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Mirrors > Home > MPE Home > Th. List > climserle | Structured version Visualization version 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 |
---|---|
clim2ser.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 | clim2ser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climserle.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
3 | climserle.3 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) | |
4 | 2, 1 | eleqtrdi 2848 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | eluzel2 12880 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
7 | climserle.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
8 | 1, 6, 7 | serfre 14068 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) |
9 | 8 | ffvelcdmda 7103 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ) |
10 | 1 | peano2uzs 12941 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍) |
11 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑗 + 1))) | |
12 | 11 | breq2d 5159 | . . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1)))) |
13 | 12 | imbi2d 340 | . . . . . . 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 413 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘𝑘))) |
16 | 13, 15 | vtoclga 3576 | . . . . . 6 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘(𝑗 + 1)))) |
17 | 16 | impcom 407 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1))) |
18 | 10, 17 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1))) |
19 | 11 | eleq1d 2823 | . . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ)) |
20 | 19 | imbi2d 340 | . . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℝ) ↔ (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ))) |
21 | 7 | expcom 413 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (𝐹‘𝑘) ∈ ℝ)) |
22 | 20, 21 | vtoclga 3576 | . . . . . . 7 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ)) |
23 | 22 | impcom 407 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ) |
24 | 10, 23 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ) |
25 | 9, 24 | addge01d 11848 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (0 ≤ (𝐹‘(𝑗 + 1)) ↔ (seq𝑀( + , 𝐹)‘𝑗) ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))) |
26 | 18, 25 | mpbid 232 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1)))) |
27 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
28 | 27, 1 | eleqtrdi 2848 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
29 | seqp1 14053 | . . . 4 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1)))) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1)))) |
31 | 26, 30 | breqtrrd 5175 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))) |
32 | 1, 2, 3, 9, 31 | climub 15694 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 ≤ cle 11293 ℤcz 12610 ℤ≥cuz 12875 seqcseq 14038 ⇝ cli 15516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fl 13828 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 |
This theorem is referenced by: isumrpcl 15875 ege2le3 16122 prmreclem6 16954 ioombl1lem4 25609 rge0scvg 33909 |
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