ( A [,] B )  |->  k )  e.  ( ( A [,] B )
-cn-> CC ) )
8479, 83subcncf 37843 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
x  e.  ( A [,] B )  |->  ( x  -  k ) )  e.  ( ( A [,] B )
-cn-> CC ) )
85 expcncf 22032 . . . . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  (
y  e.  CC  |->  ( y ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )  e.  ( CC -cn-> CC ) )
88 oveq1 6315 . . . . . . 7  |-  ( y  =  ( x  -  k )  ->  (
y ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) )  =  ( ( x  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )
8975, 84, 87, 82, 88cncfcompt2 37874 . . . . . 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 37876 . . . . 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 2549 . . . 4  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
9253, 91mulcncf 22476 . . 3  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( ( _e  ^c  -u x )  x.  ( F `  x
) ) )  e.  ( ( A [,] B ) -cn-> CC ) )
93 cniccibl 22877 . . 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 1292 . 2  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( ( _e  ^c  -u x )  x.  ( F `  x
) ) )  e.  L^1 )
952, 4, 23, 94iblss 22841 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    \ cdif 3387    C_ wss 3390   ifcif 3872   {cpr 3961    |-> cmpt 4454   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562   -oocmnf 9691   RR*cxr 9692    - cmin 9880   -ucneg 9881   NNcn 10631   NN0cn0 10893   RR+crp 11325   (,)cioo 11660   (,]cioc 11661   [,]cicc 11663   ...cfz 11810   ^cexp 12310   prod_cprod 14036   _eceu 14192   ↾t crest 15397   TopOpenctopn 15398  ℂfldccnfld 19047   -cn->ccncf 21986   volcvol 22493   L^1cibl 22654    ^c ccxp 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-prod 14037  df-ef 14198  df-e 14199  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-0p 22707  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586
This theorem is referenced by:  etransclem23  38234  etransclem46  38257
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