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Theorem etransclem2 38213
Description: Derivative of  G. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem2.xf  |-  F/_ x F
etransclem2.f  |-  ( ph  ->  F : RR --> CC )
etransclem2.dvnf  |-  ( (
ph  /\  i  e.  ( 0 ... ( R  +  1 ) ) )  ->  (
( RR  Dn
F ) `  i
) : RR --> CC )
etransclem2.g  |-  G  =  ( x  e.  RR  |->  sum_ i  e.  ( 0 ... R ) ( ( ( RR  Dn F ) `  i ) `  x
) )
Assertion
Ref Expression
etransclem2  |-  ( ph  ->  ( RR  _D  G
)  =  ( x  e.  RR  |->  sum_ i  e.  ( 0 ... R
) ( ( ( RR  Dn F ) `  ( i  +  1 ) ) `
 x ) ) )
Distinct variable groups:    i, F    R, i, x    ph, i, x
Allowed substitution hints:    F( x)    G( x, i)

Proof of Theorem etransclem2
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 etransclem2.g . . 3  |-  G  =  ( x  e.  RR  |->  sum_ i  e.  ( 0 ... R ) ( ( ( RR  Dn F ) `  i ) `  x
) )
21oveq2i 6319 . 2  |-  ( RR 
_D  G )  =  ( RR  _D  (
x  e.  RR  |->  sum_ i  e.  ( 0 ... R ) ( ( ( RR  Dn F ) `  i ) `  x
) ) )
3 eqid 2471 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
43tgioo2 21899 . . 3  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5 reelprrecn 9649 . . . 4  |-  RR  e.  { RR ,  CC }
65a1i 11 . . 3  |-  ( ph  ->  RR  e.  { RR ,  CC } )
7 reopn 37591 . . . 4  |-  RR  e.  ( topGen `  ran  (,) )
87a1i 11 . . 3  |-  ( ph  ->  RR  e.  ( topGen ` 
ran  (,) ) )
9 fzfid 12224 . . 3  |-  ( ph  ->  ( 0 ... R
)  e.  Fin )
10 fzelp1 11874 . . . . . 6  |-  ( i  e.  ( 0 ... R )  ->  i  e.  ( 0 ... ( R  +  1 ) ) )
11 etransclem2.dvnf . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... ( R  +  1 ) ) )  ->  (
( RR  Dn
F ) `  i
) : RR --> CC )
1210, 11sylan2 482 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
( RR  Dn
F ) `  i
) : RR --> CC )
13123adant3 1050 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... R
)  /\  x  e.  RR )  ->  ( ( RR  Dn F ) `  i ) : RR --> CC )
14 simp3 1032 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... R
)  /\  x  e.  RR )  ->  x  e.  RR )
1513, 14ffvelrnd 6038 . . 3  |-  ( (
ph  /\  i  e.  ( 0 ... R
)  /\  x  e.  RR )  ->  ( ( ( RR  Dn
F ) `  i
) `  x )  e.  CC )
16 fzp1elp1 11875 . . . . . 6  |-  ( i  e.  ( 0 ... R )  ->  (
i  +  1 )  e.  ( 0 ... ( R  +  1 ) ) )
17 ovex 6336 . . . . . . 7  |-  ( i  +  1 )  e. 
