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Mirrors > Home > MPE Home > Th. List > i1f1 | Structured version Visualization version Unicode version |
Description: Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
i1f1.1 |
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Ref | Expression |
---|---|
i1f1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i1f1.1 |
. . . . . 6
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2 | 1 | i1f1lem 22695 |
. . . . 5
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3 | 2 | simpli 464 |
. . . 4
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4 | 0re 9668 |
. . . . 5
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5 | 1re 9667 |
. . . . 5
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6 | prssi 4140 |
. . . . 5
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7 | 4, 5, 6 | mp2an 683 |
. . . 4
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8 | fss 5759 |
. . . 4
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9 | 3, 7, 8 | mp2an 683 |
. . 3
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10 | 9 | a1i 11 |
. 2
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11 | prfi 7871 |
. . 3
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12 | 1ex 9663 |
. . . . . . . 8
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13 | 12 | prid2 4093 |
. . . . . . 7
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14 | c0ex 9662 |
. . . . . . . 8
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15 | 14 | prid1 4092 |
. . . . . . 7
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16 | 13, 15 | keepel 3959 |
. . . . . 6
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17 | 16 | a1i 11 |
. . . . 5
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18 | 17, 1 | fmptd 6068 |
. . . 4
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19 | frn 5757 |
. . . 4
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20 | 18, 19 | syl 17 |
. . 3
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21 | ssfi 7817 |
. . 3
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22 | 11, 20, 21 | sylancr 674 |
. 2
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23 | 3, 19 | ax-mp 5 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | df-pr 3982 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | equncomi 3591 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 23, 25 | sseqtri 3475 |
. . . . . . . . . 10
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27 | ssdif 3579 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 27 | ax-mp 5 |
. . . . . . . . 9
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29 | difun2 3858 |
. . . . . . . . . 10
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30 | difss 3571 |
. . . . . . . . . 10
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31 | 29, 30 | eqsstri 3473 |
. . . . . . . . 9
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32 | 28, 31 | sstri 3452 |
. . . . . . . 8
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33 | 32 | sseli 3439 |
. . . . . . 7
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34 | elsni 4004 |
. . . . . . 7
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35 | 33, 34 | syl 17 |
. . . . . 6
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36 | 35 | sneqd 3991 |
. . . . 5
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37 | 36 | imaeq2d 5186 |
. . . 4
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38 | 2 | simpri 468 |
. . . . 5
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39 | 38 | adantr 471 |
. . . 4
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40 | 37, 39 | sylan9eqr 2517 |
. . 3
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41 | simpll 765 |
. . 3
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42 | 40, 41 | eqeltrd 2539 |
. 2
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43 | 40 | fveq2d 5891 |
. . 3
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44 | simplr 767 |
. . 3
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45 | 43, 44 | eqeltrd 2539 |
. 2
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46 | 10, 22, 42, 45 | i1fd 22687 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-rep 4528 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-inf2 8171 ax-cnex 9620 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 ax-pre-mulgt0 9641 ax-pre-sup 9642 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-fal 1460 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rmo 2756 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-int 4248 df-iun 4293 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-se 4812 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-isom 5609 df-riota 6276 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-of 6557 df-om 6719 df-1st 6819 df-2nd 6820 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-1o 7207 df-2o 7208 df-oadd 7211 df-er 7388 df-map 7499 df-pm 7500 df-en 7595 df-dom 7596 df-sdom 7597 df-fin 7598 df-sup 7981 df-inf 7982 df-oi 8050 df-card 8398 df-cda 8623 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-sub 9887 df-neg 9888 df-div 10297 df-nn 10637 df-2 10695 df-3 10696 df-n0 10898 df-z 10966 df-uz 11188 df-q 11293 df-rp 11331 df-xadd 11438 df-ioo 11667 df-ico 11669 df-icc 11670 df-fz 11813 df-fzo 11946 df-fl 12059 df-seq 12245 df-exp 12304 df-hash 12547 df-cj 13210 df-re 13211 df-im 13212 df-sqrt 13346 df-abs 13347 df-clim 13600 df-sum 13801 df-xmet 19011 df-met 19012 df-ovol 22464 df-vol 22466 df-mbf 22625 df-itg1 22626 |
This theorem is referenced by: itg11 22697 itg2const 22746 itg2addnclem 32037 |
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