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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 26077 | . . . . 5 β’ ((π β π β§ π β β) β (Polyβπ) β (Polyβπ)) | |
2 | ssrexv 4044 | . . . . 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 480 | . . 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 4066 | . 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 3983 | . . 3 β’ ((π β π β§ π β β) β π β β) | |
7 | itgoval 42455 | . . 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 42455 | . . 3 β’ (π β β β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
10 | 9 | adantl 481 | . 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 4022 | 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 395 = wceq 1533 β wcel 2098 βwrex 3062 {crab 3424 β wss 3941 βcfv 6534 βcc 11105 0cc0 11107 1c1 11108 Polycply 26062 coeffccoe 26064 degcdgr 26065 IntgOvercitgo 42451 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-map 8819 df-nn 12212 df-n0 12472 df-ply 26066 df-itgo 42453 |
This theorem is referenced by: (None) |
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