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Mirrors > Home > MPE Home > Th. List > abelth2 | Structured version Visualization version GIF version |
Description: Abel's theorem, restricted to the [0, 1] interval. (Contributed by Mario Carneiro, 2-Apr-2015.) |
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))) |
Ref | Expression |
---|---|
abelth2 | ⊢ (𝜑 → 𝐹 ∈ ((0[,]1)–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitssre 12877 | . . . . . . 7 ⊢ (0[,]1) ⊆ ℝ | |
2 | ax-resscn 10583 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
3 | 1, 2 | sstri 3924 | . . . . . 6 ⊢ (0[,]1) ⊆ ℂ |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (0[,]1) ⊆ ℂ) |
5 | 1re 10630 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
6 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ∈ (0[,]1)) | |
7 | elicc01 12844 | . . . . . . . . . 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 10939 | . . . . . . . 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 11195 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ≤ (1 − 𝑧)) |
13 | 1red 10631 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 1 ∈ ℝ) | |
14 | 8 | simp3d 1141 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → 𝑧 ≤ 1) |
15 | 9, 13, 14 | abssubge0d 14783 | . . . . . 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 14774 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘𝑧) = 𝑧) |
18 | 17 | oveq2d 7151 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − (abs‘𝑧)) = (1 − 𝑧)) |
19 | 18 | oveq2d 7151 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − (abs‘𝑧))) = (1 · (1 − 𝑧))) |
20 | 11 | recnd 10658 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 − 𝑧) ∈ ℂ) |
21 | 20 | mulid2d 10648 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − 𝑧)) = (1 − 𝑧)) |
22 | 19, 21 | eqtrd 2833 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (1 · (1 − (abs‘𝑧))) = (1 − 𝑧)) |
23 | 12, 15, 22 | 3brtr4d 5062 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (0[,]1)) → (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))) |
24 | 4, 23 | ssrabdv 4001 | . . . 4 ⊢ (𝜑 → (0[,]1) ⊆ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))}) |
25 | 24 | resmptd 5875 | . . 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 2851 | . 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 10631 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
31 | 0le1 11152 | . . . . 5 ⊢ 0 ≤ 1 | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ≤ 1) |
33 | eqid 2798 | . . . 4 ⊢ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (1 · (1 − (abs‘𝑧)))} | |
34 | eqid 2798 | . . . 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 25036 | . . 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 23502 | . . 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 2891 | 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 3110 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 − cmin 10859 ℕ0cn0 11885 [,]cicc 12729 seqcseq 13364 ↑cexp 13425 abscabs 14585 ⇝ cli 14833 Σcsu 15034 –cn→ccncf 23481 |
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 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
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 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cn 21832 df-cnp 21833 df-t1 21919 df-haus 21920 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-ulm 24972 |
This theorem is referenced by: leibpi 25528 |
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