etransclem2 - Mathbox for Glauco Siliprandi

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Theorem etransclem2 47382

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 7435	. 2 ⊢ (ℝ D 𝐺) = (ℝ D (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
3		tgioo4 25133	. . 3 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
4		eqid 2760	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5		reelprrecn 11299	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
6	5	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
7		reopn 46441	. . . 4 ⊢ ℝ ∈ (topGen‘ran (,))
8	7	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
9		fzfid 14123	. . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin)
10		fzelp1 13717	. . . . . 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 605	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
13	12	3adant3 1150	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
14		simp3 1156	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
15	13, 14	ffvelcdmd 7090	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥) ∈ ℂ)
16		fzp1elp1 13718	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → (𝑖 + 1) ∈ (0...(𝑅 + 1)))
17		ovex 7457	. . . . . . 7 ⊢ (𝑖 + 1) ∈ V
18		eleq1 2848	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → (𝑗 ∈ (0...(𝑅 + 1)) ↔ (𝑖 + 1) ∈ (0...(𝑅 + 1))))
19	18	anbi2d 642	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1)))))
20		fveq2 6890	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → ((ℝ D^𝑛 𝐹)‘𝑗) = ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)))
21	20	feq1d 6696	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ))
22	19, 21	imbi12d 347	. . . . . . 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 2848	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0...(𝑅 + 1)) ↔ 𝑗 ∈ (0...(𝑅 + 1))))
24	23	anbi2d 642	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1)))))
25		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((ℝ D^𝑛 𝐹)‘𝑖) = ((ℝ D^𝑛 𝐹)‘𝑗))
26	25	feq1d 6696	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ))
27	24, 26	imbi12d 347	. . . . . . . 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 2021	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ)
29	17, 22, 28	vtocl 3520	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
30	16, 29	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
31	30	3adant3 1150	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
32	31, 14	ffvelcdmd 7090	. . 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 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ
35		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥 D^𝑛
36		etransclem2.xf	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
37	34, 35, 36	nfov 7454	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℝ D^𝑛 𝐹)
38		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑖
39	37, 38	nffv 6900	. . . . . . . 8 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘𝑖)
40	39	dffn5f 6961	. . . . . . 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 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖) = (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
42	41	eqcomd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)) = ((ℝ D^𝑛 𝐹)‘𝑖))
43	42	oveq2d 7440	. . . 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 11264	. . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
49		cnex 11288	. . . . . . . . 9 ⊢ ℂ ∈ V
50	49	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℂ ∈ V)
51		reex 11298	. . . . . . . 8 ⊢ ℝ ∈ V
52		elpm2g 8872	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℝ ∈ V) → (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)))
53	50, 51, 52	sylancl 598	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)))
54	48, 53	mpbird 260	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
55	54	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
56		elfznn0 13761	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ₀)
57	56	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ₀)
58		dvnp1 26254	. . . . 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 1398	. . . 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 2922	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 + 1)
62	37, 61	nffv 6900	. . . . . 6 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘(𝑖 + 1))
63	62	dffn5f 6961	. . . . 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 221	. . . 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 2801	. . 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	3, 4, 6, 8, 9, 15, 32, 65	dvmptfsum 26304	. 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 2807	1 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 Ⅎwnfc 2907 Vcvv 3450 ⊆ wss 3896 {cpr 4586 ↦ cmpt 5185 dom cdm 5651 ran crn 5652 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ↑_pm cpm 8856 ℂcc 11205 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 ℕ₀cn0 12613 (,)cioo 13483 ...cfz 13646 Σcsu 15862 TopOpenctopn 17601 topGenctg 17617 ℂ_fldccnfld 21687 D cdv 26192 D^𝑛 cdvn 26193

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-icc 13490 df-fz 13647 df-fzo 13796 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-clim 15664 df-sum 15863 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196 df-dvn 26197

This theorem is used by: etransclem46 47426

W3C validator