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Mirrors > Home > ILE Home > Th. List > clim2iser2 | GIF version |
Description: The limit of an infinite series with an initial segment added. (Contributed by Jim Kingdon, 21-Aug-2021.) |
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, ℂ) ⇝ 𝐴) |
Ref | Expression |
---|---|
clim2iser2 | ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (𝐴 + (seq𝑀( + , 𝐹, ℂ)‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2081 | . 2 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1)) | |
2 | clim2ser.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
3 | clim2ser.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | syl6eleq 2171 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | peano2uz 8671 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
7 | eluzelz 8628 | . . 3 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ ℤ) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
9 | clim2ser2.5 | . 2 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ) ⇝ 𝐴) | |
10 | eluzel2 8624 | . . . . 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 | iserf 9453 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ) |
14 | 13, 2 | ffvelrnd 5324 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ ℂ) |
15 | iseqex 9433 | . . 3 ⊢ seq𝑀( + , 𝐹, ℂ) ∈ V | |
16 | 15 | a1i 9 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ∈ V) |
17 | 6, 3 | syl6eleqr 2172 | . . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍) |
18 | 3 | uztrn2 8636 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
19 | 17, 18 | sylan 277 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
20 | 19, 12 | syldan 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ) |
21 | 1, 8, 20 | iserf 9453 | . . 3 ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ):(ℤ≥‘(𝑁 + 1))⟶ℂ) |
22 | 21 | ffvelrnda 5323 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( + , 𝐹, ℂ)‘𝑗) ∈ ℂ) |
23 | addcl 7098 | . . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
24 | 23 | adantl 271 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
25 | addass 7103 | . . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) | |
26 | 25 | adantl 271 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 + 𝑥) + 𝑦) = (𝑘 + (𝑥 + 𝑦))) |
27 | simpr 108 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) | |
28 | cnex 7097 | . . . . 5 ⊢ ℂ ∈ V | |
29 | 28 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → ℂ ∈ V) |
30 | 4 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
31 | 3 | eleq2i 2145 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀)) |
32 | 31, 12 | sylan2br 282 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
33 | 32 | adantlr 460 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
34 | 24, 26, 27, 29, 30, 33 | iseqsplit 9458 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) = ((seq𝑀( + , 𝐹, ℂ)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹, ℂ)‘𝑗))) |
35 | 14 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ ℂ) |
36 | 35, 22 | addcomd 7259 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → ((seq𝑀( + , 𝐹, ℂ)‘𝑁) + (seq(𝑁 + 1)( + , 𝐹, ℂ)‘𝑗)) = ((seq(𝑁 + 1)( + , 𝐹, ℂ)‘𝑗) + (seq𝑀( + , 𝐹, ℂ)‘𝑁))) |
37 | 34, 36 | eqtrd 2113 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) = ((seq(𝑁 + 1)( + , 𝐹, ℂ)‘𝑗) + (seq𝑀( + , 𝐹, ℂ)‘𝑁))) |
38 | 1, 8, 9, 14, 16, 22, 37 | climaddc1 10167 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (𝐴 + (seq𝑀( + , 𝐹, ℂ)‘𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 Vcvv 2601 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 ℂcc 6979 1c1 6982 + caddc 6984 ℤcz 8351 ℤ≥cuz 8619 seqcseq 9431 ⇝ cli 10117 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-rp 8735 df-fz 9030 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-clim 10118 |
This theorem is referenced by: iiserex 10177 |
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