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Mathbox for Thierry Arnoux |
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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 32989 | . 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 770 | . . . 4 β’ ((π β§ (π = πΉ β§ π = πΊ)) β π = πΉ) | |
6 | simprr 772 | . . . 4 β’ ((π β§ (π = πΉ β§ π = πΊ)) β π = πΊ) | |
7 | 5, 6 | oveq12d 7380 | . . 3 β’ ((π β§ (π = πΉ β§ π = πΊ)) β (π βf π·π) = (πΉ βf π·πΊ)) |
8 | 7 | fveq2d 6851 | . 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 6862 | . 2 β’ (π β (((β*π βΎs (0[,]+β))sitgπ)β(πΉ βf π·πΊ)) β V) | |
12 | 4, 8, 9, 10, 11 | ovmpod 7512 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3448 βͺ cuni 4870 dom cdm 5638 ran crn 5639 βcfv 6501 (class class class)co 7362 βf cof 7620 0cc0 11058 +βcpnf 11193 [,]cicc 13274 βΎs cress 17119 distcds 17149 β*π cxrs 17389 measurescmeas 32834 sitmcsitm 32968 sitgcsitg 32969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-1st 7926 df-2nd 7927 df-sitm 32971 |
This theorem is referenced by: sitmcl 32991 sitmf 32992 |
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