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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 25805, 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 12919 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12646 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
3 | itg2i1fseq.3 | . . . . . 6 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) | |
4 | 3 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫1) |
5 | itg1cl 25734 | . . . . 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 7134 | . . 3 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ) |
9 | 3 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫1) |
10 | peano2nn 12276 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ) | |
11 | ffvelcdm 7101 | . . . . . 6 ⊢ ((𝑃:ℕ⟶dom ∫1 ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫1) | |
12 | 3, 10, 11 | syl2an 596 | . . . . 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 484 | . . . . . . . 8 ⊢ ((0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) → (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) | |
15 | 14 | ralimi 3081 | . . . . . . 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 6907 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘)) | |
18 | fvoveq1 7454 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1))) | |
19 | 17, 18 | breq12d 5161 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)))) |
20 | 19 | rspccva 3621 | . . . . . 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 25763 | . . . . 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 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ (∫1‘(𝑃‘(𝑘 + 1)))) |
24 | 2fveq3 6912 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (∫1‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑘))) | |
25 | fvex 6920 | . . . . . 6 ⊢ (∫1‘(𝑃‘𝑘)) ∈ V | |
26 | 24, 7, 25 | fvmpt 7016 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘𝑘) = (∫1‘(𝑃‘𝑘))) |
27 | 26 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = (∫1‘(𝑃‘𝑘))) |
28 | 2fveq3 6912 | . . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫1‘(𝑃‘𝑚)) = (∫1‘(𝑃‘(𝑘 + 1)))) | |
29 | fvex 6920 | . . . . . . 7 ⊢ (∫1‘(𝑃‘(𝑘 + 1))) ∈ V | |
30 | 28, 7, 29 | fvmpt 7016 | . . . . . 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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘(𝑘 + 1)) = (∫1‘(𝑃‘(𝑘 + 1)))) |
33 | 23, 27, 32 | 3brtr4d 5180 | . . 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 5170 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀) |
37 | 36 | ralrimiva 3144 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) |
38 | brralrspcev 5208 | . . . 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 15703 | . 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 25805 | . . 3 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < )) |
45 | 8 | frnd 6745 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ) |
46 | 7, 6 | dmmptd 6714 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ) |
47 | 1nn 12275 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
48 | ne0i 4347 | . . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅) | |
49 | 47, 48 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅) |
50 | 46, 49 | eqnetrd 3006 | . . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅) |
51 | dm0rn0 5938 | . . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅) | |
52 | 51 | necon3bii 2991 | . . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅) |
53 | 50, 52 | sylib 218 | . . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅) |
54 | ffn 6737 | . . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ) | |
55 | breq1 5151 | . . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑘) → (𝑦 ≤ 𝑧 ↔ (𝑆‘𝑘) ≤ 𝑧)) | |
56 | 55 | ralrn 7108 | . . . . . . 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 3177 | . . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)) |
59 | 39, 58 | mpbird 257 | . . . 4 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧) |
60 | supxrre 13366 | . . . 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 1370 | . . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < )) |
62 | 44, 61 | eqtrd 2775 | . 2 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ, < )) |
63 | 40, 62 | breqtrrd 5176 | 1 ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘r cofr 7696 supcsup 9478 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 ℕcn 12264 [,)cico 13386 ⇝ cli 15517 MblFncmbf 25663 ∫1citg1 25664 ∫2citg2 25665 0𝑝c0p 25718 |
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-cc 10473 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 ax-addf 11232 |
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-disj 5116 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-of 7697 df-ofr 7698 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-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 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-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 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 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cmp 23411 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-0p 25719 |
This theorem is referenced by: itg2i1fseq3 25807 itg2addlem 25808 |
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