etransclem2 - Mathbox for Glauco Siliprandi

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Theorem etransclem2 43777

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 7286	. 2 ⊢ (ℝ D 𝐺) = (ℝ D (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
3		eqid 2738	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
4	3	tgioo2 23966	. . 3 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
5		reelprrecn 10963	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
6	5	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
7		reopn 42828	. . . 4 ⊢ ℝ ∈ (topGen‘ran (,))
8	7	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
9		fzfid 13693	. . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin)
10		fzelp1 13308	. . . . . 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 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
13	12	3adant3 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
14		simp3 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
15	13, 14	ffvelrnd 6962	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥) ∈ ℂ)
16		fzp1elp1 13309	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → (𝑖 + 1) ∈ (0...(𝑅 + 1)))
17		ovex 7308	. . . . . . 7 ⊢ (𝑖 + 1) ∈ V
18		eleq1 2826	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → (𝑗 ∈ (0...(𝑅 + 1)) ↔ (𝑖 + 1) ∈ (0...(𝑅 + 1))))
19	18	anbi2d 629	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1)))))
20		fveq2 6774	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → ((ℝ D^𝑛 𝐹)‘𝑗) = ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)))
21	20	feq1d 6585	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ))
22	19, 21	imbi12d 345	. . . . . . 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 2826	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0...(𝑅 + 1)) ↔ 𝑗 ∈ (0...(𝑅 + 1))))
24	23	anbi2d 629	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1)))))
25		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((ℝ D^𝑛 𝐹)‘𝑖) = ((ℝ D^𝑛 𝐹)‘𝑗))
26	25	feq1d 6585	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ))
27	24, 26	imbi12d 345	. . . . . . . 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	chvarvv 2002	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ)
29	17, 22, 28	vtocl 3498	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
30	16, 29	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
31	30	3adant3 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
32	31, 14	ffvelrnd 6962	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥) ∈ ℂ)
33	12	ffnd 6601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖) Fn ℝ)
34		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ
35		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥 D^𝑛
36		etransclem2.xf	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
37	34, 35, 36	nfov 7305	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℝ D^𝑛 𝐹)
38		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑖
39	37, 38	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘𝑖)
40	39	dffn5f 6840	. . . . . . 7 ⊢ (((ℝ D^𝑛 𝐹)‘𝑖) Fn ℝ ↔ ((ℝ D^𝑛 𝐹)‘𝑖) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
41	33, 40	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
42	41	eqcomd 2744	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)) = ((ℝ D^𝑛 𝐹)‘𝑖))
43	42	oveq2d 7291	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (ℝ D (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
44		ax-resscn 10928	. . . . . 6 ⊢ ℝ ⊆ ℂ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ℝ ⊆ ℂ)
46		etransclem2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
47		ffdm 6630	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℂ → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
48	46, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
49		cnex 10952	. . . . . . . . 9 ⊢ ℂ ∈ V
50	49	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℂ ∈ V)
51		reex 10962	. . . . . . . 8 ⊢ ℝ ∈ V
52		elpm2g 8632	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℝ ∈ V) → (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)))
53	50, 51, 52	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)))
54	48, 53	mpbird 256	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
55	54	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
56		elfznn0 13349	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ₀)
57	56	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ₀)
58		dvnp1 25089	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑖 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
59	45, 55, 57, 58	syl3anc 1370	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
60	30	ffnd 6601	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) Fn ℝ)
61		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 + 1)
62	37, 61	nffv 6784	. . . . . 6 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘(𝑖 + 1))
63	62	dffn5f 6840	. . . . 5 ⊢ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) Fn ℝ ↔ ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))
64	60, 63	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))
65	43, 59, 64	3eqtr2d 2784	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (ℝ D (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))
66	4, 3, 6, 8, 9, 15, 32, 65	dvmptfsum 25139	. 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)))
67	2, 66	eqtrid 2790	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Ⅎwnfc 2887 Vcvv 3432 ⊆ wss 3887 {cpr 4563 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_pm cpm 8616 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 ℕ₀cn0 12233 (,)cioo 13079 ...cfz 13239 Σcsu 15397 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 D cdv 25027 D^𝑛 cdvn 25028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-dvn 25032

This theorem is referenced by: etransclem46 43821

W3C validator