![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sitmfval | Structured version Visualization version GIF version |
Description: Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
Ref | Expression |
---|---|
sitmval.d | β’ π· = (distβπ) |
sitmval.1 | β’ (π β π β π) |
sitmval.2 | β’ (π β π β βͺ ran measures) |
sitmfval.1 | β’ (π β πΉ β dom (πsitgπ)) |
sitmfval.2 | β’ (π β πΊ β dom (πsitgπ)) |
Ref | Expression |
---|---|
sitmfval | β’ (π β (πΉ(πsitmπ)πΊ) = (((β*π βΎs (0[,]+β))sitgπ)β(πΉ βf π·πΊ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sitmval.d | . . 3 β’ π· = (distβπ) | |
2 | sitmval.1 | . . 3 β’ (π β π β π) | |
3 | sitmval.2 | . . 3 β’ (π β π β βͺ ran measures) | |
4 | 1, 2, 3 | sitmval 33647 | . 2 β’ (π β (πsitmπ) = (π β dom (πsitgπ), π β dom (πsitgπ) β¦ (((β*π βΎs (0[,]+β))sitgπ)β(π βf π·π)))) |
5 | simprl 768 | . . . 4 β’ ((π β§ (π = πΉ β§ π = πΊ)) β π = πΉ) | |
6 | simprr 770 | . . . 4 β’ ((π β§ (π = πΉ β§ π = πΊ)) β π = πΊ) | |
7 | 5, 6 | oveq12d 7430 | . . 3 β’ ((π β§ (π = πΉ β§ π = πΊ)) β (π βf π·π) = (πΉ βf π·πΊ)) |
8 | 7 | fveq2d 6895 | . 2 β’ ((π β§ (π = πΉ β§ π = πΊ)) β (((β*π βΎs (0[,]+β))sitgπ)β(π βf π·π)) = (((β*π βΎs (0[,]+β))sitgπ)β(πΉ βf π·πΊ))) |
9 | sitmfval.1 | . 2 β’ (π β πΉ β dom (πsitgπ)) | |
10 | sitmfval.2 | . 2 β’ (π β πΊ β dom (πsitgπ)) | |
11 | fvexd 6906 | . 2 β’ (π β (((β*π βΎs (0[,]+β))sitgπ)β(πΉ βf π·πΊ)) β V) | |
12 | 4, 8, 9, 10, 11 | ovmpod 7563 | 1 β’ (π β (πΉ(πsitmπ)πΊ) = (((β*π βΎs (0[,]+β))sitgπ)β(πΉ βf π·πΊ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 Vcvv 3473 βͺ cuni 4908 dom cdm 5676 ran crn 5677 βcfv 6543 (class class class)co 7412 βf cof 7671 0cc0 11113 +βcpnf 11250 [,]cicc 13332 βΎs cress 17178 distcds 17211 β*π cxrs 17451 measurescmeas 33492 sitmcsitm 33626 sitgcsitg 33627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-1st 7978 df-2nd 7979 df-sitm 33629 |
This theorem is referenced by: sitmcl 33649 sitmf 33650 |
Copyright terms: Public domain | W3C validator |