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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 2738 | . 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 2849 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | peano2uz 12629 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
7 | eluzelz 12580 | . . 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 12575 | . . . . 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 13739 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
14 | 13, 2 | ffvelrnd 6955 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
15 | seqex 13711 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ V) |
17 | 6, 3 | eleqtrrdi 2850 | . . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍) |
18 | 3 | uztrn2 12589 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
19 | 17, 18 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
20 | 19, 12 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ) |
21 | 1, 8, 20 | serf 13739 | . . 3 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ) |
22 | 21 | ffvelrnda 6954 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( + , 𝐹)‘𝑗) ∈ ℂ) |
23 | 14 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ) |
24 | addcl 10941 | . . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
25 | 24 | adantl 482 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
26 | addass 10946 | . . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) | |
27 | 26 | adantl 482 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) |
28 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) | |
29 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
30 | elfzuz 13240 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
31 | 30, 3 | eleqtrrdi 2850 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
32 | 31, 12 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
33 | 32 | adantlr 712 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
34 | 25, 27, 28, 29, 33 | seqsplit 13744 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹)‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹)‘𝑗))) |
35 | 23, 22, 34 | comraddd 11177 | . 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 15332 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (𝐴 + (seq𝑀( + , 𝐹)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3430 class class class wbr 5074 ‘cfv 6427 (class class class)co 7268 ℂcc 10857 1c1 10860 + caddc 10862 ℤcz 12307 ℤ≥cuz 12570 ...cfz 13227 seqcseq 13709 ⇝ cli 15181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-sup 9189 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-n0 12222 df-z 12308 df-uz 12571 df-rp 12719 df-fz 13228 df-seq 13710 df-exp 13771 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-clim 15185 |
This theorem is referenced by: iserex 15356 abelthlem6 25583 abelthlem9 25587 leibpi 26080 |
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