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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 11227 | . . 3 β’ β β V | |
2 | 1 | elpw2 5351 | . 2 β’ (π β π« β β π β β) |
3 | fveq2 6902 | . . . . 5 β’ (π = π β (Polyβπ ) = (Polyβπ)) | |
4 | 3 | rexeqdv 3324 | . . . 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 3438 | . . 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 42614 | . . 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 5338 | . . 3 β’ {π₯ β β β£ βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} β V |
8 | 5, 6, 7 | fvmpt 7010 | . 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 394 = wceq 1533 β wcel 2098 βwrex 3067 {crab 3430 β wss 3949 π« cpw 4606 βcfv 6553 βcc 11144 0cc0 11146 1c1 11147 Polycply 26138 coeffccoe 26140 degcdgr 26141 IntgOvercitgo 42612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-cnex 11202 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-itgo 42614 |
This theorem is referenced by: aaitgo 42617 itgoss 42618 itgocn 42619 |
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