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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > itgoval | Structured version Visualization version GIF version |
Description: Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
itgoval | β’ (π β β β (IntgOverβπ) = {π₯ β β β£ βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 11139 | . . 3 β’ β β V | |
2 | 1 | elpw2 5307 | . 2 β’ (π β π« β β π β β) |
3 | fveq2 6847 | . . . . 5 β’ (π = π β (Polyβπ ) = (Polyβπ)) | |
4 | 3 | rexeqdv 3317 | . . . 4 β’ (π = π β (βπ β (Polyβπ )((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1) β βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1))) |
5 | 4 | rabbidv 3418 | . . 3 β’ (π = π β {π₯ β β β£ βπ β (Polyβπ )((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} = {π₯ β β β£ βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) |
6 | df-itgo 41515 | . . 3 β’ IntgOver = (π β π« β β¦ {π₯ β β β£ βπ β (Polyβπ )((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
7 | 1 | rabex 5294 | . . 3 β’ {π₯ β β β£ βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} β V |
8 | 5, 6, 7 | fvmpt 6953 | . 2 β’ (π β π« β β (IntgOverβπ) = {π₯ β β β£ βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) |
9 | 2, 8 | sylbir 234 | 1 β’ (π β β β (IntgOverβπ) = {π₯ β β β£ βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3074 {crab 3410 β wss 3915 π« cpw 4565 βcfv 6501 βcc 11056 0cc0 11058 1c1 11059 Polycply 25561 coeffccoe 25563 degcdgr 25564 IntgOvercitgo 41513 |
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-sep 5261 ax-nul 5268 ax-pr 5389 ax-cnex 11114 |
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-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 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-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-iota 6453 df-fun 6503 df-fv 6509 df-itgo 41515 |
This theorem is referenced by: aaitgo 41518 itgoss 41519 itgocn 41520 |
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