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Mirrors > Home > MPE Home > Th. List > clim2ser | Structured version Visualization version 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 2737 | . 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 2848 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | peano2uz 12497 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
7 | eluzelz 12448 | . . 3 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ ℤ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
9 | clim2ser.5 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) | |
10 | eluzel2 12443 | . . . . 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 13604 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
14 | 13, 2 | ffvelrnd 6905 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
15 | seqex 13576 | . . 3 ⊢ seq(𝑁 + 1)( + , 𝐹) ∈ V | |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ∈ V) |
17 | 13 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
18 | 6, 3 | eleqtrrdi 2849 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍) |
19 | 3 | uztrn2 12457 | . . . 4 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍) |
20 | 18, 19 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍) |
21 | 17, 20 | ffvelrnd 6905 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℂ) |
22 | addcl 10811 | . . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
23 | 22 | adantl 485 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
24 | addass 10816 | . . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) | |
25 | 24 | adantl 485 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) |
26 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) | |
27 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
28 | elfzuz 13108 | . . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
29 | 28, 3 | eleqtrrdi 2849 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
30 | 29, 12 | sylan2 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
31 | 30 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
32 | 23, 25, 26, 27, 31 | seqsplit 13609 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗))) |
33 | 32 | oveq1d 7228 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → ((seq𝑀( + , 𝐹)‘𝑗) − (seq𝑀( + , 𝐹)‘𝑁)) = (((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗)) − (seq𝑀( + , 𝐹)‘𝑁))) |
34 | 14 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
35 | 3 | uztrn2 12457 | . . . . . . . 8 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
36 | 18, 35 | sylan 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
37 | 36, 12 | syldan 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ) |
38 | 1, 8, 37 | serf 13604 | . . . . 5 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ) |
39 | 38 | ffvelrnda 6904 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( + , 𝐹)‘𝑗) ∈ ℂ) |
40 | 34, 39 | pncan2d 11191 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗)) − (seq𝑀( + , 𝐹)‘𝑁)) = (seq(𝑁 + 1)( + , 𝐹)‘𝑗)) |
41 | 33, 40 | eqtr2d 2778 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( + , 𝐹)‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑗) − (seq𝑀( + , 𝐹)‘𝑁))) |
42 | 1, 8, 9, 14, 16, 21, 41 | climsubc1 15199 | 1 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ (𝐴 − (seq𝑀( + , 𝐹)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3408 class class class wbr 5053 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 1c1 10730 + caddc 10732 − cmin 11062 ℤcz 12176 ℤ≥cuz 12438 ...cfz 13095 seqcseq 13574 ⇝ cli 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 |
This theorem is referenced by: iserex 15220 ege2le3 15651 abelthlem9 25332 stirlinglem7 43296 stirlinglem11 43300 stirlinglem12 43301 |
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