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Mirrors > Home > MPE Home > Th. List > iblneg | Structured version Visualization version Unicode version |
Description: The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
itgcnval.1 | |
itgcnval.2 |
Ref | Expression |
---|---|
iblneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgcnval.2 | . . . . . . . . . 10 | |
2 | iblmbf 23534 | . . . . . . . . . 10 MblFn | |
3 | 1, 2 | syl 17 | . . . . . . . . 9 MblFn |
4 | itgcnval.1 | . . . . . . . . 9 | |
5 | 3, 4 | mbfmptcl 23404 | . . . . . . . 8 |
6 | 5 | renegd 13949 | . . . . . . 7 |
7 | 6 | breq2d 4665 | . . . . . 6 |
8 | 7, 6 | ifbieq1d 4109 | . . . . 5 |
9 | 8 | mpteq2dva 4744 | . . . 4 |
10 | 5 | iblcn 23565 | . . . . . . . 8 |
11 | 1, 10 | mpbid 222 | . . . . . . 7 |
12 | 11 | simpld 475 | . . . . . 6 |
13 | 5 | recld 13934 | . . . . . . 7 |
14 | 13 | iblre 23560 | . . . . . 6 |
15 | 12, 14 | mpbid 222 | . . . . 5 |
16 | 15 | simprd 479 | . . . 4 |
17 | 9, 16 | eqeltrd 2701 | . . 3 |
18 | 6 | negeqd 10275 | . . . . . . . 8 |
19 | 13 | recnd 10068 | . . . . . . . . 9 |
20 | 19 | negnegd 10383 | . . . . . . . 8 |
21 | 18, 20 | eqtrd 2656 | . . . . . . 7 |
22 | 21 | breq2d 4665 | . . . . . 6 |
23 | 22, 21 | ifbieq1d 4109 | . . . . 5 |
24 | 23 | mpteq2dva 4744 | . . . 4 |
25 | 15 | simpld 475 | . . . 4 |
26 | 24, 25 | eqeltrd 2701 | . . 3 |
27 | 5 | negcld 10379 | . . . . 5 |
28 | 27 | recld 13934 | . . . 4 |
29 | 28 | iblre 23560 | . . 3 |
30 | 17, 26, 29 | mpbir2and 957 | . 2 |
31 | 5 | imnegd 13950 | . . . . . . 7 |
32 | 31 | breq2d 4665 | . . . . . 6 |
33 | 32, 31 | ifbieq1d 4109 | . . . . 5 |
34 | 33 | mpteq2dva 4744 | . . . 4 |
35 | 11 | simprd 479 | . . . . . 6 |
36 | 5 | imcld 13935 | . . . . . . 7 |
37 | 36 | iblre 23560 | . . . . . 6 |
38 | 35, 37 | mpbid 222 | . . . . 5 |
39 | 38 | simprd 479 | . . . 4 |
40 | 34, 39 | eqeltrd 2701 | . . 3 |
41 | 31 | negeqd 10275 | . . . . . . . 8 |
42 | 36 | recnd 10068 | . . . . . . . . 9 |
43 | 42 | negnegd 10383 | . . . . . . . 8 |
44 | 41, 43 | eqtrd 2656 | . . . . . . 7 |
45 | 44 | breq2d 4665 | . . . . . 6 |
46 | 45, 44 | ifbieq1d 4109 | . . . . 5 |
47 | 46 | mpteq2dva 4744 | . . . 4 |
48 | 38 | simpld 475 | . . . 4 |
49 | 47, 48 | eqeltrd 2701 | . . 3 |
50 | 27 | imcld 13935 | . . . 4 |
51 | 50 | iblre 23560 | . . 3 |
52 | 40, 49, 51 | mpbir2and 957 | . 2 |
53 | 27 | iblcn 23565 | . 2 |
54 | 30, 52, 53 | mpbir2and 957 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 cif 4086 class class class wbr 4653 cmpt 4729 cfv 5888 cc0 9936 cle 10075 cneg 10267 cre 13837 cim 13838 MblFncmbf 23383 cibl 23386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-0p 23437 |
This theorem is referenced by: itgneg 23570 iblsub 23588 itgsub 23592 iblsubnc 33471 itgsubnc 33472 |
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