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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 2913 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
6 | 1, 2, 5 | serfre 13393 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) |
7 | isumsup.2 | . . . 4 ⊢ 𝐺 = seq𝑀( + , 𝐹) | |
8 | 7 | feq1i 6499 | . . 3 ⊢ (𝐺:𝑍⟶ℝ ↔ seq𝑀( + , 𝐹):𝑍⟶ℝ) |
9 | 6, 8 | sylibr 236 | . 2 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) |
10 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
11 | 10, 1 | eleqtrdi 2923 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
12 | eluzelz 12247 | . . . . 5 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
13 | uzid 12252 | . . . . 5 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
14 | peano2uz 12295 | . . . . 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 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝜑) | |
17 | elfzuz 12898 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
18 | 17, 1 | eleqtrrdi 2924 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ∈ 𝑍) |
19 | 16, 18, 5 | syl2an 597 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℝ) |
20 | 1 | peano2uzs 12296 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍) |
21 | 20 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍) |
22 | elfzuz 12898 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1)) → 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) | |
23 | 1 | uztrn2 12256 | . . . . . 6 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍) |
24 | 21, 22, 23 | syl2an 597 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 𝑘 ∈ 𝑍) |
25 | isumsup.6 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) | |
26 | 25, 3 | breqtrrd 5086 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
27 | 26 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
28 | 24, 27 | syldan 593 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 0 ≤ (𝐹‘𝑘)) |
29 | 11, 15, 19, 28 | sermono 13396 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))) |
30 | 7 | fveq1i 6665 | . . 3 ⊢ (𝐺‘𝑗) = (seq𝑀( + , 𝐹)‘𝑗) |
31 | 7 | fveq1i 6665 | . . 3 ⊢ (𝐺‘(𝑗 + 1)) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)) |
32 | 29, 30, 31 | 3brtr4g 5092 | . 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 15020 | 1 ⊢ (𝜑 → 𝐺 ⇝ sup(ran 𝐺, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 class class class wbr 5058 ran crn 5550 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 supcsup 8898 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 seqcseq 13363 ⇝ cli 14835 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 |
This theorem is referenced by: isumsup 15196 ovoliunlem1 24097 ioombl1lem4 24156 uniioombllem2 24178 uniioombllem6 24183 sge0isum 42703 |
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