etransclem2 - Mathbox for Glauco Siliprandi

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Theorem etransclem2 40453

Description: Derivative of 𝐺. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem2.xf	⊢ Ⅎ𝑥𝐹
etransclem2.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
etransclem2.dvnf	⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
etransclem2.g	⊢ 𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))

**Assertion**
Ref	Expression
etransclem2	⊢ (𝜑 → (ℝ D 𝐺) = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))

Distinct variable groups: 𝑖,𝐹 𝑅,𝑖,𝑥 𝜑,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 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
2	1	oveq2i 6661	. 2 ⊢ (ℝ D 𝐺) = (ℝ D (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
3		eqid 2622	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
4	3	tgioo2 22606	. . 3 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
5		reelprrecn 10028	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
6	5	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
7		reopn 39501	. . . 4 ⊢ ℝ ∈ (topGen‘ran (,))
8	7	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
9		fzfid 12772	. . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin)
10		fzelp1 12393	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ (0...(𝑅 + 1)))
11		etransclem2.dvnf	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
12	10, 11	sylan2 491	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
13	12	3adant3 1081	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
14		simp3 1063	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
15	13, 14	ffvelrnd 6360	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥) ∈ ℂ)
16		fzp1elp1 12394	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → (𝑖 + 1) ∈ (0...(𝑅 + 1)))
17		ovex 6678	. . . . . . 7 ⊢ (𝑖 + 1) ∈ V
18		eleq1 2689	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → (𝑗 ∈ (0...(𝑅 + 1)) ↔ (𝑖 + 1) ∈ (0...(𝑅 + 1))))
19	18	anbi2d 740	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1)))))
20		fveq2 6191	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → ((ℝ D^𝑛 𝐹)‘𝑗) = ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)))
21	20	feq1d 6030	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ))
22	19, 21	imbi12d 334	. . . . . . 7 ⊢ (𝑗 = (𝑖 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)))
23		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0...(𝑅 + 1)) ↔ 𝑗 ∈ (0...(𝑅 + 1))))
24	23	anbi2d 740	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1)))))
25		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((ℝ D^𝑛 𝐹)‘𝑖) = ((ℝ D^𝑛 𝐹)‘𝑗))
26	25	feq1d 6030	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ))
27	24, 26	imbi12d 334	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ)))
28	27, 11	chvarv 2263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ)
29	17, 22, 28	vtocl 3259	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
30	16, 29	sylan2 491	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
31	30	3adant3 1081	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
32	31, 14	ffvelrnd 6360	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥) ∈ ℂ)
33		ffn 6045	. . . . . . . 8 ⊢ (((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ → ((ℝ D^𝑛 𝐹)‘𝑖) Fn ℝ)
34	12, 33	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖) Fn ℝ)
35		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ
36		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥 D^𝑛
37		etransclem2.xf	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
38	35, 36, 37	nfov 6676	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℝ D^𝑛 𝐹)
39		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑖
40	38, 39	nffv 6198	. . . . . . . 8 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘𝑖)
41	40	dffn5f 6252	. . . . . . 7 ⊢ (((ℝ D^𝑛 𝐹)‘𝑖) Fn ℝ ↔ ((ℝ D^𝑛 𝐹)‘𝑖) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
42	34, 41	sylib 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
43	42	eqcomd 2628	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)) = ((ℝ D^𝑛 𝐹)‘𝑖))
44	43	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (ℝ D (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
45		ax-resscn 9993	. . . . . 6 ⊢ ℝ ⊆ ℂ
46	45	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ℝ ⊆ ℂ)
47		etransclem2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
48		ffdm 6062	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℂ → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
49	47, 48	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
50		cnex 10017	. . . . . . . . 9 ⊢ ℂ ∈ V
51	50	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℂ ∈ V)
52		reex 10027	. . . . . . . 8 ⊢ ℝ ∈ V
53		elpm2g 7874	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℝ ∈ V) → (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)))
54	51, 52, 53	sylancl 694	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)))
55	49, 54	mpbird 247	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
56	55	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
57		elfznn0 12433	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ₀)
58	57	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ₀)
59		dvnp1 23688	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑖 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
60	46, 56, 58, 59	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
61		ffn 6045	. . . . . 6 ⊢ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) Fn ℝ)
62	30, 61	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) Fn ℝ)
63		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 + 1)
64	38, 63	nffv 6198	. . . . . 6 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘(𝑖 + 1))
65	64	dffn5f 6252	. . . . 5 ⊢ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) Fn ℝ ↔ ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))
66	62, 65	sylib 208	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))
67	44, 60, 66	3eqtr2d 2662	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (ℝ D (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))
68	4, 3, 6, 8, 9, 15, 32, 67	dvmptfsum 23738	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))) = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))
69	2, 68	syl5eq 2668	1 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 Vcvv 3200 ⊆ 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 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℕ₀cn0 11292 (,)cioo 12175 ...cfz 12326 Σcsu 14416 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 D cdv 23627 D^𝑛 cdvn 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

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