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Mirrors > Home > ILE Home > Th. List > Mathboxes > cvgcmp2n | GIF version |
Description: A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.) |
Ref | Expression |
---|---|
cvgcmp2n.cl | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) |
cvgcmp2n.ge0 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘)) |
cvgcmp2n.lt | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘))) |
Ref | Expression |
---|---|
cvgcmp2n | ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 9557 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 9274 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
3 | cvgcmp2n.cl | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) | |
4 | 3 | recnd 7980 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℂ) |
5 | 1, 2, 4 | serf 10467 | . 2 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℂ) |
6 | 2rp 9652 | . . 3 ⊢ 2 ∈ ℝ+ | |
7 | 6 | a1i 9 | . 2 ⊢ (𝜑 → 2 ∈ ℝ+) |
8 | 3 | adantlr 477 | . . . 4 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) |
9 | cvgcmp2n.ge0 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘)) | |
10 | 9 | adantlr 477 | . . . 4 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘)) |
11 | cvgcmp2n.lt | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘))) | |
12 | 11 | adantlr 477 | . . . 4 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘))) |
13 | simprl 529 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ ℕ) | |
14 | simprr 531 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑚)) | |
15 | 8, 10, 12, 13, 14 | cvgcmp2nlemabs 14611 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (abs‘((seq1( + , 𝐺)‘𝑛) − (seq1( + , 𝐺)‘𝑚))) < (2 / 𝑚)) |
16 | 15 | ralrimivva 2559 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq1( + , 𝐺)‘𝑛) − (seq1( + , 𝐺)‘𝑚))) < (2 / 𝑚)) |
17 | 5, 7, 16 | climcvg1n 11349 | 1 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4001 dom cdm 4624 ‘cfv 5213 (class class class)co 5870 ℝcr 7805 0cc0 7806 1c1 7807 + caddc 7809 < clt 7986 ≤ cle 7987 − cmin 8122 / cdiv 8623 ℕcn 8913 2c2 8964 ℤ≥cuz 9522 ℝ+crp 9647 seqcseq 10438 ↑cexp 10512 abscabs 10997 ⇝ cli 11277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4116 ax-sep 4119 ax-nul 4127 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-iinf 4585 ax-cnex 7897 ax-resscn 7898 ax-1cn 7899 ax-1re 7900 ax-icn 7901 ax-addcl 7902 ax-addrcl 7903 ax-mulcl 7904 ax-mulrcl 7905 ax-addcom 7906 ax-mulcom 7907 ax-addass 7908 ax-mulass 7909 ax-distr 7910 ax-i2m1 7911 ax-0lt1 7912 ax-1rid 7913 ax-0id 7914 ax-rnegex 7915 ax-precex 7916 ax-cnre 7917 ax-pre-ltirr 7918 ax-pre-ltwlin 7919 ax-pre-lttrn 7920 ax-pre-apti 7921 ax-pre-ltadd 7922 ax-pre-mulgt0 7923 ax-pre-mulext 7924 ax-arch 7925 ax-caucvg 7926 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-iun 3887 df-br 4002 df-opab 4063 df-mpt 4064 df-tr 4100 df-id 4291 df-po 4294 df-iso 4295 df-iord 4364 df-on 4366 df-ilim 4367 df-suc 4369 df-iom 4588 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-f1 5218 df-fo 5219 df-f1o 5220 df-fv 5221 df-isom 5222 df-riota 5826 df-ov 5873 df-oprab 5874 df-mpo 5875 df-1st 6136 df-2nd 6137 df-recs 6301 df-irdg 6366 df-frec 6387 df-1o 6412 df-oadd 6416 df-er 6530 df-en 6736 df-dom 6737 df-fin 6738 df-pnf 7988 df-mnf 7989 df-xr 7990 df-ltxr 7991 df-le 7992 df-sub 8124 df-neg 8125 df-reap 8526 df-ap 8533 df-div 8624 df-inn 8914 df-2 8972 df-3 8973 df-4 8974 df-n0 9171 df-z 9248 df-uz 9523 df-q 9614 df-rp 9648 df-ico 9888 df-fz 10003 df-fzo 10136 df-seqfrec 10439 df-exp 10513 df-ihash 10747 df-cj 10842 df-re 10843 df-im 10844 df-rsqrt 10998 df-abs 10999 df-clim 11278 df-sumdc 11353 |
This theorem is referenced by: trilpolemclim 14615 |
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