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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 25068 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
2 | ffvelcdm 7037 | . . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,]+∞)) | |
3 | 0xr 11209 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
4 | pnfxr 11216 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
5 | elicc1 13315 | . . . . . . . 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 1147 | . . . . . 6 ⊢ ((𝐹‘𝑦) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝑦)) |
8 | 2, 7 | syl 17 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → 0 ≤ (𝐹‘𝑦)) |
9 | 8 | ralrimiva 3144 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦)) |
10 | 0re 11164 | . . . . . 6 ⊢ 0 ∈ ℝ | |
11 | fnconstg 6735 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) Fn ℝ) |
13 | ffn 6673 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ) | |
14 | reex 11149 | . . . . . 6 ⊢ ℝ ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V) |
16 | inidm 4183 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
17 | c0ex 11156 | . . . . . . 7 ⊢ 0 ∈ V | |
18 | 17 | fvconst2 7158 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((ℝ × {0})‘𝑦) = 0) |
19 | 18 | adantl 483 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → ((ℝ × {0})‘𝑦) = 0) |
20 | eqidd 2738 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦)) | |
21 | 12, 13, 15, 15, 16, 19, 20 | ofrfval 7632 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((ℝ × {0}) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
22 | 9, 21 | mpbird 257 | . . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) ∘r ≤ 𝐹) |
23 | i1f0 25067 | . . . 4 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
24 | itg2ub 25114 | . . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (ℝ × {0}) ∈ dom ∫1 ∧ (ℝ × {0}) ∘r ≤ 𝐹) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) | |
25 | 23, 24 | mp3an2 1450 | . . 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 5146 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3065 Vcvv 3448 {csn 4591 class class class wbr 5110 × cxp 5636 dom cdm 5638 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∘r cofr 7621 ℝcr 11057 0cc0 11058 +∞cpnf 11193 ℝ*cxr 11195 ≤ cle 11197 [,]cicc 13274 ∫1citg1 24995 ∫2citg2 24996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-ofr 7623 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 df-mbf 24999 df-itg1 25000 df-itg2 25001 |
This theorem is referenced by: itg2lecl 25119 itg2const2 25122 itg2seq 25123 itg2monolem2 25132 itg2monolem3 25133 itg2gt0 25141 |
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