Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > volun | Structured version Visualization version Unicode version |
Description: The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.) |
Ref | Expression |
---|---|
volun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1064 | . . . . . 6 | |
2 | mblss 23299 | . . . . . . . 8 | |
3 | 1, 2 | syl 17 | . . . . . . 7 |
4 | simpl2 1065 | . . . . . . . 8 | |
5 | mblss 23299 | . . . . . . . 8 | |
6 | 4, 5 | syl 17 | . . . . . . 7 |
7 | 3, 6 | unssd 3789 | . . . . . 6 |
8 | readdcl 10019 | . . . . . . . 8 | |
9 | 8 | adantl 482 | . . . . . . 7 |
10 | simprl 794 | . . . . . . . 8 | |
11 | simprr 796 | . . . . . . . 8 | |
12 | ovolun 23267 | . . . . . . . 8 | |
13 | 3, 10, 6, 11, 12 | syl22anc 1327 | . . . . . . 7 |
14 | ovollecl 23251 | . . . . . . 7 | |
15 | 7, 9, 13, 14 | syl3anc 1326 | . . . . . 6 |
16 | mblsplit 23300 | . . . . . 6 | |
17 | 1, 7, 15, 16 | syl3anc 1326 | . . . . 5 |
18 | simpl3 1066 | . . . . . 6 | |
19 | indir 3875 | . . . . . . . . . 10 | |
20 | inidm 3822 | . . . . . . . . . . . 12 | |
21 | incom 3805 | . . . . . . . . . . . 12 | |
22 | 20, 21 | uneq12i 3765 | . . . . . . . . . . 11 |
23 | unabs 3854 | . . . . . . . . . . 11 | |
24 | 22, 23 | eqtri 2644 | . . . . . . . . . 10 |
25 | 19, 24 | eqtri 2644 | . . . . . . . . 9 |
26 | 25 | a1i 11 | . . . . . . . 8 |
27 | 26 | fveq2d 6195 | . . . . . . 7 |
28 | 21 | eqeq1i 2627 | . . . . . . . . . . 11 |
29 | disj3 4021 | . . . . . . . . . . 11 | |
30 | 28, 29 | bitr3i 266 | . . . . . . . . . 10 |
31 | 30 | biimpi 206 | . . . . . . . . 9 |
32 | uncom 3757 | . . . . . . . . . . 11 | |
33 | 32 | difeq1i 3724 | . . . . . . . . . 10 |
34 | difun2 4048 | . . . . . . . . . 10 | |
35 | 33, 34 | eqtri 2644 | . . . . . . . . 9 |
36 | 31, 35 | syl6reqr 2675 | . . . . . . . 8 |
37 | 36 | fveq2d 6195 | . . . . . . 7 |
38 | 27, 37 | oveq12d 6668 | . . . . . 6 |
39 | 18, 38 | syl 17 | . . . . 5 |
40 | 17, 39 | eqtrd 2656 | . . . 4 |
41 | 40 | ex 450 | . . 3 |
42 | mblvol 23298 | . . . . . 6 | |
43 | 42 | eleq1d 2686 | . . . . 5 |
44 | mblvol 23298 | . . . . . 6 | |
45 | 44 | eleq1d 2686 | . . . . 5 |
46 | 43, 45 | bi2anan9 917 | . . . 4 |
47 | 46 | 3adant3 1081 | . . 3 |
48 | unmbl 23305 | . . . . . 6 | |
49 | mblvol 23298 | . . . . . 6 | |
50 | 48, 49 | syl 17 | . . . . 5 |
51 | 42, 44 | oveqan12d 6669 | . . . . 5 |
52 | 50, 51 | eqeq12d 2637 | . . . 4 |
53 | 52 | 3adant3 1081 | . . 3 |
54 | 41, 47, 53 | 3imtr4d 283 | . 2 |
55 | 54 | imp 445 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cdif 3571 cun 3572 cin 3573 wss 3574 c0 3915 class class class wbr 4653 cdm 5114 cfv 5888 (class class class)co 6650 cr 9935 caddc 9939 cle 10075 covol 23231 cvol 23232 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-iun 4522 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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 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-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-ovol 23233 df-vol 23234 |
This theorem is referenced by: volinun 23314 volfiniun 23315 volsup 23324 ovolioo 23336 ismblfin 33450 volioc 40188 volico 40200 |
Copyright terms: Public domain | W3C validator |