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Mirrors > Home > MPE Home > Th. List > itgge0 | Structured version Visualization version GIF version |
Description: The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
itgge0.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
itgge0.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
itgge0.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
itgge0 | ⊢ (𝜑 → 0 ≤ ∫𝐴𝐵 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgz 24632 | . 2 ⊢ ∫𝐴0 d𝑥 = 0 | |
2 | fconstmpt 5596 | . . . 4 ⊢ (𝐴 × {0}) = (𝑥 ∈ 𝐴 ↦ 0) | |
3 | itgge0.1 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
4 | iblmbf 24619 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
6 | itgge0.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
7 | 5, 6 | mbfdm2 24488 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom vol) |
8 | ibl0 24638 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 × {0}) ∈ 𝐿1) |
10 | 2, 9 | eqeltrrid 2836 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝐿1) |
11 | 0red 10801 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ) | |
12 | itgge0.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) | |
13 | 10, 3, 11, 6, 12 | itgle 24661 | . 2 ⊢ (𝜑 → ∫𝐴0 d𝑥 ≤ ∫𝐴𝐵 d𝑥) |
14 | 1, 13 | eqbrtrrid 5075 | 1 ⊢ (𝜑 → 0 ≤ ∫𝐴𝐵 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 {csn 4527 class class class wbr 5039 ↦ cmpt 5120 × cxp 5534 dom cdm 5536 ℝcr 10693 0cc0 10694 ≤ cle 10833 volcvol 24314 MblFncmbf 24465 𝐿1cibl 24468 ∫citg 24469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xadd 12670 df-ioo 12904 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 df-xmet 20310 df-met 20311 df-ovol 24315 df-vol 24316 df-mbf 24470 df-itg1 24471 df-itg2 24472 df-ibl 24473 df-itg 24474 df-0p 24521 |
This theorem is referenced by: itgabs 24686 areaf 25798 fdvposle 32247 itgabsnc 35532 fourierdlem47 43312 |
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