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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 11190 | . . 3 β’ β β V | |
2 | 1 | elpw2 5338 | . 2 β’ (π β π« β β π β β) |
3 | fveq2 6884 | . . . . 5 β’ (π = π β (Polyβπ ) = (Polyβπ)) | |
4 | 3 | rexeqdv 3320 | . . . 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 3434 | . . 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 42461 | . . 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 5325 | . . 3 β’ {π₯ β β β£ βπ β (Polyβπ)((πβπ₯) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} β V |
8 | 5, 6, 7 | fvmpt 6991 | . 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 395 = wceq 1533 β wcel 2098 βwrex 3064 {crab 3426 β wss 3943 π« cpw 4597 βcfv 6536 βcc 11107 0cc0 11109 1c1 11110 Polycply 26068 coeffccoe 26070 degcdgr 26071 IntgOvercitgo 42459 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-cnex 11165 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-itgo 42461 |
This theorem is referenced by: aaitgo 42464 itgoss 42465 itgocn 42466 |
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