Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omef | Structured version Visualization version Unicode version |
Description: An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
omef.o | OutMeas |
omef.x |
Ref | Expression |
---|---|
omef |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omef.o | . . . . 5 OutMeas | |
2 | isome 40708 | . . . . . 6 OutMeas OutMeas Σ^ | |
3 | 1, 2 | syl 17 | . . . . 5 OutMeas Σ^ |
4 | 1, 3 | mpbid 222 | . . . 4 Σ^ |
5 | 4 | simplld 791 | . . 3 |
6 | 5 | simplld 791 | . 2 |
7 | simp-4r 807 | . . . . 5 Σ^ | |
8 | 4, 7 | syl 17 | . . . 4 |
9 | omef.x | . . . . 5 | |
10 | 9 | pweqi 4162 | . . . 4 |
11 | 8, 10 | syl6reqr 2675 | . . 3 |
12 | 11 | feq2d 6031 | . 2 |
13 | 6, 12 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 c0 3915 cpw 4158 cuni 4436 class class class wbr 4653 cdm 5114 cres 5116 wf 5884 cfv 5888 (class class class)co 6650 com 7065 cdom 7953 cc0 9936 cpnf 10071 cle 10075 cicc 12178 Σ^csumge0 40579 OutMeascome 40703 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ome 40704 |
This theorem is referenced by: omecl 40717 omeunle 40730 omeiunle 40731 caratheodory 40742 |
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