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Mirrors > Home > MPE Home > Th. List > isumsup2 | Structured version Visualization version GIF version |
Description: An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.) |
Ref | Expression |
---|---|
isumsup.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumsup.2 | ⊢ 𝐺 = seq𝑀( + , 𝐹) |
isumsup.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumsup.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumsup.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
isumsup.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) |
isumsup.7 | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐺‘𝑗) ≤ 𝑥) |
Ref | Expression |
---|---|
isumsup2 | ⊢ (𝜑 → 𝐺 ⇝ sup(ran 𝐺, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumsup.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumsup.3 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isumsup.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
4 | isumsup.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) | |
5 | 3, 4 | eqeltrd 2838 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
6 | 1, 2, 5 | serfre 14068 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) |
7 | isumsup.2 | . . . 4 ⊢ 𝐺 = seq𝑀( + , 𝐹) | |
8 | 7 | feq1i 6727 | . . 3 ⊢ (𝐺:𝑍⟶ℝ ↔ seq𝑀( + , 𝐹):𝑍⟶ℝ) |
9 | 6, 8 | sylibr 234 | . 2 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) |
10 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
11 | 10, 1 | eleqtrdi 2848 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
12 | eluzelz 12885 | . . . . 5 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
13 | uzid 12890 | . . . . 5 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
14 | peano2uz 12940 | . . . . 5 ⊢ (𝑗 ∈ (ℤ≥‘𝑗) → (𝑗 + 1) ∈ (ℤ≥‘𝑗)) | |
15 | 11, 12, 13, 14 | 4syl 19 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ (ℤ≥‘𝑗)) |
16 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝜑) | |
17 | elfzuz 13556 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
18 | 17, 1 | eleqtrrdi 2849 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ∈ 𝑍) |
19 | 16, 18, 5 | syl2an 596 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℝ) |
20 | 1 | peano2uzs 12941 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍) |
21 | 20 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍) |
22 | elfzuz 13556 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1)) → 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) | |
23 | 1 | uztrn2 12894 | . . . . . 6 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍) |
24 | 21, 22, 23 | syl2an 596 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 𝑘 ∈ 𝑍) |
25 | isumsup.6 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) | |
26 | 25, 3 | breqtrrd 5175 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
27 | 26 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
28 | 24, 27 | syldan 591 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 0 ≤ (𝐹‘𝑘)) |
29 | 11, 15, 19, 28 | sermono 14071 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))) |
30 | 7 | fveq1i 6907 | . . 3 ⊢ (𝐺‘𝑗) = (seq𝑀( + , 𝐹)‘𝑗) |
31 | 7 | fveq1i 6907 | . . 3 ⊢ (𝐺‘(𝑗 + 1)) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)) |
32 | 29, 30, 31 | 3brtr4g 5181 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ≤ (𝐺‘(𝑗 + 1))) |
33 | isumsup.7 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐺‘𝑗) ≤ 𝑥) | |
34 | 1, 2, 9, 32, 33 | climsup 15702 | 1 ⊢ (𝜑 → 𝐺 ⇝ sup(ran 𝐺, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 class class class wbr 5147 ran crn 5689 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supcsup 9477 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 < clt 11292 ≤ cle 11293 ℤcz 12610 ℤ≥cuz 12875 ...cfz 13543 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-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 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-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 |
This theorem is referenced by: isumsup 15879 ovoliunlem1 25550 ioombl1lem4 25609 uniioombllem2 25631 uniioombllem6 25636 sge0isum 46382 |
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