_V
18 eleq1 2537 . . . . . . . . 9  |-  ( j  =  ( i  +  1 )  ->  (
j  e.  ( 0 ... ( R  + 
1 ) )  <->  ( i  +  1 )  e.  ( 0 ... ( R  +  1 ) ) ) )
1918anbi2d 718 . . . . . . . 8  |-  ( j  =  ( i  +  1 )  ->  (
( ph  /\  j  e.  ( 0 ... ( R  +  1 ) ) )  <->  ( ph  /\  ( i  +  1 )  e.  ( 0 ... ( R  + 
1 ) ) ) ) )
20 fveq2 5879 . . . . . . . . 9  |-  ( j  =  ( i  +  1 )  ->  (
( RR  Dn
F ) `  j
)  =  ( ( RR  Dn F ) `  ( i  +  1 ) ) )
2120feq1d 5724 . . . . . . . 8  |-  ( j  =  ( i  +  1 )  ->  (
( ( RR  Dn F ) `  j ) : RR --> CC 
<->  ( ( RR  Dn F ) `  ( i  +  1 ) ) : RR --> CC ) )
2219, 21imbi12d 327 . . . . . . 7  |-  ( j  =  ( i  +  1 )  ->  (
( ( ph  /\  j  e.  ( 0 ... ( R  + 
1 ) ) )  ->  ( ( RR  Dn F ) `
 j ) : RR --> CC )  <->  ( ( ph  /\  ( i  +  1 )  e.  ( 0 ... ( R  +  1 ) ) )  ->  ( ( RR  Dn F ) `
 ( i  +  1 ) ) : RR --> CC ) ) )
23 eleq1 2537 . . . . . . . . . 10  |-  ( i  =  j  ->  (
i  e.  ( 0 ... ( R  + 
1 ) )  <->  j  e.  ( 0 ... ( R  +  1 ) ) ) )
2423anbi2d 718 . . . . . . . . 9  |-  ( i  =  j  ->  (
( ph  /\  i  e.  ( 0 ... ( R  +  1 ) ) )  <->  ( ph  /\  j  e.  ( 0 ... ( R  + 
1 ) ) ) ) )
25 fveq2 5879 . . . . . . . . . 10  |-  ( i  =  j  ->  (
( RR  Dn
F ) `  i
)  =  ( ( RR  Dn F ) `  j ) )
2625feq1d 5724 . . . . . . . . 9  |-  ( i  =  j  ->  (
( ( RR  Dn F ) `  i ) : RR --> CC 
<->  ( ( RR  Dn F ) `  j ) : RR --> CC ) )
2724, 26imbi12d 327 . . . . . . . 8  |-  ( i  =  j  ->  (
( ( ph  /\  i  e.  ( 0 ... ( R  + 
1 ) ) )  ->  ( ( RR  Dn F ) `
 i ) : RR --> CC )  <->  ( ( ph  /\  j  e.  ( 0 ... ( R  +  1 ) ) )  ->  ( ( RR  Dn F ) `
 j ) : RR --> CC ) ) )
2827, 11chvarv 2120 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... ( R  +  1 ) ) )  ->  (
( RR  Dn
F ) `  j
) : RR --> CC )
2917, 22, 28vtocl 3086 . . . . . 6  |-  ( (
ph  /\  ( i  +  1 )  e.  ( 0 ... ( R  +  1 ) ) )  ->  (
( RR  Dn
F ) `  (
i  +  1 ) ) : RR --> CC )
3016, 29sylan2 482 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
( RR  Dn
F ) `  (
i  +  1 ) ) : RR --> CC )
31303adant3 1050 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... R
)  /\  x  e.  RR )  ->  ( ( RR  Dn F ) `  ( i  +  1 ) ) : RR --> CC )
3231, 14ffvelrnd 6038 . . 3  |-  ( (
ph  /\  i  e.  ( 0 ... R
)  /\  x  e.  RR )  ->  ( ( ( RR  Dn
F ) `  (
i  +  1 ) ) `  x )  e.  CC )
33 ffn 5739 . . . . . . . 8  |-  ( ( ( RR  Dn
F ) `  i
) : RR --> CC  ->  ( ( RR  Dn
F ) `  i
)  Fn  RR )
3412, 33syl 17 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
( RR  Dn
F ) `  i
)  Fn  RR )
35 nfcv 2612 . . . . . . . . . 10  |-  F/_ x RR
36 nfcv 2612 . . . . . . . . . 10  |-  F/_ x  Dn
37 etransclem2.xf . . . . . . . . . 10  |-  F/_ x F
3835, 36, 37nfov 6334 . . . . . . . . 9  |-  F/_ x
( RR  Dn
F )
39 nfcv 2612 . . . . . . . . 9  |-  F/_ x
i
4038, 39nffv 5886 . . . . . . . 8  |-  F/_ x
( ( RR  Dn F ) `  i )
4140dffn5f 5935 . . . . . . 7  |-  ( ( ( RR  Dn
F ) `  i
)  Fn  RR  <->  ( ( RR  Dn F ) `
 i )  =  ( x  e.  RR  |->  ( ( ( RR  Dn F ) `
 i ) `  x ) ) )
4234, 41sylib 201 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
( RR  Dn
F ) `  i
)  =  ( x  e.  RR  |->  ( ( ( RR  Dn
F ) `  i
) `  x )
) )
4342eqcomd 2477 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
x  e.  RR  |->  ( ( ( RR  Dn F ) `  i ) `  x
) )  =  ( ( RR  Dn
F ) `  i
) )
4443oveq2d 6324 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  ( RR  _D  ( x  e.  RR  |->  ( ( ( RR  Dn F ) `  i ) `
 x ) ) )  =  ( RR 
_D  ( ( RR  Dn F ) `
 i ) ) )
45 ax-resscn 9614 . . . . . 6  |-  RR  C_  CC
