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Mirrors > Home > MPE Home > Th. List > itg2eqa | Structured version Visualization version GIF version |
Description: Approximate equality of integrals. If 𝐹 = 𝐺 for almost all 𝑥, then ∫2𝐹 = ∫2𝐺. (Contributed by Mario Carneiro, 12-Aug-2014.) |
Ref | Expression |
---|---|
itg2lea.1 | ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) |
itg2lea.2 | ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) |
itg2lea.3 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
itg2lea.4 | ⊢ (𝜑 → (vol*‘𝐴) = 0) |
itg2eqa.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
Ref | Expression |
---|---|
itg2eqa | ⊢ (𝜑 → (∫2‘𝐹) = (∫2‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg2lea.1 | . . 3 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) | |
2 | itg2cl 25241 | . . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) ∈ ℝ*) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ*) |
4 | itg2lea.2 | . . 3 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) | |
5 | itg2cl 25241 | . . 3 ⊢ (𝐺:ℝ⟶(0[,]+∞) → (∫2‘𝐺) ∈ ℝ*) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ*) |
7 | itg2lea.3 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
8 | itg2lea.4 | . . 3 ⊢ (𝜑 → (vol*‘𝐴) = 0) | |
9 | iccssxr 13403 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
10 | eldifi 4125 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
11 | ffvelcdm 7080 | . . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,]+∞)) | |
12 | 1, 10, 11 | syl2an 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ∈ (0[,]+∞)) |
13 | 9, 12 | sselid 3979 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ∈ ℝ*) |
14 | 13 | xrleidd 13127 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐹‘𝑥)) |
15 | itg2eqa.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = (𝐺‘𝑥)) | |
16 | 14, 15 | breqtrd 5173 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) |
17 | 1, 4, 7, 8, 16 | itg2lea 25253 | . 2 ⊢ (𝜑 → (∫2‘𝐹) ≤ (∫2‘𝐺)) |
18 | 15, 14 | eqbrtrrd 5171 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺‘𝑥) ≤ (𝐹‘𝑥)) |
19 | 4, 1, 7, 8, 18 | itg2lea 25253 | . 2 ⊢ (𝜑 → (∫2‘𝐺) ≤ (∫2‘𝐹)) |
20 | 3, 6, 17, 19 | xrletrid 13130 | 1 ⊢ (𝜑 → (∫2‘𝐹) = (∫2‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3944 ⊆ wss 3947 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 +∞cpnf 11241 ℝ*cxr 11243 ≤ cle 11245 [,]cicc 13323 vol*covol 24970 ∫2citg2 25124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-rest 17364 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-bases 22440 df-cmp 22882 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 |
This theorem is referenced by: itgeqa 25322 |
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