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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 24036 | . . 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 24036 | . . 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 12635 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
10 | eldifi 3994 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
11 | ffvelrn 6674 | . . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,]+∞)) | |
12 | 1, 10, 11 | syl2an 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ∈ (0[,]+∞)) |
13 | 9, 12 | sseldi 3857 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ∈ ℝ*) |
14 | 13 | xrleidd 12362 | . . . 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 4955 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) |
17 | 1, 4, 7, 8, 16 | itg2lea 24048 | . 2 ⊢ (𝜑 → (∫2‘𝐹) ≤ (∫2‘𝐺)) |
18 | 15, 14 | eqbrtrrd 4953 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺‘𝑥) ≤ (𝐹‘𝑥)) |
19 | 4, 1, 7, 8, 18 | itg2lea 24048 | . 2 ⊢ (𝜑 → (∫2‘𝐺) ≤ (∫2‘𝐹)) |
20 | 3, 6, 17, 19 | xrletrid 12365 | 1 ⊢ (𝜑 → (∫2‘𝐹) = (∫2‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∖ cdif 3827 ⊆ wss 3830 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ℝcr 10334 0cc0 10335 +∞cpnf 10471 ℝ*cxr 10473 ≤ cle 10475 [,]cicc 12557 vol*covol 23766 ∫2citg2 23920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-disj 4898 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-ofr 7228 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fi 8670 df-sup 8701 df-inf 8702 df-oi 8769 df-dju 9124 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-ioo 12558 df-ico 12560 df-icc 12561 df-fz 12709 df-fzo 12850 df-fl 12977 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-sum 14904 df-rest 16552 df-topgen 16573 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-top 21206 df-topon 21223 df-bases 21258 df-cmp 21699 df-ovol 23768 df-vol 23769 df-mbf 23923 df-itg1 23924 df-itg2 23925 |
This theorem is referenced by: itgeqa 24117 |
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