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Mirrors > Home > MPE Home > Th. List > itg0 | Structured version Unicode version |
Description: The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
itg0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2452 |
. . 3
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2 | 1 | dfitg 21375 |
. 2
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3 | ifan 3938 |
. . . . . . . . . . 11
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4 | noel 3744 |
. . . . . . . . . . . 12
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5 | 4 | iffalsei 3903 |
. . . . . . . . . . 11
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6 | 3, 5 | eqtri 2481 |
. . . . . . . . . 10
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7 | 6 | mpteq2i 4478 |
. . . . . . . . 9
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8 | fconstmpt 4985 |
. . . . . . . . 9
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9 | 7, 8 | eqtr4i 2484 |
. . . . . . . 8
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10 | 9 | fveq2i 5797 |
. . . . . . 7
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11 | itg20 21343 |
. . . . . . 7
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12 | 10, 11 | eqtri 2481 |
. . . . . 6
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13 | 12 | oveq2i 6206 |
. . . . 5
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14 | ax-icn 9447 |
. . . . . . 7
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15 | elfznn0 11593 |
. . . . . . 7
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16 | expcl 11995 |
. . . . . . 7
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17 | 14, 15, 16 | sylancr 663 |
. . . . . 6
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18 | 17 | mul01d 9674 |
. . . . 5
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19 | 13, 18 | syl5eq 2505 |
. . . 4
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20 | 19 | sumeq2i 13289 |
. . 3
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21 | fzfi 11906 |
. . . . 5
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22 | 21 | olci 391 |
. . . 4
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23 | sumz 13312 |
. . . 4
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24 | 22, 23 | ax-mp 5 |
. . 3
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25 | 20, 24 | eqtri 2481 |
. 2
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26 | 2, 25 | eqtri 2481 |
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 1954 ax-ext 2431 ax-rep 4506 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 ax-inf2 7953 ax-cnex 9444 ax-resscn 9445 ax-1cn 9446 ax-icn 9447 ax-addcl 9448 ax-addrcl 9449 ax-mulcl 9450 ax-mulrcl 9451 ax-mulcom 9452 ax-addass 9453 ax-mulass 9454 ax-distr 9455 ax-i2m1 9456 ax-1ne0 9457 ax-1rid 9458 ax-rnegex 9459 ax-rrecex 9460 ax-cnre 9461 ax-pre-lttri 9462 ax-pre-lttrn 9463 ax-pre-ltadd 9464 ax-pre-mulgt0 9465 ax-pre-sup 9466 ax-addf 9467 |
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 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-nel 2648 df-ral 2801 df-rex 2802 df-reu 2803 df-rmo 2804 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-pss 3447 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4195 df-int 4232 df-iun 4276 df-disj 4366 df-br 4396 df-opab 4454 df-mpt 4455 df-tr 4489 df-eprel 4735 df-id 4739 df-po 4744 df-so 4745 df-fr 4782 df-se 4783 df-we 4784 df-ord 4825 df-on 4826 df-lim 4827 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-isom 5530 df-riota 6156 df-ov 6198 df-oprab 6199 df-mpt2 6200 df-of 6425 df-ofr 6426 df-om 6582 df-1st 6682 df-2nd 6683 df-recs 6937 df-rdg 6971 df-1o 7025 df-2o 7026 df-oadd 7029 df-er 7206 df-map 7321 df-pm 7322 df-en 7416 df-dom 7417 df-sdom 7418 df-fin 7419 df-sup 7797 df-oi 7830 df-card 8215 df-cda 8443 df-pnf 9526 df-mnf 9527 df-xr 9528 df-ltxr 9529 df-le 9530 df-sub 9703 df-neg 9704 df-div 10100 df-nn 10429 df-2 10486 df-3 10487 df-n0 10686 df-z 10753 df-uz 10968 df-q 11060 df-rp 11098 df-xadd 11196 df-ioo 11410 df-ico 11412 df-icc 11413 df-fz 11550 df-fzo 11661 df-fl 11754 df-seq 11919 df-exp 11978 df-hash 12216 df-cj 12701 df-re 12702 df-im 12703 df-sqr 12837 df-abs 12838 df-clim 13079 df-sum 13277 df-xmet 17930 df-met 17931 df-ovol 21075 df-vol 21076 df-mbf 21227 df-itg1 21228 df-itg2 21229 df-itg 21231 df-0p 21276 |
This theorem is referenced by: itgsplitioo 21443 ditg0 21456 ditgneg 21460 ftc2 21644 ftc2nc 28619 areacirc 28632 |
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