4645a1i 11 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  RR  C_  CC )
47 etransclem2.f . . . . . . . 8  |-  ( ph  ->  F : RR --> CC )
48 ffdm 5755 . . . . . . . 8  |-  ( F : RR --> CC  ->  ( F : dom  F --> CC  /\  dom  F  C_  RR ) )
4947, 48syl 17 . . . . . . 7  |-  ( ph  ->  ( F : dom  F --> CC  /\  dom  F  C_  RR ) )
50 cnex 9638 . . . . . . . . 9  |-  CC  e.  _V
5150a1i 11 . . . . . . . 8  |-  ( ph  ->  CC  e.  _V )
52 reex 9648 . . . . . . . 8  |-  RR  e.  _V
53 elpm2g 7506 . . . . . . . 8  |-  ( ( CC  e.  _V  /\  RR  e.  _V )  -> 
( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F  C_  RR ) ) )
5451, 52, 53sylancl 675 . . . . . . 7  |-  ( ph  ->  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F  C_  RR ) ) )
5549, 54mpbird 240 . . . . . 6  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
5655adantr 472 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  F  e.  ( CC  ^pm  RR ) )
57 elfznn0 11913 . . . . . 6  |-  ( i  e.  ( 0 ... R )  ->  i  e.  NN0 )
5857adantl 473 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  i  e.  NN0 )
59 dvnp1 22958 . . . . 5  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR )  /\  i  e.  NN0 )  ->  ( ( RR  Dn F ) `
 ( i  +  1 ) )  =  ( RR  _D  (
( RR  Dn
F ) `  i
) ) )
6046, 56, 58, 59syl3anc 1292 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
( RR  Dn
F ) `  (
i  +  1 ) )  =  ( RR 
_D  ( ( RR  Dn F ) `
 i ) ) )
61 ffn 5739 . . . . . 6  |-  ( ( ( RR  Dn
F ) `  (
i  +  1 ) ) : RR --> CC  ->  ( ( RR  Dn
F ) `  (
i  +  1 ) )  Fn  RR )
6230, 61syl 17 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
( RR  Dn
F ) `  (
i  +  1 ) )  Fn  RR )
63 nfcv 2612 . . . . . . 7  |-  F/_ x
( i  +  1 )
6438, 63nffv 5886 . . . . . 6  |-  F/_ x
( ( RR  Dn F ) `  ( i  +  1 ) )
6564dffn5f 5935 . . . . 5  |-  ( ( ( RR  Dn
F ) `  (
i  +  1 ) )  Fn  RR  <->  ( ( RR  Dn F ) `
 ( i  +  1 ) )  =  ( x  e.  RR  |->  ( ( ( RR  Dn F ) `
 ( i  +  1 ) ) `  x ) ) )
6662, 65sylib 201 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  (
( RR  Dn
F ) `  (
i  +  1 ) )  =  ( x  e.  RR  |->  ( ( ( RR  Dn
F ) `  (
i  +  1 ) ) `  x ) ) )
6744, 60, 663eqtr2d 2511 . . 3  |-  ( (
ph  /\  i  e.  ( 0 ... R
) )  ->  ( RR  _D  ( x  e.  RR  |->  ( ( ( RR  Dn F ) `  i ) `
 x ) ) )  =  ( x  e.  RR  |->  ( ( ( RR  Dn
F ) `  (
i  +  1 ) ) `  x ) ) )
684, 3, 6, 8, 9, 15, 32, 67dvmptfsum 23006 . 2  |-  ( ph  ->  ( RR  _D  (
x  e.  RR  |->  sum_ i  e.  ( 0 ... R ) ( ( ( RR  Dn F ) `  i ) `  x
) ) )  =  ( x  e.  RR  |->  sum_ i  e.  ( 0 ... R ) ( ( ( RR  Dn F ) `  ( i  +  1 ) ) `  x
) ) )
692, 68syl5eq 2517 1  |-  ( ph  ->  ( RR  _D  G
)  =  ( x  e.  RR  |->  sum_ i  e.  ( 0 ... R
) ( ( ( RR  Dn F ) `  ( i  +  1 ) ) `
 x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   F/_wnfc 2599   _Vcvv 3031    C_ wss 3390   {cpr 3961    |-> cmpt 4454   dom cdm 4839   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^pm cpm 7491   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   NN0cn0 10893   (,)cioo 11660   ...cfz 11810   sum_csu 13829   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047    _D cdv 22897    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
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-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-ioo 11664  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  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:  etransclem46  38257
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