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Mirrors > Home > ILE Home > Th. List > climserle | GIF version |
Description: The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
Ref | Expression |
---|---|
clim2iser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climserle.2 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
climserle.3 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) |
climserle.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climserle.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climserle | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2iser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climserle.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
3 | climserle.3 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) | |
4 | 2, 1 | eleqtrdi 2233 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | eluzel2 9355 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
7 | climserle.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
8 | 1, 6, 7 | serfre 10279 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) |
9 | 8 | ffvelrnda 5563 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ) |
10 | 1 | peano2uzs 9406 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍) |
11 | fveq2 5429 | . . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑗 + 1))) | |
12 | 11 | breq2d 3949 | . . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1)))) |
13 | 12 | imbi2d 229 | . . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → 0 ≤ (𝐹‘𝑘)) ↔ (𝜑 → 0 ≤ (𝐹‘(𝑗 + 1))))) |
14 | climserle.5 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) | |
15 | 14 | expcom 115 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘𝑘))) |
16 | 13, 15 | vtoclga 2755 | . . . . . 6 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘(𝑗 + 1)))) |
17 | 16 | impcom 124 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1))) |
18 | 10, 17 | sylan2 284 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1))) |
19 | 11 | eleq1d 2209 | . . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ)) |
20 | 19 | imbi2d 229 | . . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℝ) ↔ (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ))) |
21 | 7 | expcom 115 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (𝐹‘𝑘) ∈ ℝ)) |
22 | 20, 21 | vtoclga 2755 | . . . . . . 7 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ)) |
23 | 22 | impcom 124 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ) |
24 | 10, 23 | sylan2 284 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ) |
25 | 9, 24 | addge01d 8319 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (0 ≤ (𝐹‘(𝑗 + 1)) ↔ (seq𝑀( + , 𝐹)‘𝑗) ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))) |
26 | 18, 25 | mpbid 146 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1)))) |
27 | 1 | eleq2i 2207 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀)) |
28 | 27 | biimpi 119 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀)) |
29 | 28 | adantl 275 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
30 | simpll 519 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝜑) | |
31 | 1 | eleq2i 2207 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀)) |
32 | 31 | biimpri 132 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ 𝑍) |
33 | 32 | adantl 275 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ 𝑍) |
34 | 30, 33, 7 | syl2anc 409 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ) |
35 | readdcl 7770 | . . . . 5 ⊢ ((𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑘 + 𝑣) ∈ ℝ) | |
36 | 35 | adantl 275 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ) |
37 | 29, 34, 36 | seq3p1 10266 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1)))) |
38 | 26, 37 | breqtrrd 3964 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))) |
39 | 1, 2, 3, 9, 38 | climub 11145 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 ≤ cle 7825 ℤcz 9078 ℤ≥cuz 9350 seqcseq 10249 ⇝ cli 11079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-fz 9822 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 |
This theorem is referenced by: isumrpcl 11295 ege2le3 11414 |
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