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Theorem abelth2 25141

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 12936	. . . . . . 7 ⊢ (0[,]1) ⊆ ℝ
2		ax-resscn 10637	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3903	. . . . . 6 ⊢ (0[,]1) ⊆ ℂ
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → (0[,]1) ⊆ ℂ)
5		1re 10684	. . . . . . . 8 ⊢ 1 ∈ ℝ
6		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ∈ (0[,]1))
7		elicc01 12903	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
8	6, 7	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
9	8	simp1d 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ∈ ℝ)
10		resubcl 10993	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (1 − 𝑧) ∈ ℝ)
11	5, 9, 10	sylancr 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ∈ ℝ)
12	11	leidd 11249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ≤ (1 − 𝑧))
13		1red 10685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 1 ∈ ℝ)
14	8	simp3d 1141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ≤ 1)
15	9, 13, 14	abssubge0d 14844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘(1 − 𝑧)) = (1 − 𝑧))
16	8	simp2d 1140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 0 ≤ 𝑧)
17	9, 16	absidd 14835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘𝑧) = 𝑧)
18	17	oveq2d 7171	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − (abs‘𝑧)) = (1 − 𝑧))
19	18	oveq2d 7171	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − (abs‘𝑧))) = (1 · (1 − 𝑧)))
20	11	recnd 10712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ∈ ℂ)
21	20	mulid2d 10702	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − 𝑧)) = (1 − 𝑧))
22	19, 21	eqtrd 2793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − (abs‘𝑧))) = (1 − 𝑧))
23	12, 15, 22	3brtr4d 5067	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧))))
24	4, 23	ssrabdv 3980	. . . 4 ⊢ (𝜑 → (0[,]1) ⊆ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))})
25	24	resmptd 5884	. . 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 2811	. 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 10685	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
31		0le1 11206	. . . . 5 ⊢ 0 ≤ 1
32	31	a1i 11	. . . 4 ⊢ (𝜑 → 0 ≤ 1)
33		eqid 2758	. . . 4 ⊢ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))}
34		eqid 2758	. . . 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 25140	. . 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 23603	. . 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 2853	1 ⊢ (𝜑 → 𝐹 ∈ ((0[,]1)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3074 ⊆ wss 3860 class class class wbr 5035 ↦ cmpt 5115 dom cdm 5527 ↾ cres 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7155 ℂcc 10578 ℝcr 10579 0cc0 10580 1c1 10581 + caddc 10583 · cmul 10585 ≤ cle 10719 − cmin 10913 ℕ₀cn0 11939 [,]cicc 12787 seqcseq 13423 ↑cexp 13484 abscabs 14646 ⇝ cli 14894 Σcsu 15095 –cn→ccncf 23582

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-inf2 9142 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658 ax-addf 10659 ax-mulf 10660

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-se 5487 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-2o 8118 df-er 8304 df-map 8423 df-pm 8424 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-fi 8913 df-sup 8944 df-inf 8945 df-oi 9012 df-card 9406 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-z 12026 df-dec 12143 df-uz 12288 df-q 12394 df-rp 12436 df-xneg 12553 df-xadd 12554 df-xmul 12555 df-ico 12790 df-icc 12791 df-fz 12945 df-fzo 13088 df-fl 13216 df-seq 13424 df-exp 13485 df-hash 13746 df-shft 14479 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-limsup 14881 df-clim 14898 df-rlim 14899 df-sum 15096 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-starv 16643 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-unif 16651 df-hom 16652 df-cco 16653 df-rest 16759 df-topn 16760 df-0g 16778 df-gsum 16779 df-topgen 16780 df-pt 16781 df-prds 16784 df-xrs 16838 df-qtop 16843 df-imas 16844 df-xps 16846 df-mre 16920 df-mrc 16921 df-acs 16923 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-submnd 18028 df-mulg 18297 df-cntz 18519 df-cmn 18980 df-psmet 20163 df-xmet 20164 df-met 20165 df-bl 20166 df-mopn 20167 df-cnfld 20172 df-top 21599 df-topon 21616 df-topsp 21638 df-bases 21651 df-cld 21724 df-ntr 21725 df-cn 21932 df-cnp 21933 df-t1 22019 df-haus 22020 df-tx 22267 df-hmeo 22460 df-xms 23027 df-ms 23028 df-tms 23029 df-cncf 23584 df-ulm 25076

This theorem is referenced by: leibpi 25632

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