Theorem etransclem34 40485

Description: The N-th derivative of F is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem34.s	|- ( ph -> S e. { RR , CC } )
etransclem34.a	|- ( ph -> X e. ( ( TopOpen ` ℂfld ) ↾t S ) )
etransclem34.p	|- ( ph -> P e. NN )
etransclem34.m	|- ( ph -> M e. NN0 )
etransclem34.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( x - k ) ^ P ) ) )
etransclem34.n	|- ( ph -> N e. NN0 )
etransclem34.h	|- H = ( k e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) )
etransclem34.c	|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = n } )

Assertion

Ref	Expression
etransclem34	|- ( ph -> ( ( S Dn F ) ` N ) e. ( X -cn-> CC ) )

Distinct variable groups:   x, k   C, c, k, x   F, c   H, c, k, n, x   M, c, k, x, n   N, c, k, x, n   P, k, x   S, c, k, n, x   X, c, k, x, n   ph, c, k, x, n

Allowed substitution hints:   C( n)   P( n, c)   F( x, k, n)

Proof of Theorem etransclem34

Dummy variables j y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem34.s	. . 3 |- ( ph -> S e. { RR , CC } )
2		etransclem34.a	. . 3 |- ( ph -> X e. ( ( TopOpen ` ℂfld ) ↾t S ) )
3		etransclem34.p	. . 3 |- ( ph -> P e. NN )
4		etransclem34.m	. . 3 |- ( ph -> M e. NN0 )
5		etransclem34.f	. . 3 |- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( x - k ) ^ P ) ) )
6		etransclem34.n	. . 3 |- ( ph -> N e. NN0 )
7		etransclem34.h	. . 3 |- H = ( k e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) )
8		etransclem34.c	. . 3 |- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = n } )
9	1, 2, 3, 4, 5, 6, 7, 8	etransclem30 40481	. 2 |- ( ph -> ( ( S Dn F ) ` N ) = ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) x. prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) ) )
10	1, 2	dvdmsscn 40151	. . 3 |- ( ph -> X C_ CC )
11	8, 6	etransclem16 40467	. . 3 |- ( ph -> ( C ` N ) e. Fin )
12	10	adantr 481	. . . . 5 |- ( ( ph /\ c e. ( C ` N ) ) -> X C_ CC )
13	6	faccld 13071	. . . . . . . 8 |- ( ph -> ( ! ` N ) e. NN )
14	13	nncnd 11036	. . . . . . 7 |- ( ph -> ( ! ` N ) e. CC )
15	14	adantr 481	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ! ` N ) e. CC )
16		fzfid 12772	. . . . . . 7 |- ( ( ph /\ c e. ( C ` N ) ) -> ( 0 ... M ) e. Fin )
17		fzssnn0 39533	. . . . . . . . . 10 |- ( 0 ... N ) C_ NN0
18		ssrab2 3687	. . . . . . . . . . . . 13 |- { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } C_ ( ( 0 ... N ) ^m ( 0 ... M ) )
19		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( C ` N ) )
20	8, 6	etransclem12 40463	. . . . . . . . . . . . . . 15 |- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } )
21	20	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ c e. ( C ` N ) ) -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } )
22	19, 21	eleqtrd 2703	. . . . . . . . . . . . 13 |- ( ( ph /\ c e. ( C ` N ) ) -> c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } )
23	18, 22	sseldi 3601	. . . . . . . . . . . 12 |- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) )
24		elmapi 7879	. . . . . . . . . . . 12 |- ( c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) )
25	23, 24	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ c e. ( C ` N ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) )
26	25	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( c ` k ) e. ( 0 ... N ) )
27	17, 26	sseldi 3601	. . . . . . . . 9 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( c ` k ) e. NN0 )
28	27	faccld 13071	. . . . . . . 8 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ! ` ( c ` k ) ) e. NN )
29	28	nncnd 11036	. . . . . . 7 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ! ` ( c ` k ) ) e. CC )
30	16, 29	fprodcl 14682	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) e. CC )
31	28	nnne0d 11065	. . . . . . 7 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ! ` ( c ` k ) ) =/= 0 )
32	16, 29, 31	fprodn0 14709	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) =/= 0 )
33	15, 30, 32	divcld 10801	. . . . 5 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) e. CC )
34		ssid 3624	. . . . . 6 |- CC C_ CC
35	34	a1i 11	. . . . 5 |- ( ( ph /\ c e. ( C ` N ) ) -> CC C_ CC )
36	12, 33, 35	constcncfg 40084	. . . 4 |- ( ( ph /\ c e. ( C ` N ) ) -> ( x e. X |-> ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) ) e. ( X -cn-> CC ) )
37	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> S e. { RR , CC } )
38	2	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> X e. ( ( TopOpen ` ℂfld ) ↾t S ) )
39	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> P e. NN )
40		etransclem5 40456	. . . . . . . . 9 |- ( k e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) = ( j e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
41	7, 40	eqtri 2644	. . . . . . . 8 |- H = ( j e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
42		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> k e. ( 0 ... M ) )
43	37, 38, 39, 41, 42, 27	etransclem20 40471	. . . . . . 7 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) : X --> CC )
44	43	3adant2 1080	. . . . . 6 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ x e. X /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) : X --> CC )
45		simp2 1062	. . . . . 6 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ x e. X /\ k e. ( 0 ... M ) ) -> x e. X )
46	44, 45	ffvelrnd 6360	. . . . 5 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ x e. X /\ k e. ( 0 ... M ) ) -> ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) e. CC )
47	43	feqmptd 6249	. . . . . 6 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) = ( x e. X |-> ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) )
48	37, 38, 39, 41, 42, 27	etransclem22 40473	. . . . . 6 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) e. ( X -cn-> CC ) )
49	47, 48	eqeltrrd 2702	. . . . 5 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( x e. X |-> ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) e. ( X -cn-> CC ) )
50	12, 16, 46, 49	fprodcncf 40114	. . . 4 |- ( ( ph /\ c e. ( C ` N ) ) -> ( x e. X |-> prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) e. ( X -cn-> CC ) )
51	36, 50	mulcncf 23215	. . 3 |- ( ( ph /\ c e. ( C ` N ) ) -> ( x e. X |-> ( ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) x. prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) ) e. ( X -cn-> CC ) )
52	10, 11, 51	fsumcncf 40091	. 2 |- ( ph -> ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) x. prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) ) e. ( X -cn-> CC ) )
53	9, 52	eqeltrd 2701	1 |- ( ph -> ( ( S Dn F ) ` N ) e. ( X -cn-> CC ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ifcif 4086   {cpr 4179   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ...cfz 12326   ^cexp 12860   !cfa 13060   sum_csu 14416   prod_cprod 14635   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   -cn->ccncf 22679   Dncdvn 23628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-prod 14636  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-dvn 23632

This theorem is referenced by:  etransclem40  40491