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Theorem etransclem17 38228
Description: The  N-th derivative of  H. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem17.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
etransclem17.x  |-  ( ph  ->  X  e.  ( (
TopOpen ` fld )t  S ) )
etransclem17.p  |-  ( ph  ->  P  e.  NN )
etransclem17.1  |-  H  =  ( j  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
etransclem17.J  |-  ( ph  ->  J  e.  ( 0 ... M ) )
etransclem17.n  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
etransclem17  |-  ( ph  ->  ( ( S  Dn ( H `  J ) ) `  N )  =  ( x  e.  X  |->  if ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  < 
N ,  0 ,  ( ( ( ! `
 if ( J  =  0 ,  ( P  -  1 ) ,  P ) )  /  ( ! `  ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N
) ) )  x.  ( ( x  -  J ) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) ) ) )
Distinct variable groups:    j, J, x    j, M, x    x, N    P, j, x    x, S    j, X, x    ph, j, x
Allowed substitution hints:    S( j)    H( x, j)    N( j)

Proof of Theorem etransclem17
StepHypRef Expression
1 etransclem17.1 . . . . . 6  |-  H  =  ( j  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
2 etransclem17.s . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  { RR ,  CC } )
3 etransclem17.x . . . . . . . . . . . . . 14  |-  ( ph  ->  X  e.  ( (
TopOpen ` fld )t  S ) )
42, 3dvdmsscn 37908 . . . . . . . . . . . . 13  |-  ( ph  ->  X  C_  CC )
54sselda 3418 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
65adantlr 729 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  ( 0 ... M
) )  /\  x  e.  X )  ->  x  e.  CC )
7 elfzelz 11826 . . . . . . . . . . . . 13  |-  ( j  e.  ( 0 ... M )  ->  j  e.  ZZ )
87zcnd 11064 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... M )  ->  j  e.  CC )
98ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  ( 0 ... M
) )  /\  x  e.  X )  ->  j  e.  CC )
106, 9negsubd 10011 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 0 ... M
) )  /\  x  e.  X )  ->  (
x  +  -u j
)  =  ( x  -  j ) )
1110eqcomd 2477 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 0 ... M
) )  /\  x  e.  X )  ->  (
x  -  j )  =  ( x  +  -u j ) )
1211oveq1d 6323 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0 ... M
) )  /\  x  e.  X )  ->  (
( x  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) )  =  ( ( x  +  -u j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )
1312mpteq2dva 4482 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... M
) )  ->  (
x  e.  X  |->  ( ( x  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( x  e.  X  |->  ( ( x  +  -u j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
1413mpteq2dva 4482 . . . . . 6  |-  ( ph  ->  ( j  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )  =  ( j  e.  ( 0 ... M
)  |->  ( x  e.  X  |->  ( ( x  +  -u j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) ) )
151, 14syl5eq 2517 . . . . 5  |-  ( ph  ->  H  =  ( j  e.  ( 0 ... M )  |->  ( x  e.  X  |->  ( ( x  +  -u j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) ) ) )
16 negeq 9887 . . . . . . . . 9  |-  ( j  =  J  ->  -u j  =  -u J )
1716oveq2d 6324 . . . . . . . 8  |-  ( j  =  J  ->  (
x  +  -u j
)  =  ( x  +  -u J ) )
18 eqeq1 2475 . . . . . . . . 9  |-  ( j  =  J  ->  (
j  =  0  <->  J  =  0 ) )
1918ifbid 3894 . . . . . . . 8  |-  ( j  =  J  ->  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  =  if ( J  =  0 ,  ( P  -  1 ) ,  P ) )
2017, 19oveq12d 6326 . . . . . . 7  |-  ( j  =  J  ->  (
( x  +  -u j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) )  =  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) )
2120mpteq2dv 4483 . . . . . 6  |-  ( j  =  J  ->  (
x  e.  X  |->  ( ( x  +  -u j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) )  =  ( x  e.  X  |->  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
2221adantl 473 . . . . 5  |-  ( (
ph  /\  j  =  J )  ->  (
x  e.  X  |->  ( ( x  +  -u j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) )  =  ( x  e.  X  |->  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
23 etransclem17.J . . . . 5  |-  ( ph  ->  J  e.  ( 0 ... M ) )
24 mptexg 6151 . . . . . 6  |-  ( X  e.  ( ( TopOpen ` fld )t  S
)  ->  ( x  e.  X  |->  ( ( x  +  -u J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) )  e.  _V )
253, 24syl 17 . . . . 5  |-  ( ph  ->  ( x  e.  X  |->  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) )  e. 
