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Mirrors > Home > MPE Home > Th. List > itg2ge0 | Structured version Visualization version GIF version |
Description: The integral of a nonnegative real function is greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg2ge0 | ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg10 23892 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
2 | ffvelrn 6621 | . . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,]+∞)) | |
3 | 0xr 10423 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
4 | pnfxr 10430 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
5 | elicc1 12531 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐹‘𝑦) ∈ (0[,]+∞) ↔ ((𝐹‘𝑦) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ≤ +∞))) | |
6 | 3, 4, 5 | mp2an 682 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ (0[,]+∞) ↔ ((𝐹‘𝑦) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ≤ +∞)) |
7 | 6 | simp2bi 1137 | . . . . . 6 ⊢ ((𝐹‘𝑦) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝑦)) |
8 | 2, 7 | syl 17 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → 0 ≤ (𝐹‘𝑦)) |
9 | 8 | ralrimiva 3147 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦)) |
10 | 0re 10378 | . . . . . 6 ⊢ 0 ∈ ℝ | |
11 | fnconstg 6343 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) Fn ℝ) |
13 | ffn 6291 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ) | |
14 | reex 10363 | . . . . . 6 ⊢ ℝ ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V) |
16 | inidm 4042 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
17 | c0ex 10370 | . . . . . . 7 ⊢ 0 ∈ V | |
18 | 17 | fvconst2 6741 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((ℝ × {0})‘𝑦) = 0) |
19 | 18 | adantl 475 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → ((ℝ × {0})‘𝑦) = 0) |
20 | eqidd 2778 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦)) | |
21 | 12, 13, 15, 15, 16, 19, 20 | ofrfval 7182 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((ℝ × {0}) ∘𝑟 ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
22 | 9, 21 | mpbird 249 | . . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) ∘𝑟 ≤ 𝐹) |
23 | i1f0 23891 | . . . 4 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
24 | itg2ub 23937 | . . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (ℝ × {0}) ∈ dom ∫1 ∧ (ℝ × {0}) ∘𝑟 ≤ 𝐹) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) | |
25 | 23, 24 | mp3an2 1522 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (ℝ × {0}) ∘𝑟 ≤ 𝐹) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) |
26 | 22, 25 | mpdan 677 | . 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) |
27 | 1, 26 | syl5eqbrr 4922 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ∀wral 3089 Vcvv 3397 {csn 4397 class class class wbr 4886 × cxp 5353 dom cdm 5355 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ∘𝑟 cofr 7173 ℝcr 10271 0cc0 10272 +∞cpnf 10408 ℝ*cxr 10410 ≤ cle 10412 [,]cicc 12490 ∫1citg1 23819 ∫2citg2 23820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xadd 12258 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-xmet 20135 df-met 20136 df-ovol 23668 df-vol 23669 df-mbf 23823 df-itg1 23824 df-itg2 23825 |
This theorem is referenced by: itg2lecl 23942 itg2const2 23945 itg2seq 23946 itg2monolem2 23955 itg2monolem3 23956 itg2gt0 23964 |
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