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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 3995 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍) |
3 | isumless.6 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) | |
4 | 3 | recnd 11287 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
5 | 2, 4 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | 5 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
7 | isumless.1 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
8 | 7 | eqimssi 4056 | . . . . 5 ⊢ 𝑍 ⊆ (ℤ≥‘𝑀) |
9 | 8 | orci 865 | . . . 4 ⊢ (𝑍 ⊆ (ℤ≥‘𝑀) ∨ 𝑍 ∈ Fin) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑍 ⊆ (ℤ≥‘𝑀) ∨ 𝑍 ∈ Fin)) |
11 | sumss2 15759 | . . 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 838 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0)) |
13 | isumless.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
14 | eleq1w 2822 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴)) | |
15 | fveq2 6907 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
16 | 14, 15 | ifbieq1d 4555 | . . . . . 6 ⊢ (𝑗 = 𝑘 → if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0)) |
17 | eqid 2735 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0)) = (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0)) | |
18 | fvex 6920 | . . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V | |
19 | c0ex 11253 | . . . . . . 7 ⊢ 0 ∈ V | |
20 | 18, 19 | ifex 4581 | . . . . . 6 ⊢ if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) ∈ V |
21 | 16, 17, 20 | fvmpt 7016 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0)) |
22 | 21 | adantl 481 | . . . 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 4550 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) = if(𝑘 ∈ 𝐴, 𝐵, 0)) |
25 | 22, 24 | eqtrd 2775 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) |
26 | 0re 11261 | . . . 4 ⊢ 0 ∈ ℝ | |
27 | ifcl 4576 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ) | |
28 | 3, 26, 27 | sylancl 586 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ) |
29 | isumless.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵) | |
30 | leid 11355 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ 𝐵) | |
31 | breq1 5151 | . . . . . 6 ⊢ (𝐵 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (𝐵 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)) | |
32 | breq1 5151 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (0 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)) | |
33 | 31, 32 | ifboth 4570 | . . . . 5 ⊢ ((𝐵 ≤ 𝐵 ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵) |
34 | 30, 33 | sylan 580 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵) |
35 | 3, 29, 34 | syl2anc 584 | . . 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 15762 | . . 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 15877 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ Σ𝑘 ∈ 𝑍 𝐵) |
40 | 12, 39 | eqbrtrd 5170 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 + caddc 11156 ≤ cle 11294 ℤcz 12611 ℤ≥cuz 12876 seqcseq 14039 ⇝ cli 15517 Σcsu 15719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 |
This theorem is referenced by: isumltss 15881 climcnds 15884 harmonic 15892 mertenslem1 15917 prmreclem5 16954 ovoliunlem1 25551 ovoliun2 25555 esumpcvgval 34059 eulerpartlems 34342 geomcau 37746 |
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