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Mirrors > Home > MPE Home > Th. List > clim2ser2 | Structured version Visualization version GIF version |
Description: The limit of an infinite series with an initial segment added. (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) |
clim2ser2.5 | ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ 𝐴) |
Ref | Expression |
---|---|
clim2ser2 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (𝐴 + (seq𝑀( + , 𝐹)‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 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 2923 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | peano2uz 12295 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
7 | eluzelz 12247 | . . 3 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ ℤ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
9 | clim2ser2.5 | . 2 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ 𝐴) | |
10 | eluzel2 12242 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
11 | 4, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | clim2ser.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
13 | 3, 11, 12 | serf 13392 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
14 | 13, 2 | ffvelrnd 6846 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
15 | seqex 13365 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ V) |
17 | 6, 3 | eleqtrrdi 2924 | . . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍) |
18 | 3 | uztrn2 12256 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
19 | 17, 18 | sylan 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
20 | 19, 12 | syldan 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ) |
21 | 1, 8, 20 | serf 13392 | . . 3 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ) |
22 | 21 | ffvelrnda 6845 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( + , 𝐹)‘𝑗) ∈ ℂ) |
23 | 14 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
24 | addcl 10613 | . . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
25 | 24 | adantl 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
26 | addass 10618 | . . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) | |
27 | 26 | adantl 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) |
28 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) | |
29 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
30 | elfzuz 12898 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
31 | 30, 3 | eleqtrrdi 2924 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
32 | 31, 12 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
33 | 32 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
34 | 25, 27, 28, 29, 33 | seqsplit 13397 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗))) |
35 | 23, 22, 34 | comraddd 10848 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑗) = ((seq(𝑁 + 1)( + , 𝐹)‘𝑗) + (seq𝑀( + , 𝐹)‘𝑁))) |
36 | 1, 8, 9, 14, 16, 22, 35 | climaddc1 14985 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (𝐴 + (seq𝑀( + , 𝐹)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 seqcseq 13363 ⇝ cli 14835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 |
This theorem is referenced by: iserex 15007 abelthlem6 25018 abelthlem9 25022 leibpi 25514 |
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