etransclem2 - Mathbox for Glauco Siliprandi

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Theorem etransclem2 44938

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 7416	. 2 ⊢ (ℝ D 𝐺) = (ℝ D (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
3		eqid 2732	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
4	3	tgioo2 24310	. . 3 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
5		reelprrecn 11198	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
6	5	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
7		reopn 43985	. . . 4 ⊢ ℝ ∈ (topGen‘ran (,))
8	7	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
9		fzfid 13934	. . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin)
10		fzelp1 13549	. . . . . 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 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
14		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
15	13, 14	ffvelcdmd 7084	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥) ∈ ℂ)
16		fzp1elp1 13550	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → (𝑖 + 1) ∈ (0...(𝑅 + 1)))
17		ovex 7438	. . . . . . 7 ⊢ (𝑖 + 1) ∈ V
18		eleq1 2821	. . . . . . . . 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 6888	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → ((ℝ D^𝑛 𝐹)‘𝑗) = ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)))
21	20	feq1d 6699	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ))
22	19, 21	imbi12d 344	. . . . . . 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 2821	. . . . . . . . . 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 6888	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((ℝ D^𝑛 𝐹)‘𝑖) = ((ℝ D^𝑛 𝐹)‘𝑗))
26	25	feq1d 6699	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ))
27	24, 26	imbi12d 344	. . . . . . . 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 3549	. . . . . 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 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
32	31, 14	ffvelcdmd 7084	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥) ∈ ℂ)
33	12	ffnd 6715	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖) Fn ℝ)
34		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ
35		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑥 D^𝑛
36		etransclem2.xf	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
37	34, 35, 36	nfov 7435	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℝ D^𝑛 𝐹)
38		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑖
39	37, 38	nffv 6898	. . . . . . . 8 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘𝑖)
40	39	dffn5f 6960	. . . . . . 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 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)) = ((ℝ D^𝑛 𝐹)‘𝑖))
43	42	oveq2d 7421	. . . 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 11163	. . . . . 6 ⊢ ℝ ⊆ ℂ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ℝ ⊆ ℂ)
46		etransclem2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
47		ffdm 6744	. . . . . . . 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 11187	. . . . . . . . 9 ⊢ ℂ ∈ V
50	49	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℂ ∈ V)
51		reex 11197	. . . . . . . 8 ⊢ ℝ ∈ V
52		elpm2g 8834	. . . . . . . 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 13590	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ₀)
57	56	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ₀)
58		dvnp1 25433	. . . . 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 1371	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
60	30	ffnd 6715	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) Fn ℝ)
61		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 + 1)
62	37, 61	nffv 6898	. . . . . 6 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘(𝑖 + 1))
63	62	dffn5f 6960	. . . . 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 2778	. . 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 25483	. 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 2784	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 1087 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2883 Vcvv 3474 ⊆ wss 3947 {cpr 4629 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_pm cpm 8817 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ℕ₀cn0 12468 (,)cioo 13320 ...cfz 13480 Σcsu 15628 TopOpenctopn 17363 topGenctg 17379 ℂ_fldccnfld 20936 D cdv 25371 D^𝑛 cdvn 25372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-dvn 25376

This theorem is referenced by: etransclem46 44982

W3C validator