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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 25637 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
2 | ffvelcdm 7096 | . . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,]+∞)) | |
3 | 0xr 11299 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
4 | pnfxr 11306 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
5 | elicc1 13408 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐹‘𝑦) ∈ (0[,]+∞) ↔ ((𝐹‘𝑦) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ≤ +∞))) | |
6 | 3, 4, 5 | mp2an 690 | . . . . . . 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 3143 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦)) |
10 | 0re 11254 | . . . . . 6 ⊢ 0 ∈ ℝ | |
11 | fnconstg 6790 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) Fn ℝ) |
13 | ffn 6727 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ) | |
14 | reex 11237 | . . . . . 6 ⊢ ℝ ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V) |
16 | inidm 4221 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
17 | c0ex 11246 | . . . . . . 7 ⊢ 0 ∈ V | |
18 | 17 | fvconst2 7222 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((ℝ × {0})‘𝑦) = 0) |
19 | 18 | adantl 480 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → ((ℝ × {0})‘𝑦) = 0) |
20 | eqidd 2729 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦)) | |
21 | 12, 13, 15, 15, 16, 19, 20 | ofrfval 7701 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((ℝ × {0}) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
22 | 9, 21 | mpbird 256 | . . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) ∘r ≤ 𝐹) |
23 | i1f0 25636 | . . . 4 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
24 | itg2ub 25683 | . . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (ℝ × {0}) ∈ dom ∫1 ∧ (ℝ × {0}) ∘r ≤ 𝐹) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) | |
25 | 23, 24 | mp3an2 1445 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (ℝ × {0}) ∘r ≤ 𝐹) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) |
26 | 22, 25 | mpdan 685 | . 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫1‘(ℝ × {0})) ≤ (∫2‘𝐹)) |
27 | 1, 26 | eqbrtrrid 5188 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 {csn 4632 class class class wbr 5152 × cxp 5680 dom cdm 5682 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∘r cofr 7690 ℝcr 11145 0cc0 11146 +∞cpnf 11283 ℝ*cxr 11285 ≤ cle 11287 [,]cicc 13367 ∫1citg1 25564 ∫2citg2 25565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 df-itg2 25570 |
This theorem is referenced by: itg2lecl 25688 itg2const2 25691 itg2seq 25692 itg2monolem2 25701 itg2monolem3 25702 itg2gt0 25710 |
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