Theorem etransclem2 40453

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.

Step	Hyp	Ref	Expression
1		etransclem2.g	. . 3 |- G = ( x e. RR |-> sum_ i e. ( 0 ... R ) ( ( ( RR Dn F ) ` i ) ` x ) )
2	1	oveq2i 6661	. 2 |- ( RR _D G ) = ( RR _D ( x e. RR |-> sum_ i e. ( 0 ... R ) ( ( ( RR Dn F ) ` i ) ` x ) ) )
3		eqid 2622	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4	3	tgioo2 22606	. . 3 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
5		reelprrecn 10028	. . . 4 |- RR e. { RR , CC }
6	5	a1i 11	. . 3 |- ( ph -> RR e. { RR , CC } )
7		reopn 39501	. . . 4 |- RR e. ( topGen ` ran (,) )
8	7	a1i 11	. . 3 |- ( ph -> RR e. ( topGen ` ran (,) ) )
9		fzfid 12772	. . 3 |- ( ph -> ( 0 ... R ) e. Fin )
10		fzelp1 12393	. . . . . 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 )
12	10, 11	sylan2 491	. . . . 5 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( RR Dn F ) ` i ) : RR --> CC )
13	12	3adant3 1081	. . . 4 |- ( ( ph /\ i e. ( 0 ... R ) /\ x e. RR ) -> ( ( RR Dn F ) ` i ) : RR --> CC )
14		simp3 1063	. . . 4 |- ( ( ph /\ i e. ( 0 ... R ) /\ x e. RR ) -> x e. RR )
15	13, 14	ffvelrnd 6360	. . 3 |- ( ( ph /\ i e. ( 0 ... R ) /\ x e. RR ) -> ( ( ( RR Dn F ) ` i ) ` x ) e. CC )
16		fzp1elp1 12394	. . . . . 6 |- ( i e. ( 0 ... R ) -> ( i + 1 ) e. ( 0 ... ( R + 1 ) ) )
17		ovex 6678	. . . . . . 7 |- ( i + 1 ) e. _V
18		eleq1 2689	. . . . . . . . 9 |- ( j = ( i + 1 ) -> ( j e. ( 0 ... ( R + 1 ) ) <-> ( i + 1 ) e. ( 0 ... ( R + 1 ) ) ) )
19	18	anbi2d 740	. . . . . . . 8 |- ( j = ( i + 1 ) -> ( ( ph /\ j e. ( 0 ... ( R + 1 ) ) ) <-> ( ph /\ ( i + 1 ) e. ( 0 ... ( R + 1 ) ) ) ) )
20		fveq2 6191	. . . . . . . . 9 |- ( j = ( i + 1 ) -> ( ( RR Dn F ) ` j ) = ( ( RR Dn F ) ` ( i + 1 ) ) )
21	20	feq1d 6030	. . . . . . . 8 |- ( j = ( i + 1 ) -> ( ( ( RR Dn F ) ` j ) : RR --> CC <-> ( ( RR Dn F ) ` ( i + 1 ) ) : RR --> CC ) )
22	19, 21	imbi12d 334	. . . . . . 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 2689	. . . . . . . . . 10 |- ( i = j -> ( i e. ( 0 ... ( R + 1 ) ) <-> j e. ( 0 ... ( R + 1 ) ) ) )
24	23	anbi2d 740	. . . . . . . . 9 |- ( i = j -> ( ( ph /\ i e. ( 0 ... ( R + 1 ) ) ) <-> ( ph /\ j e. ( 0 ... ( R + 1 ) ) ) ) )
25		fveq2 6191	. . . . . . . . . 10 |- ( i = j -> ( ( RR Dn F ) ` i ) = ( ( RR Dn F ) ` j ) )
26	25	feq1d 6030	. . . . . . . . 9 |- ( i = j -> ( ( ( RR Dn F ) ` i ) : RR --> CC <-> ( ( RR Dn F ) ` j ) : RR --> CC ) )
27	24, 26	imbi12d 334	. . . . . . . 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 ) ) )
28	27, 11	chvarv 2263	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... ( R + 1 ) ) ) -> ( ( RR Dn F ) ` j ) : RR --> CC )
29	17, 22, 28	vtocl 3259	. . . . . 6 |- ( ( ph /\ ( i + 1 ) e. ( 0 ... ( R + 1 ) ) ) -> ( ( RR Dn F ) ` ( i + 1 ) ) : RR --> CC )
30	16, 29	sylan2 491	. . . . 5 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( RR Dn F ) ` ( i + 1 ) ) : RR --> CC )
31	30	3adant3 1081	. . . 4 |- ( ( ph /\ i e. ( 0 ... R ) /\ x e. RR ) -> ( ( RR Dn F ) ` ( i + 1 ) ) : RR --> CC )
32	31, 14	ffvelrnd 6360	. . 3 |- ( ( ph /\ i e. ( 0 ... R ) /\ x e. RR ) -> ( ( ( RR Dn F ) ` ( i + 1 ) ) ` x ) e. CC )
33		ffn 6045	. . . . . . . 8 |- ( ( ( RR Dn F ) ` i ) : RR --> CC -> ( ( RR Dn F ) ` i ) Fn RR )
34	12, 33	syl 17	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( RR Dn F ) ` i ) Fn RR )
35		nfcv 2764	. . . . . . . . . 10 |- F/_ x RR
36		nfcv 2764	. . . . . . . . . 10 |- F/_ x Dn
37		etransclem2.xf	. . . . . . . . . 10 |- F/_ x F
