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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > itgoss | Structured version Visualization version GIF version |
Description: An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
itgoss | β’ ((π β π β§ π β β) β (IntgOverβπ) β (IntgOverβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyss 25576 | . . . . 5 β’ ((π β π β§ π β β) β (Polyβπ) β (Polyβπ)) | |
2 | ssrexv 4012 | . . . . 5 β’ ((Polyβπ) β (Polyβπ) β (βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1) β βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1))) | |
3 | 1, 2 | syl 17 | . . . 4 β’ ((π β π β§ π β β) β (βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1) β βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1))) |
4 | 3 | adantr 482 | . . 3 β’ (((π β π β§ π β β) β§ π β β) β (βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1) β βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1))) |
5 | 4 | ss2rabdv 4034 | . 2 β’ ((π β π β§ π β β) β {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} β {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) |
6 | sstr 3953 | . . 3 β’ ((π β π β§ π β β) β π β β) | |
7 | itgoval 41531 | . . 3 β’ (π β β β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
8 | 6, 7 | syl 17 | . 2 β’ ((π β π β§ π β β) β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) |
9 | itgoval 41531 | . . 3 β’ (π β β β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
10 | 9 | adantl 483 | . 2 β’ ((π β π β§ π β β) β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) |
11 | 5, 8, 10 | 3sstr4d 3992 | 1 β’ ((π β π β§ π β β) β (IntgOverβπ) β (IntgOverβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3070 {crab 3406 β wss 3911 βcfv 6497 βcc 11054 0cc0 11056 1c1 11057 Polycply 25561 coeffccoe 25563 degcdgr 25564 IntgOvercitgo 41527 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-1cn 11114 ax-addcl 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-map 8770 df-nn 12159 df-n0 12419 df-ply 25565 df-itgo 41529 |
This theorem is referenced by: (None) |
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