| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > itg2i1fseq2 | Structured version Visualization version GIF version | ||
| Description: In an extension to the results of itg2i1fseq 25790, 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 12921 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
| 2 | 1zzd 12648 | . . 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 25720 | . . . . 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 12278 | . . . . . 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 3083 | . . . . . . 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 6906 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘)) | |
| 18 | fvoveq1 7454 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1))) | |
| 19 | 17, 18 | breq12d 5156 | . . . . . . 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 25748 | . . . . 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 1373 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ (∫1‘(𝑃‘(𝑘 + 1)))) |
| 24 | 2fveq3 6911 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (∫1‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑘))) | |
| 25 | fvex 6919 | . . . . . 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 6911 | . . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫1‘(𝑃‘𝑚)) = (∫1‘(𝑃‘(𝑘 + 1)))) | |
| 29 | fvex 6919 | . . . . . . 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 5175 | . . 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 5165 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀) |
| 37 | 36 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) |
| 38 | brralrspcev 5203 | . . . 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 15706 | . 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 25790 | . . 3 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < )) |
| 45 | 8 | frnd 6744 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ) |
| 46 | 7, 6 | dmmptd 6713 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ) |
| 47 | 1nn 12277 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
| 48 | ne0i 4341 | . . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅) | |
| 49 | 47, 48 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅) |
| 50 | 46, 49 | eqnetrd 3008 | . . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅) |
| 51 | dm0rn0 5935 | . . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅) | |
| 52 | 51 | necon3bii 2993 | . . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅) |
| 53 | 50, 52 | sylib 218 | . . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅) |
| 54 | ffn 6736 | . . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ) | |
| 55 | breq1 5146 | . . . . . . . 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 3179 | . . . . 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 13369 | . . . 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 1373 | . . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < )) |
| 62 | 44, 61 | eqtrd 2777 | . 2 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ, < )) |
| 63 | 40, 62 | breqtrrd 5171 | 1 ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘r cofr 7696 supcsup 9480 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 [,)cico 13389 ⇝ cli 15520 MblFncmbf 25649 ∫1citg1 25650 ∫2citg2 25651 0𝑝c0p 25704 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-0p 25705 |
| This theorem is referenced by: itg2i1fseq3 25792 itg2addlem 25793 |
| Copyright terms: Public domain | W3C validator |