38	35, 36, 37	nfov 6676	. . . . . . . . 9 |- F/_ x ( RR Dn F )
39		nfcv 2764	. . . . . . . . 9 |- F/_ x i
40	38, 39	nffv 6198	. . . . . . . 8 |- F/_ x ( ( RR Dn F ) ` i )
41	40	dffn5f 6252	. . . . . . 7 |- ( ( ( RR Dn F ) ` i ) Fn RR <-> ( ( RR Dn F ) ` i ) = ( x e. RR |-> ( ( ( RR Dn F ) ` i ) ` x ) ) )
42	34, 41	sylib 208	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( RR Dn F ) ` i ) = ( x e. RR |-> ( ( ( RR Dn F ) ` i ) ` x ) ) )
43	42	eqcomd 2628	. . . . 5 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( x e. RR |-> ( ( ( RR Dn F ) ` i ) ` x ) ) = ( ( RR Dn F ) ` i ) )
44	43	oveq2d 6666	. . . 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 9993	. . . . . 6 |- RR C_ CC
46	45	a1i 11	. . . . 5 |- ( ( ph /\ i e. ( 0 ... R ) ) -> RR C_ CC )
47		etransclem2.f	. . . . . . . 8 |- ( ph -> F : RR --> CC )
48		ffdm 6062	. . . . . . . 8 |- ( F : RR --> CC -> ( F : dom F --> CC /\ dom F C_ RR ) )
49	47, 48	syl 17	. . . . . . 7 |- ( ph -> ( F : dom F --> CC /\ dom F C_ RR ) )
50		cnex 10017	. . . . . . . . 9 |- CC e. _V
51	50	a1i 11	. . . . . . . 8 |- ( ph -> CC e. _V )
52		reex 10027	. . . . . . . 8 |- RR e. _V
53		elpm2g 7874	. . . . . . . 8 |- ( ( CC e. _V /\ RR e. _V ) -> ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) ) )
54	51, 52, 53	sylancl 694	. . . . . . 7 |- ( ph -> ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) ) )
55	49, 54	mpbird 247	. . . . . 6 |- ( ph -> F e. ( CC ^pm RR ) )
56	55	adantr 481	. . . . 5 |- ( ( ph /\ i e. ( 0 ... R ) ) -> F e. ( CC ^pm RR ) )
57		elfznn0 12433	. . . . . 6 |- ( i e. ( 0 ... R ) -> i e. NN0 )
58	57	adantl 482	. . . . 5 |- ( ( ph /\ i e. ( 0 ... R ) ) -> i e. NN0 )
59		dvnp1 23688	. . . . 5 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) /\ i e. NN0 ) -> ( ( RR Dn F ) ` ( i + 1 ) ) = ( RR _D ( ( RR Dn F ) ` i ) ) )
60	46, 56, 58, 59	syl3anc 1326	. . . 4 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( RR Dn F ) ` ( i + 1 ) ) = ( RR _D ( ( RR Dn F ) ` i ) ) )
61		ffn 6045	. . . . . 6 |- ( ( ( RR Dn F ) ` ( i + 1 ) ) : RR --> CC -> ( ( RR Dn F ) ` ( i + 1 ) ) Fn RR )
62	30, 61	syl 17	. . . . 5 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( RR Dn F ) ` ( i + 1 ) ) Fn RR )
63		nfcv 2764	. . . . . . 7 |- F/_ x ( i + 1 )
64	38, 63	nffv 6198	. . . . . 6 |- F/_ x ( ( RR Dn F ) ` ( i + 1 ) )
65	64	dffn5f 6252	. . . . 5 |- ( ( ( RR Dn F ) ` ( i + 1 ) ) Fn RR <-> ( ( RR Dn F ) ` ( i + 1 ) ) = ( x e. RR |-> ( ( ( RR Dn F ) ` ( i + 1 ) ) ` x ) ) )
66	62, 65	sylib 208	. . . 4 |- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( RR Dn F ) ` ( i + 1 ) ) = ( x e. RR |-> ( ( ( RR Dn F ) ` ( i + 1 ) ) ` x ) ) )
67	44, 60, 66	3eqtr2d 2662	. . 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 ) ) )
68	4, 3, 6, 8, 9, 15, 32, 67	dvmptfsum 23738	. 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 ) ) )
69	2, 68	syl5eq 2668	1 |- ( ph -> ( RR _D G ) = ( x e. RR |-> sum_ i e. ( 0 ... R ) ( ( ( RR Dn F ) ` ( i + 1 ) ) ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   C_ wss 3574   {cpr 4179   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   (,)cioo 12175   ...cfz 12326   sum_csu 14416   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   _D cdv 23627   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

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-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  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-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:  etransclem46  40497