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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 2915 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
6 | 1, 2, 5 | serfre 13402 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) |
7 | isumsup.2 | . . . 4 ⊢ 𝐺 = seq𝑀( + , 𝐹) | |
8 | 7 | feq1i 6507 | . . 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 2925 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
12 | eluzelz 12256 | . . . . 5 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
13 | uzid 12261 | . . . . 5 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
14 | peano2uz 12304 | . . . . 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 12907 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
18 | 17, 1 | eleqtrrdi 2926 | . . . . 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 12305 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍) |
21 | 20 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍) |
22 | elfzuz 12907 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1)) → 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) | |
23 | 1 | uztrn2 12265 | . . . . . 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 5096 | . . . . . 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 13405 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))) |
30 | 7 | fveq1i 6673 | . . 3 ⊢ (𝐺‘𝑗) = (seq𝑀( + , 𝐹)‘𝑗) |
31 | 7 | fveq1i 6673 | . . 3 ⊢ (𝐺‘(𝑗 + 1)) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)) |
32 | 29, 30, 31 | 3brtr4g 5102 | . 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 15028 | 1 ⊢ (𝜑 → 𝐺 ⇝ sup(ran 𝐺, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 class class class wbr 5068 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supcsup 8906 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 ℤcz 11984 ℤ≥cuz 12246 ...cfz 12895 seqcseq 13372 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 |
This theorem is referenced by: isumsup 15204 ovoliunlem1 24105 ioombl1lem4 24164 uniioombllem2 24186 uniioombllem6 24191 sge0isum 42716 |
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