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Mirrors > Home > MPE Home > Th. List > itg2i1fseq3 | Structured version Visualization version GIF version |
Description: Special case of itg2i1fseq2 25144: if the integral of 𝐹 is a real number, then the standard limit relation holds on the integrals of simple functions approaching 𝐹. (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))) |
itg2i1fseq3.7 | ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) |
Ref | Expression |
---|---|
itg2i1fseq3 | ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg2i1fseq.1 | . 2 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
2 | itg2i1fseq.2 | . 2 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) | |
3 | itg2i1fseq.3 | . 2 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) | |
4 | itg2i1fseq.4 | . 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) | |
5 | itg2i1fseq.5 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) | |
6 | itg2i1fseq.6 | . 2 ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) | |
7 | itg2i1fseq3.7 | . 2 ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) | |
8 | icossicc 13362 | . . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
9 | fss 6689 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞)) | |
10 | 2, 8, 9 | sylancl 587 | . . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) |
11 | 10 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℝ⟶(0[,]+∞)) |
12 | 3 | ffvelcdmda 7039 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫1) |
13 | 1, 2, 3, 4, 5 | itg2i1fseqle 25142 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘r ≤ 𝐹) |
14 | itg2ub 25121 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑃‘𝑘) ∈ dom ∫1 ∧ (𝑃‘𝑘) ∘r ≤ 𝐹) → (∫1‘(𝑃‘𝑘)) ≤ (∫2‘𝐹)) | |
15 | 11, 12, 13, 14 | syl3anc 1372 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ (∫2‘𝐹)) |
16 | 1, 2, 3, 4, 5, 6, 7, 15 | itg2i1fseq2 25144 | 1 ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ⊆ wss 3914 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5637 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ∘r cofr 7620 ℝcr 11058 0cc0 11059 1c1 11060 + caddc 11062 +∞cpnf 11194 ≤ cle 11198 ℕcn 12161 [,)cico 13275 [,]cicc 13276 ⇝ cli 15375 MblFncmbf 25001 ∫1citg1 25002 ∫2citg2 25003 0𝑝c0p 25056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cc 10379 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-disj 5075 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-ofr 7622 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-oadd 8420 df-omul 8421 df-er 8654 df-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-dju 9845 df-card 9883 df-acn 9886 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-rlim 15380 df-sum 15580 df-rest 17312 df-topgen 17333 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-bases 22319 df-cmp 22761 df-ovol 24851 df-vol 24852 df-mbf 25006 df-itg1 25007 df-itg2 25008 df-0p 25057 |
This theorem is referenced by: itg2addlem 25146 |
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