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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 2851 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 5 | eluzel2 12883 | . . . . 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 14072 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) |
| 9 | 8 | ffvelcdmda 7104 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ) |
| 10 | 1 | peano2uzs 12944 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍) |
| 11 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑗 + 1))) | |
| 12 | 11 | breq2d 5155 | . . . . . . . 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 3577 | . . . . . 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 2826 | . . . . . . . . 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 3577 | . . . . . . 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 11851 | . . . 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 2851 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
| 29 | seqp1 14057 | . . . 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 5171 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))) |
| 32 | 1, 2, 3, 9, 31 | climub 15698 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 seqcseq 14042 ⇝ cli 15520 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fl 13832 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 |
| This theorem is referenced by: isumrpcl 15879 ege2le3 16126 prmreclem6 16959 ioombl1lem4 25596 rge0scvg 33948 |
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