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Mirrors > Home > MPE Home > Th. List > coemul | Structured version Visualization version GIF version |
Description: A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
coefv0.1 | β’ π΄ = (coeffβπΉ) |
coeadd.2 | β’ π΅ = (coeffβπΊ) |
Ref | Expression |
---|---|
coemul | β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ π β β0) β ((coeffβ(πΉ βf Β· πΊ))βπ) = Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coefv0.1 | . . . . . 6 β’ π΄ = (coeffβπΉ) | |
2 | coeadd.2 | . . . . . 6 β’ π΅ = (coeffβπΊ) | |
3 | eqid 2728 | . . . . . 6 β’ (degβπΉ) = (degβπΉ) | |
4 | eqid 2728 | . . . . . 6 β’ (degβπΊ) = (degβπΊ) | |
5 | 1, 2, 3, 4 | coemullem 26197 | . . . . 5 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ)) β ((coeffβ(πΉ βf Β· πΊ)) = (π β β0 β¦ Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) β§ (degβ(πΉ βf Β· πΊ)) β€ ((degβπΉ) + (degβπΊ)))) |
6 | 5 | simpld 494 | . . . 4 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ)) β (coeffβ(πΉ βf Β· πΊ)) = (π β β0 β¦ Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π))))) |
7 | 6 | fveq1d 6899 | . . 3 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ)) β ((coeffβ(πΉ βf Β· πΊ))βπ) = ((π β β0 β¦ Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π))))βπ)) |
8 | oveq2 7428 | . . . . 5 β’ (π = π β (0...π) = (0...π)) | |
9 | fvoveq1 7443 | . . . . . . 7 β’ (π = π β (π΅β(π β π)) = (π΅β(π β π))) | |
10 | 9 | oveq2d 7436 | . . . . . 6 β’ (π = π β ((π΄βπ) Β· (π΅β(π β π))) = ((π΄βπ) Β· (π΅β(π β π)))) |
11 | 10 | adantr 480 | . . . . 5 β’ ((π = π β§ π β (0...π)) β ((π΄βπ) Β· (π΅β(π β π))) = ((π΄βπ) Β· (π΅β(π β π)))) |
12 | 8, 11 | sumeq12dv 15685 | . . . 4 β’ (π = π β Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π))) = Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) |
13 | eqid 2728 | . . . 4 β’ (π β β0 β¦ Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) = (π β β0 β¦ Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) | |
14 | sumex 15667 | . . . 4 β’ Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π))) β V | |
15 | 12, 13, 14 | fvmpt 7005 | . . 3 β’ (π β β0 β ((π β β0 β¦ Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π))))βπ) = Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) |
16 | 7, 15 | sylan9eq 2788 | . 2 β’ (((πΉ β (Polyβπ) β§ πΊ β (Polyβπ)) β§ π β β0) β ((coeffβ(πΉ βf Β· πΊ))βπ) = Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) |
17 | 16 | 3impa 1108 | 1 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ π β β0) β ((coeffβ(πΉ βf Β· πΊ))βπ) = Ξ£π β (0...π)((π΄βπ) Β· (π΅β(π β π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5148 β¦ cmpt 5231 βcfv 6548 (class class class)co 7420 βf cof 7683 0cc0 11139 + caddc 11142 Β· cmul 11144 β€ cle 11280 β cmin 11475 β0cn0 12503 ...cfz 13517 Ξ£csu 15665 Polycply 26131 coeffccoe 26133 degcdgr 26134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-0p 25612 df-ply 26135 df-coe 26137 df-dgr 26138 |
This theorem is referenced by: coemulhi 26201 coemulc 26202 vieta1lem2 26259 plymulx0 34179 |
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