_V )
2615, 22, 23, 25fvmptd 5969 . . . 4  |-  ( ph  ->  ( H `  J
)  =  ( x  e.  X  |->  ( ( x  +  -u J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
2726oveq2d 6324 . . 3  |-  ( ph  ->  ( S  Dn
( H `  J
) )  =  ( S  Dn ( x  e.  X  |->  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) ) ) )
2827fveq1d 5881 . 2  |-  ( ph  ->  ( ( S  Dn ( H `  J ) ) `  N )  =  ( ( S  Dn
( x  e.  X  |->  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) ) ) `
 N ) )
29 etransclem17.n . . 3  |-  ( ph  ->  N  e.  NN0 )
30 elfzelz 11826 . . . . . . 7  |-  ( J  e.  ( 0 ... M )  ->  J  e.  ZZ )
3130zcnd 11064 . . . . . 6  |-  ( J  e.  ( 0 ... M )  ->  J  e.  CC )
3223, 31syl 17 . . . . 5  |-  ( ph  ->  J  e.  CC )
3332negcld 9992 . . . 4  |-  ( ph  -> 
-u J  e.  CC )
34 etransclem17.p . . . . . 6  |-  ( ph  ->  P  e.  NN )
35 nnm1nn0 10935 . . . . . 6  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
3634, 35syl 17 . . . . 5  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
3734nnnn0d 10949 . . . . 5  |-  ( ph  ->  P  e.  NN0 )
3836, 37ifcld 3915 . . . 4  |-  ( ph  ->  if ( J  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
39 eqid 2471 . . . 4  |-  ( x  e.  X  |->  ( ( x  +  -u J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( x  e.  X  |->  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) )
402, 3, 33, 38, 39dvnxpaek 37914 . . 3  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( ( S  Dn ( x  e.  X  |->  ( ( x  +  -u J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) ) ) `  N )  =  ( x  e.  X  |->  if ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  < 
N ,  0 ,  ( ( ( ! `
 if ( J  =  0 ,  ( P  -  1 ) ,  P ) )  /  ( ! `  ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N
) ) )  x.  ( ( x  +  -u J ) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) ) ) )
4129, 40mpdan 681 . 2  |-  ( ph  ->  ( ( S  Dn ( x  e.  X  |->  ( ( x  +  -u J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) ) ) `
 N )  =  ( x  e.  X  |->  if ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  <  N ,  0 ,  ( ( ( ! `  if ( J  =  0 ,  ( P  -  1 ) ,  P ) )  /  ( ! `
 ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N ) ) )  x.  ( ( x  +  -u J
) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) ) ) )
4232adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  J  e.  CC )
435, 42negsubd 10011 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
x  +  -u J
)  =  ( x  -  J ) )
4443oveq1d 6323 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( x  +  -u J ) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) )  =  ( ( x  -  J ) ^ ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N ) ) )
4544oveq2d 6324 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( ! `  if ( J  =  0 ,  ( P  - 
1 ) ,  P
) )  /  ( ! `  ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) )  x.  (
( x  +  -u J ) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) )  =  ( ( ( ! `  if ( J  =  0 ,  ( P  - 
1 ) ,  P
) )  /  ( ! `  ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) )  x.  (
( x  -  J
) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) )
4645ifeq2d 3891 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  if ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  <  N ,  0 ,  ( ( ( ! `  if ( J  =  0 ,  ( P  - 
1 ) ,  P
) )  /  ( ! `  ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) )  x.  (
( x  +  -u J ) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) )  =  if ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  <  N ,  0 ,  ( ( ( ! `  if ( J  =  0 ,  ( P  -  1 ) ,  P ) )  /  ( ! `
 ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N ) ) )  x.  ( ( x  -  J ) ^ ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N ) ) ) ) )
4746mpteq2dva 4482 . 2  |-  ( ph  ->  ( x  e.  X  |->  if ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  <  N ,  0 ,  ( ( ( ! `  if ( J  =  0 ,  ( P  -  1 ) ,  P ) )  /  ( ! `
 ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N ) ) )  x.  ( ( x  +  -u J
) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) ) )  =  ( x  e.  X  |->  if ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  <  N , 
0 ,  ( ( ( ! `  if ( J  =  0 ,  ( P  - 
1 ) ,  P
) )  /  ( ! `  ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) )  x.  (
( x  -  J
) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) ) ) )
4828, 41, 473eqtrd 2509 1  |-  ( ph  ->  ( ( S  Dn ( H `  J ) ) `  N )  =  ( x  e.  X  |->  if ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  < 
N ,  0 ,  ( ( ( ! `
 if ( J  =  0 ,  ( P  -  1 ) ,  P ) )  /  ( ! `  ( if ( J  =  0 ,  ( P  -  1 ) ,  P )  -  N
) ) )  x.  ( ( x  -  J ) ^ ( if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  -  N ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   ifcif 3872   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   NN0cn0 10893   ...cfz 11810   ^cexp 12310   !cfa 12497   ↾t crest 15397   TopOpenctopn 15398  ℂfldccnfld 19047    Dncdvn 22898
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-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-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-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-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-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-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-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-fac 12498  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  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-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-limc 22900  df-dv 22901  df-dvn 22902
This theorem is referenced by:  etransclem19  38230  etransclem20  38231  etransclem21  38232  etransclem22  38233
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