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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 25610 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
2 | ffvelcdm 7085 | . . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,]+∞)) | |
3 | 0xr 11285 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
4 | pnfxr 11292 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
5 | elicc1 13394 | . . . . . . . 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 1144 | . . . . . 6 ⊢ ((𝐹‘𝑦) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝑦)) |
8 | 2, 7 | syl 17 | . . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑦 ∈ ℝ) → 0 ≤ (𝐹‘𝑦)) |
9 | 8 | ralrimiva 3142 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦)) |
10 | 0re 11240 | . . . . . 6 ⊢ 0 ∈ ℝ | |
11 | fnconstg 6779 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (ℝ × {0}) Fn ℝ) |
13 | ffn 6716 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ) | |
14 | reex 11223 | . . . . . 6 ⊢ ℝ ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V) |
16 | inidm 4214 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
17 | c0ex 11232 | . . . . . . 7 ⊢ 0 ∈ V | |
18 | 17 | fvconst2 7210 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((ℝ × {0})‘𝑦) = 0) |
19 | 18 | adantl 481 | . . . . 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 7689 | . . . 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 25609 | . . . 4 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
24 | itg2ub 25656 | . . . 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 5178 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 {csn 4624 class class class wbr 5142 × cxp 5670 dom cdm 5672 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∘r cofr 7678 ℝcr 11131 0cc0 11132 +∞cpnf 11269 ℝ*cxr 11271 ≤ cle 11273 [,]cicc 13353 ∫1citg1 25537 ∫2citg2 25538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-oi 9527 df-dju 9918 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-xadd 13119 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 df-xmet 21265 df-met 21266 df-ovol 25386 df-vol 25387 df-mbf 25541 df-itg1 25542 df-itg2 25543 |
This theorem is referenced by: itg2lecl 25661 itg2const2 25664 itg2seq 25665 itg2monolem2 25674 itg2monolem3 25675 itg2gt0 25683 |
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