![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isumless | Structured version Visualization version GIF version |
Description: A finite sum of nonnegative numbers is less than or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumless.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumless.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumless.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
isumless.4 | ⊢ (𝜑 → 𝐴 ⊆ 𝑍) |
isumless.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) |
isumless.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) |
isumless.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵) |
isumless.8 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isumless | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumless.4 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑍) | |
2 | 1 | sselda 3976 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍) |
3 | isumless.6 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) | |
4 | 3 | recnd 11274 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
5 | 2, 4 | syldan 589 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | 5 | ralrimiva 3135 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
7 | isumless.1 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
8 | 7 | eqimssi 4037 | . . . . 5 ⊢ 𝑍 ⊆ (ℤ≥‘𝑀) |
9 | 8 | orci 863 | . . . 4 ⊢ (𝑍 ⊆ (ℤ≥‘𝑀) ∨ 𝑍 ∈ Fin) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑍 ⊆ (ℤ≥‘𝑀) ∨ 𝑍 ∈ Fin)) |
11 | sumss2 15708 | . . 3 ⊢ (((𝐴 ⊆ 𝑍 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) ∧ (𝑍 ⊆ (ℤ≥‘𝑀) ∨ 𝑍 ∈ Fin)) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0)) | |
12 | 1, 6, 10, 11 | syl21anc 836 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0)) |
13 | isumless.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
14 | eleq1w 2808 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴)) | |
15 | fveq2 6896 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
16 | 14, 15 | ifbieq1d 4554 | . . . . . 6 ⊢ (𝑗 = 𝑘 → if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0)) |
17 | eqid 2725 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0)) = (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0)) | |
18 | fvex 6909 | . . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V | |
19 | c0ex 11240 | . . . . . . 7 ⊢ 0 ∈ V | |
20 | 18, 19 | ifex 4580 | . . . . . 6 ⊢ if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) ∈ V |
21 | 16, 17, 20 | fvmpt 7004 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0)) |
22 | 21 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0)) |
23 | isumless.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
24 | 23 | ifeq1d 4549 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) = if(𝑘 ∈ 𝐴, 𝐵, 0)) |
25 | 22, 24 | eqtrd 2765 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) |
26 | 0re 11248 | . . . 4 ⊢ 0 ∈ ℝ | |
27 | ifcl 4575 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ) | |
28 | 3, 26, 27 | sylancl 584 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ) |
29 | isumless.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵) | |
30 | leid 11342 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ 𝐵) | |
31 | breq1 5152 | . . . . . 6 ⊢ (𝐵 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (𝐵 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)) | |
32 | breq1 5152 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (0 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)) | |
33 | 31, 32 | ifboth 4569 | . . . . 5 ⊢ ((𝐵 ≤ 𝐵 ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵) |
34 | 30, 33 | sylan 578 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵) |
35 | 3, 29, 34 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵) |
36 | isumless.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
37 | 7, 13, 36, 1, 25, 5 | fsumcvg3 15711 | . . 3 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))) ∈ dom ⇝ ) |
38 | isumless.8 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
39 | 7, 13, 25, 28, 23, 3, 35, 37, 38 | isumle 15826 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ Σ𝑘 ∈ 𝑍 𝐵) |
40 | 12, 39 | eqbrtrd 5171 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 ifcif 4530 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ‘cfv 6549 Fincfn 8964 ℂcc 11138 ℝcr 11139 0cc0 11140 + caddc 11143 ≤ cle 11281 ℤcz 12591 ℤ≥cuz 12855 seqcseq 14002 ⇝ cli 15464 Σcsu 15668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 |
This theorem is referenced by: isumltss 15830 climcnds 15833 harmonic 15841 mertenslem1 15866 prmreclem5 16892 ovoliunlem1 25475 ovoliun2 25479 esumpcvgval 33828 eulerpartlems 34111 geomcau 37363 |
Copyright terms: Public domain | W3C validator |