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Mirrors > Home > MPE Home > Th. List > itgge0 | Structured version Unicode version |
Description: The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
itgge0.1 |
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itgge0.2 |
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itgge0.3 |
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Ref | Expression |
---|---|
itgge0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgz 21384 |
. 2
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2 | fconstmpt 4983 |
. . . 4
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3 | itgge0.1 |
. . . . . . 7
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4 | iblmbf 21371 |
. . . . . . 7
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5 | 3, 4 | syl 16 |
. . . . . 6
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6 | itgge0.2 |
. . . . . 6
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7 | 5, 6 | mbfdm2 21242 |
. . . . 5
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8 | ibl0 21390 |
. . . . 5
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9 | 7, 8 | syl 16 |
. . . 4
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10 | 2, 9 | syl5eqelr 2544 |
. . 3
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11 | 0red 9491 |
. . 3
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12 | itgge0.3 |
. . 3
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13 | 10, 3, 11, 6, 12 | itgle 21413 |
. 2
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14 | 1, 13 | syl5eqbrr 4427 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-inf2 7951 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 ax-pre-sup 9464 ax-addf 9465 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-disj 4364 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-se 4781 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-isom 5528 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-of 6423 df-ofr 6424 df-om 6580 df-1st 6680 df-2nd 6681 df-recs 6935 df-rdg 6969 df-1o 7023 df-2o 7024 df-oadd 7027 df-er 7204 df-map 7319 df-pm 7320 df-en 7414 df-dom 7415 df-sdom 7416 df-fin 7417 df-sup 7795 df-oi 7828 df-card 8213 df-cda 8441 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-div 10098 df-nn 10427 df-2 10484 df-3 10485 df-4 10486 df-n0 10684 df-z 10751 df-uz 10966 df-q 11058 df-rp 11096 df-xadd 11194 df-ioo 11408 df-ico 11410 df-icc 11411 df-fz 11548 df-fzo 11659 df-fl 11752 df-mod 11819 df-seq 11917 df-exp 11976 df-hash 12214 df-cj 12699 df-re 12700 df-im 12701 df-sqr 12835 df-abs 12836 df-clim 13077 df-sum 13275 df-xmet 17928 df-met 17929 df-ovol 21073 df-vol 21074 df-mbf 21225 df-itg1 21226 df-itg2 21227 df-ibl 21228 df-itg 21229 df-0p 21274 |
This theorem is referenced by: itgabs 21438 areaf 22481 itgabsnc 28602 |
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