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Mirrors > Home > MPE Home > Th. List > mblvol | Structured version Visualization version GIF version |
Description: The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.) |
Ref | Expression |
---|---|
mblvol | ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | volres 23103 | . . 3 ⊢ vol = (vol* ↾ dom vol) | |
2 | 1 | fveq1i 6104 | . 2 ⊢ (vol‘𝐴) = ((vol* ↾ dom vol)‘𝐴) |
3 | fvres 6117 | . 2 ⊢ (𝐴 ∈ dom vol → ((vol* ↾ dom vol)‘𝐴) = (vol*‘𝐴)) | |
4 | 2, 3 | syl5eq 2656 | 1 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 vol*covol 23038 volcvol 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fv 5812 df-vol 23041 |
This theorem is referenced by: volss 23108 volun 23120 volinun 23121 volfiniun 23122 voliunlem3 23127 volsup 23131 iccvolcl 23142 ovolioo 23143 ioovolcl 23144 uniioovol 23153 uniioombllem4 23160 volcn 23180 volivth 23181 vitalilem4 23186 i1fima2 23252 i1fd 23254 i1f0rn 23255 itg1val2 23257 itg1ge0 23259 itg11 23264 i1fadd 23268 i1fmul 23269 itg1addlem2 23270 itg1addlem4 23272 i1fres 23278 itg10a 23283 itg1ge0a 23284 itg1climres 23287 mbfi1fseqlem4 23291 itg2const2 23314 itg2gt0 23333 itg2cnlem2 23335 ftc1a 23604 ftc1lem4 23606 itgulm 23966 areaf 24488 cntnevol 29618 volmeas 29621 mblfinlem3 32618 mblfinlem4 32619 ismblfin 32620 voliunnfl 32623 volsupnfl 32624 itg2addnclem 32631 itg2addnclem2 32632 itg2gt0cn 32635 ftc1cnnclem 32653 ftc1anclem7 32661 areacirc 32675 arearect 36820 areaquad 36821 volioo 38840 vol0 38851 volge0 38853 volsn 38859 volicc 38891 vonvol 39552 |
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