etransclem2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem2		Structured version Visualization version GIF version

Theorem etransclem2 46841

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 7422	. 2 ⊢ (ℝ D 𝐺) = (ℝ D (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)))
3		tgioo4 24930	. . 3 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
4		eqid 2769	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5		reelprrecn 11191	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
6	5	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
7		reopn 45899	. . . 4 ⊢ ℝ ∈ (topGen‘ran (,))
8	7	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
9		fzfid 14008	. . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin)
10		fzelp1 13603	. . . . . 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 604	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
13	12	3adant3 1148	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑖):ℝ⟶ℂ)
14		simp3 1154	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
15	13, 14	ffvelcdmd 7081	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥) ∈ ℂ)
16		fzp1elp1 13604	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → (𝑖 + 1) ∈ (0...(𝑅 + 1)))
17		ovex 7444	. . . . . . 7 ⊢ (𝑖 + 1) ∈ V
18		eleq1 2857	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → (𝑗 ∈ (0...(𝑅 + 1)) ↔ (𝑖 + 1) ∈ (0...(𝑅 + 1))))
19	18	anbi2d 641	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1)))))
20		fveq2 6882	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 + 1) → ((ℝ D^𝑛 𝐹)‘𝑗) = ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)))
21	20	feq1d 6688	. . . . . . . 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 2857	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0...(𝑅 + 1)) ↔ 𝑗 ∈ (0...(𝑅 + 1))))
24	23	anbi2d 641	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0...(𝑅 + 1))) ↔ (𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1)))))
25		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((ℝ D^𝑛 𝐹)‘𝑖) = ((ℝ D^𝑛 𝐹)‘𝑗))
26	25	feq1d 6688	. . . . . . . . 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 2016	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘𝑗):ℝ⟶ℂ)
29	17, 22, 28	vtocl 3534	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...(𝑅 + 1))) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
30	16, 29	sylan2 604	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
31	30	3adant3 1148	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)):ℝ⟶ℂ)
32	31, 14	ffvelcdmd 7081	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅) ∧ 𝑥 ∈ ℝ) → (((ℝ D^𝑛 𝐹)‘(𝑖 + 1))‘𝑥) ∈ ℂ)
33	12	ffnd 6707	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘𝑖) Fn ℝ)
34		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ
35		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑥 D^𝑛
36		etransclem2.xf	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
37	34, 35, 36	nfov 7441	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℝ D^𝑛 𝐹)
38		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑖
39	37, 38	nffv 6892	. . . . . . . 8 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘𝑖)
40	39	dffn5f 6953	. . . . . . 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 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ ℝ ↦ (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥)) = ((ℝ D^𝑛 𝐹)‘𝑖))
43	42	oveq2d 7427	. . . 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 11156	. . . . . 6 ⊢ ℝ ⊆ ℂ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ℝ ⊆ ℂ)
46		etransclem2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
47		ffdm 6736	. . . . . . . 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 11180	. . . . . . . . 9 ⊢ ℂ ∈ V
50	49	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℂ ∈ V)
51		reex 11190	. . . . . . . 8 ⊢ ℝ ∈ V
52		elpm2g 8840	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℝ ∈ V) → (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)))
53	50, 51, 52	sylancl 597	. . . . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
56		elfznn0 13647	. . . . . 6 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ₀)
57	56	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ₀)
58		dvnp1 26052	. . . . 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 1396	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑖)))
60	30	ffnd 6707	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D^𝑛 𝐹)‘(𝑖 + 1)) Fn ℝ)
61		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 + 1)
62	37, 61	nffv 6892	. . . . . 6 ⊢ Ⅎ𝑥((ℝ D^𝑛 𝐹)‘(𝑖 + 1))
63	62	dffn5f 6953	. . . . 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 2810	. . 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 26102	. 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 2816	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 Ⅎwnfc 2916 Vcvv 3463 ⊆ wss 3913 {cpr 4596 ↦ cmpt 5196 dom cdm 5662 ran crn 5663 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ↑_pm cpm 8824 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 ℕ₀cn0 12503 (,)cioo 13371 ...cfz 13534 Σcsu 15736 TopOpenctopn 17473 topGenctg 17489 ℂ_fldccnfld 21490 D cdv 25990 D^𝑛 cdvn 25991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-icc 13378 df-fz 13535 df-fzo 13682 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-sum 15737 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-fbas 21487 df-fg 21488 df-cnfld 21491 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cld 23144 df-ntr 23145 df-cls 23146 df-nei 23223 df-lp 23261 df-perf 23262 df-cn 23352 df-cnp 23353 df-haus 23440 df-tx 23687 df-hmeo 23880 df-fil 23971 df-fm 24063 df-flim 24064 df-flf 24065 df-xms 24445 df-ms 24446 df-tms 24447 df-cncf 25005 df-limc 25993 df-dv 25994 df-dvn 25995

This theorem is referenced by: etransclem46 46885

W3C validator