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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 24292 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
2 | ffvelrn 6826 | . . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,]+∞)) | |
3 | 0xr 10677 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
4 | pnfxr 10684 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
5 | elicc1 12770 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐹‘𝑦) ∈ (0[,]+∞) ↔ ((𝐹‘𝑦) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ≤ +∞))) | |
6 | 3, 4, 5 | mp2an 691 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ (0[,]+∞) ↔ ((𝐹‘𝑦) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ≤ +∞)) |
7 | 6 | simp2bi 1143 | . . . . . 6 ⊢ ((𝐹‘𝑦) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝑦)) |
8 | 2, 7 | syl 17 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → 0 ≤ (𝐹‘𝑦)) |
9 | 8 | ralrimiva 3149 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦)) |
10 | 0re 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
11 | fnconstg 6541 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) Fn ℝ) |
13 | ffn 6487 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ) | |
14 | reex 10617 | . . . . . 6 ⊢ ℝ ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V) |
16 | inidm 4145 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
17 | c0ex 10624 | . . . . . . 7 ⊢ 0 ∈ V | |
18 | 17 | fvconst2 6943 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((ℝ × {0})‘𝑦) = 0) |
19 | 18 | adantl 485 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → ((ℝ × {0})‘𝑦) = 0) |
20 | eqidd 2799 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦)) | |
21 | 12, 13, 15, 15, 16, 19, 20 | ofrfval 7397 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((ℝ × {0}) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
22 | 9, 21 | mpbird 260 | . . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) ∘r ≤ 𝐹) |
23 | i1f0 24291 | . . . 4 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
24 | itg2ub 24337 | . . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (ℝ × {0}) ∈ dom ∫1 ∧ (ℝ × {0}) ∘r ≤ 𝐹) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) | |
25 | 23, 24 | mp3an2 1446 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (ℝ × {0}) ∘r ≤ 𝐹) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) |
26 | 22, 25 | mpdan 686 | . 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) |
27 | 1, 26 | eqbrtrrid 5066 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 {csn 4525 class class class wbr 5030 × cxp 5517 dom cdm 5519 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘r cofr 7388 ℝcr 10525 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 ≤ cle 10665 [,]cicc 12729 ∫1citg1 24219 ∫2citg2 24220 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 |
This theorem is referenced by: itg2lecl 24342 itg2const2 24345 itg2seq 24346 itg2monolem2 24355 itg2monolem3 24356 itg2gt0 24364 |
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