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Mirrors > Home > ILE Home > Th. List > clim2ser | GIF version |
Description: The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
clim2ser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
clim2ser.2 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
clim2ser.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
clim2ser.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) |
Ref | Expression |
---|---|
clim2ser | ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ (𝐴 − (seq𝑀( + , 𝐹)‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . 2 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1)) | |
2 | clim2ser.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
3 | clim2ser.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrdi 2230 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | peano2uz 9371 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
7 | eluzelz 9328 | . . 3 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ ℤ) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
9 | clim2ser.5 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) | |
10 | eluzel2 9324 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
11 | 4, 10 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | clim2ser.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
13 | 3, 11, 12 | serf 10240 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
14 | 13, 2 | ffvelrnd 5549 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
15 | seqex 10213 | . . 3 ⊢ seq(𝑁 + 1)( + , 𝐹) ∈ V | |
16 | 15 | a1i 9 | . 2 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ∈ V) |
17 | 13 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
18 | 6, 3 | eleqtrrdi 2231 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍) |
19 | 3 | uztrn2 9336 | . . . 4 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍) |
20 | 18, 19 | sylan 281 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍) |
21 | 17, 20 | ffvelrnd 5549 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℂ) |
22 | addcl 7738 | . . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
23 | 22 | adantl 275 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
24 | addass 7743 | . . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) | |
25 | 24 | adantl 275 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) |
26 | simpr 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) | |
27 | 4 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
28 | 3 | eleq2i 2204 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀)) |
29 | 28, 12 | sylan2br 286 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
30 | 29 | adantlr 468 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
31 | 23, 25, 26, 27, 30 | seq3split 10245 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗))) |
32 | 31 | oveq1d 5782 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → ((seq𝑀( + , 𝐹)‘𝑗) − (seq𝑀( + , 𝐹)‘𝑁)) = (((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗)) − (seq𝑀( + , 𝐹)‘𝑁))) |
33 | 14 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
34 | 3 | uztrn2 9336 | . . . . . . . 8 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
35 | 18, 34 | sylan 281 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
36 | 35, 12 | syldan 280 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ) |
37 | 1, 8, 36 | serf 10240 | . . . . 5 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ) |
38 | 37 | ffvelrnda 5548 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( + , 𝐹)‘𝑗) ∈ ℂ) |
39 | 33, 38 | pncan2d 8068 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗)) − (seq𝑀( + , 𝐹)‘𝑁)) = (seq(𝑁 + 1)( + , 𝐹)‘𝑗)) |
40 | 32, 39 | eqtr2d 2171 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( + , 𝐹)‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑗) − (seq𝑀( + , 𝐹)‘𝑁))) |
41 | 1, 8, 9, 14, 16, 21, 40 | climsubc1 11094 | 1 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ (𝐴 − (seq𝑀( + , 𝐹)‘𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 Vcvv 2681 class class class wbr 3924 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 1c1 7614 + caddc 7616 − cmin 7926 ℤcz 9047 ℤ≥cuz 9319 seqcseq 10211 ⇝ cli 11040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-rp 9435 df-fz 9784 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 |
This theorem is referenced by: iserex 11101 ege2le3 11366 |
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