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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iblempty | Structured version Visualization version Unicode version |
Description: The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iblempty |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbf0 37834 |
. 2
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2 | fconstmpt 4878 |
. . . . . . 7
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3 | 2 | eqcomi 2460 |
. . . . . 6
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4 | 3 | fveq2i 5868 |
. . . . 5
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5 | itg20 22695 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 4, 5 | eqtri 2473 |
. . . 4
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7 | 0re 9643 |
. . . 4
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8 | 6, 7 | eqeltri 2525 |
. . 3
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9 | 8 | rgenw 2749 |
. 2
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10 | noel 3735 |
. . . . . . . . 9
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11 | 10 | intnanr 926 |
. . . . . . . 8
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12 | 11 | iffalsei 3891 |
. . . . . . 7
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13 | 12 | eqcomi 2460 |
. . . . . 6
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14 | 13 | a1i 11 |
. . . . 5
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15 | 14 | mpteq2dva 4489 |
. . . 4
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16 | eqidd 2452 |
. . . 4
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17 | dm0 5048 |
. . . . 5
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18 | 17 | a1i 11 |
. . . 4
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19 | 10 | intnan 925 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | pm2.21i 135 |
. . . 4
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21 | 15, 16, 18, 20 | isibl 22723 |
. . 3
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22 | 21 | trud 1453 |
. 2
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23 | 1, 9, 22 | mpbir2an 931 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-inf2 8146 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-pre-sup 9617 ax-addf 9618 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-fal 1450 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-disj 4374 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-se 4794 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-isom 5591 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-of 6531 df-ofr 6532 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-2o 7183 df-oadd 7186 df-er 7363 df-map 7474 df-pm 7475 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-sup 7956 df-inf 7957 df-oi 8025 df-card 8373 df-cda 8598 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-div 10270 df-nn 10610 df-2 10668 df-3 10669 df-n0 10870 df-z 10938 df-uz 11160 df-q 11265 df-rp 11303 df-xadd 11410 df-ioo 11639 df-ico 11641 df-icc 11642 df-fz 11785 df-fzo 11916 df-fl 12028 df-seq 12214 df-exp 12273 df-hash 12516 df-cj 13162 df-re 13163 df-im 13164 df-sqrt 13298 df-abs 13299 df-clim 13552 df-sum 13753 df-xmet 18963 df-met 18964 df-ovol 22416 df-vol 22418 df-mbf 22577 df-itg1 22578 df-itg2 22579 df-ibl 22580 df-0p 22628 |
This theorem is referenced by: itgvol0 37845 |
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