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Mirrors > Home > MPE Home > Th. List > itg2i1fseq2 | Structured version Visualization version GIF version |
Description: In an extension to the results of itg2i1fseq 24283, if there is an upper bound on the integrals of the simple functions approaching 𝐹, then ∫2𝐹 is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.) |
Ref | Expression |
---|---|
itg2i1fseq.1 | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
itg2i1fseq.2 | ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) |
itg2i1fseq.3 | ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) |
itg2i1fseq.4 | ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) |
itg2i1fseq.5 | ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) |
itg2i1fseq.6 | ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) |
itg2i1fseq2.7 | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
itg2i1fseq2.8 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ 𝑀) |
Ref | Expression |
---|---|
itg2i1fseq2 | ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12269 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12001 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
3 | itg2i1fseq.3 | . . . . . 6 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) | |
4 | 3 | ffvelrnda 6843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫1) |
5 | itg1cl 24213 | . . . . 5 ⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (∫1‘(𝑃‘𝑚)) ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫1‘(𝑃‘𝑚)) ∈ ℝ) |
7 | itg2i1fseq.6 | . . . 4 ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) | |
8 | 6, 7 | fmptd 6870 | . . 3 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ) |
9 | 3 | ffvelrnda 6843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫1) |
10 | peano2nn 11638 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ) | |
11 | ffvelrn 6841 | . . . . . 6 ⊢ ((𝑃:ℕ⟶dom ∫1 ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫1) | |
12 | 3, 10, 11 | syl2an 595 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫1) |
13 | itg2i1fseq.4 | . . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) | |
14 | simpr 485 | . . . . . . . 8 ⊢ ((0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) → (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) | |
15 | 14 | ralimi 3157 | . . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) → ∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) |
16 | 13, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) |
17 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘)) | |
18 | fvoveq1 7168 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1))) | |
19 | 17, 18 | breq12d 5070 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)))) |
20 | 19 | rspccva 3619 | . . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1))) |
21 | 16, 20 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1))) |
22 | itg1le 24241 | . . . . 5 ⊢ (((𝑃‘𝑘) ∈ dom ∫1 ∧ (𝑃‘(𝑘 + 1)) ∈ dom ∫1 ∧ (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1))) → (∫1‘(𝑃‘𝑘)) ≤ (∫1‘(𝑃‘(𝑘 + 1)))) | |
23 | 9, 12, 21, 22 | syl3anc 1363 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ (∫1‘(𝑃‘(𝑘 + 1)))) |
24 | 2fveq3 6668 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (∫1‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑘))) | |
25 | fvex 6676 | . . . . . 6 ⊢ (∫1‘(𝑃‘𝑘)) ∈ V | |
26 | 24, 7, 25 | fvmpt 6761 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘𝑘) = (∫1‘(𝑃‘𝑘))) |
27 | 26 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = (∫1‘(𝑃‘𝑘))) |
28 | 2fveq3 6668 | . . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫1‘(𝑃‘𝑚)) = (∫1‘(𝑃‘(𝑘 + 1)))) | |
29 | fvex 6676 | . . . . . . 7 ⊢ (∫1‘(𝑃‘(𝑘 + 1))) ∈ V | |
30 | 28, 7, 29 | fvmpt 6761 | . . . . . 6 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑆‘(𝑘 + 1)) = (∫1‘(𝑃‘(𝑘 + 1)))) |
31 | 10, 30 | syl 17 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘(𝑘 + 1)) = (∫1‘(𝑃‘(𝑘 + 1)))) |
32 | 31 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘(𝑘 + 1)) = (∫1‘(𝑃‘(𝑘 + 1)))) |
33 | 23, 27, 32 | 3brtr4d 5089 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ (𝑆‘(𝑘 + 1))) |
34 | itg2i1fseq2.7 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
35 | itg2i1fseq2.8 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ 𝑀) | |
36 | 27, 35 | eqbrtrd 5079 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀) |
37 | 36 | ralrimiva 3179 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) |
38 | brralrspcev 5117 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧) | |
39 | 34, 37, 38 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧) |
40 | 1, 2, 8, 33, 39 | climsup 15014 | . 2 ⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ, < )) |
41 | itg2i1fseq.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
42 | itg2i1fseq.2 | . . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) | |
43 | itg2i1fseq.5 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) | |
44 | 41, 42, 3, 13, 43, 7 | itg2i1fseq 24283 | . . 3 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < )) |
45 | 8 | frnd 6514 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ) |
46 | 7, 6 | dmmptd 6486 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ) |
47 | 1nn 11637 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
48 | ne0i 4297 | . . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅) | |
49 | 47, 48 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅) |
50 | 46, 49 | eqnetrd 3080 | . . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅) |
51 | dm0rn0 5788 | . . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅) | |
52 | 51 | necon3bii 3065 | . . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅) |
53 | 50, 52 | sylib 219 | . . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅) |
54 | ffn 6507 | . . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ) | |
55 | breq1 5060 | . . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑘) → (𝑦 ≤ 𝑧 ↔ (𝑆‘𝑘) ≤ 𝑧)) | |
56 | 55 | ralrn 6846 | . . . . . . 7 ⊢ (𝑆 Fn ℕ → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)) |
57 | 8, 54, 56 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)) |
58 | 57 | rexbidv 3294 | . . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)) |
59 | 39, 58 | mpbird 258 | . . . 4 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧) |
60 | supxrre 12708 | . . . 4 ⊢ ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧) → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < )) | |
61 | 45, 53, 59, 60 | syl3anc 1363 | . . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < )) |
62 | 44, 61 | eqtrd 2853 | . 2 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ, < )) |
63 | 40, 62 | breqtrrd 5085 | 1 ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ran crn 5549 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘r cofr 7397 supcsup 8892 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 ℕcn 11626 [,)cico 12728 ⇝ cli 14829 MblFncmbf 24142 ∫1citg1 24143 ∫2citg2 24144 0𝑝c0p 24197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-rest 16684 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 df-cmp 21923 df-ovol 23992 df-vol 23993 df-mbf 24147 df-itg1 24148 df-itg2 24149 df-0p 24198 |
This theorem is referenced by: itg2i1fseq3 24285 itg2addlem 24286 |
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