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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 24078 | . 2 ⊢ ∫𝐴0 d𝑥 = 0 | |
2 | fconstmpt 5458 | . . . 4 ⊢ (𝐴 × {0}) = (𝑥 ∈ 𝐴 ↦ 0) | |
3 | itgge0.1 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
4 | iblmbf 24065 | . . . . . . 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 23935 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom vol) |
8 | ibl0 24084 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 × {0}) ∈ 𝐿1) |
10 | 2, 9 | syl5eqelr 2865 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝐿1) |
11 | 0red 10437 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ) | |
12 | itgge0.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) | |
13 | 10, 3, 11, 6, 12 | itgle 24107 | . 2 ⊢ (𝜑 → ∫𝐴0 d𝑥 ≤ ∫𝐴𝐵 d𝑥) |
14 | 1, 13 | syl5eqbrr 4959 | 1 ⊢ (𝜑 → 0 ≤ ∫𝐴𝐵 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2050 {csn 4435 class class class wbr 4923 ↦ cmpt 5002 × cxp 5399 dom cdm 5401 ℝcr 10328 0cc0 10329 ≤ cle 10469 volcvol 23761 MblFncmbf 23912 𝐿1cibl 23915 ∫citg 23916 |
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 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-addf 10408 |
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 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-disj 4892 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-ofr 7222 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-sup 8695 df-inf 8696 df-oi 8763 df-dju 9118 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-n0 11702 df-z 11788 df-uz 12053 df-q 12157 df-rp 12199 df-xadd 12319 df-ioo 12552 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-mod 13047 df-seq 13179 df-exp 13239 df-hash 13500 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-clim 14700 df-sum 14898 df-xmet 20234 df-met 20235 df-ovol 23762 df-vol 23763 df-mbf 23917 df-itg1 23918 df-itg2 23919 df-ibl 23920 df-itg 23921 df-0p 23968 |
This theorem is referenced by: itgabs 24132 areaf 25235 fdvposle 31520 itgabsnc 34402 fourierdlem47 41869 |
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