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Theorem abelth2 26402

Description: Abel's theorem, restricted to the [0, 1] interval. (Contributed by Mario Carneiro, 2-Apr-2015.)

**Hypotheses**
Ref	Expression
abelth2.1	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
abelth2.2	⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
abelth2.3	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)))

**Assertion**
Ref	Expression
abelth2	⊢ (𝜑 → 𝐹 ∈ ((0[,]1)–cn→ℂ))

Distinct variable groups: 𝑥,𝑛,𝐴 𝜑,𝑛,𝑥 Allowed substitution hints: 𝐹(𝑥,𝑛)

Proof of Theorem abelth2 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unitssre 13514	. . . . . . 7 ⊢ (0[,]1) ⊆ ℝ
2		ax-resscn 11184	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3968	. . . . . 6 ⊢ (0[,]1) ⊆ ℂ
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → (0[,]1) ⊆ ℂ)
5		1re 11233	. . . . . . . 8 ⊢ 1 ∈ ℝ
6		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ∈ (0[,]1))
7		elicc01 13481	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
8	6, 7	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
9	8	simp1d 1142	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ∈ ℝ)
10		resubcl 11545	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (1 − 𝑧) ∈ ℝ)
11	5, 9, 10	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ∈ ℝ)
12	11	leidd 11801	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ≤ (1 − 𝑧))
13		1red 11234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 1 ∈ ℝ)
14	8	simp3d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ≤ 1)
15	9, 13, 14	abssubge0d 15448	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘(1 − 𝑧)) = (1 − 𝑧))
16	8	simp2d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 0 ≤ 𝑧)
17	9, 16	absidd 15439	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘𝑧) = 𝑧)
18	17	oveq2d 7419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − (abs‘𝑧)) = (1 − 𝑧))
19	18	oveq2d 7419	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − (abs‘𝑧))) = (1 · (1 − 𝑧)))
20	11	recnd 11261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ∈ ℂ)
21	20	mullidd 11251	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − 𝑧)) = (1 − 𝑧))
22	19, 21	eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − (abs‘𝑧))) = (1 − 𝑧))
23	12, 15, 22	3brtr4d 5151	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧))))
24	4, 23	ssrabdv 4049	. . . 4 ⊢ (𝜑 → (0[,]1) ⊆ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))})
25	24	resmptd 6027	. . 3 ⊢ (𝜑 → ((𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (0[,]1)) = (𝑥 ∈ (0[,]1) ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))))
26		abelth2.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)))
27	25, 26	eqtr4di 2788	. 2 ⊢ (𝜑 → ((𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (0[,]1)) = 𝐹)
28		abelth2.1	. . . 4 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
29		abelth2.2	. . . 4 ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
30		1red 11234	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
31		0le1 11758	. . . . 5 ⊢ 0 ≤ 1
32	31	a1i 11	. . . 4 ⊢ (𝜑 → 0 ≤ 1)
33		eqid 2735	. . . 4 ⊢ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))}
34		eqid 2735	. . . 4 ⊢ (𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) = (𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)))
35	28, 29, 30, 32, 33, 34	abelth 26401	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) ∈ ({𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))}–cn→ℂ))
36		rescncf 24839	. . 3 ⊢ ((0[,]1) ⊆ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} → ((𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) ∈ ({𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))}–cn→ℂ) → ((𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ)))
37	24, 35, 36	sylc 65	. 2 ⊢ (𝜑 → ((𝑥 ∈ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ))
38	27, 37	eqeltrrd 2835	1 ⊢ (𝜑 → 𝐹 ∈ ((0[,]1)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 {crab 3415 ⊆ wss 3926 class class class wbr 5119 ↦ cmpt 5201 dom cdm 5654 ↾ cres 5656 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 ≤ cle 11268 − cmin 11464 ℕ₀cn0 12499 [,]cicc 13363 seqcseq 14017 ↑cexp 14077 abscabs 15251 ⇝ cli 15498 Σcsu 15700 –cn→ccncf 24818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-seq 14018 df-exp 14078 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cn 23163 df-cnp 23164 df-t1 23250 df-haus 23251 df-tx 23498 df-hmeo 23691 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-ulm 26336

This theorem is referenced by: leibpi 26902

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