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Mirrors > Home > MPE Home > Th. List > coe1mul | Structured version Visualization version GIF version |
Description: The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
coe1mul.s | โข ๐ = (Poly1โ๐ ) |
coe1mul.t | โข โ = (.rโ๐) |
coe1mul.u | โข ยท = (.rโ๐ ) |
coe1mul.b | โข ๐ต = (Baseโ๐) |
Ref | Expression |
---|---|
coe1mul | โข ((๐ โ Ring โง ๐น โ ๐ต โง ๐บ โ ๐ต) โ (coe1โ(๐น โ ๐บ)) = (๐ โ โ0 โฆ (๐ ฮฃg (๐ฅ โ (0...๐) โฆ (((coe1โ๐น)โ๐ฅ) ยท ((coe1โ๐บ)โ(๐ โ ๐ฅ))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 โข (๐ โ Ring โ ๐ โ Ring) | |
2 | coe1mul.s | . . 3 โข ๐ = (Poly1โ๐ ) | |
3 | coe1mul.b | . . 3 โข ๐ต = (Baseโ๐) | |
4 | 2, 3 | ply1bascl 21946 | . 2 โข (๐น โ ๐ต โ ๐น โ (Baseโ(PwSer1โ๐ ))) |
5 | 2, 3 | ply1bascl 21946 | . 2 โข (๐บ โ ๐ต โ ๐บ โ (Baseโ(PwSer1โ๐ ))) |
6 | eqid 2732 | . . 3 โข (PwSer1โ๐ ) = (PwSer1โ๐ ) | |
7 | eqid 2732 | . . . . 5 โข (1o mPoly ๐ ) = (1o mPoly ๐ ) | |
8 | eqid 2732 | . . . . 5 โข (1o mPwSer ๐ ) = (1o mPwSer ๐ ) | |
9 | coe1mul.t | . . . . . 6 โข โ = (.rโ๐) | |
10 | 2, 7, 9 | ply1mulr 21968 | . . . . 5 โข โ = (.rโ(1o mPoly ๐ )) |
11 | 7, 8, 10 | mplmulr 21786 | . . . 4 โข โ = (.rโ(1o mPwSer ๐ )) |
12 | eqid 2732 | . . . . 5 โข (.rโ(PwSer1โ๐ )) = (.rโ(PwSer1โ๐ )) | |
13 | 6, 8, 12 | psr1mulr 21965 | . . . 4 โข (.rโ(PwSer1โ๐ )) = (.rโ(1o mPwSer ๐ )) |
14 | 11, 13 | eqtr4i 2763 | . . 3 โข โ = (.rโ(PwSer1โ๐ )) |
15 | coe1mul.u | . . 3 โข ยท = (.rโ๐ ) | |
16 | eqid 2732 | . . 3 โข (Baseโ(PwSer1โ๐ )) = (Baseโ(PwSer1โ๐ )) | |
17 | 6, 14, 15, 16 | coe1mul2 22011 | . 2 โข ((๐ โ Ring โง ๐น โ (Baseโ(PwSer1โ๐ )) โง ๐บ โ (Baseโ(PwSer1โ๐ ))) โ (coe1โ(๐น โ ๐บ)) = (๐ โ โ0 โฆ (๐ ฮฃg (๐ฅ โ (0...๐) โฆ (((coe1โ๐น)โ๐ฅ) ยท ((coe1โ๐บ)โ(๐ โ ๐ฅ))))))) |
18 | 1, 4, 5, 17 | syl3an 1160 | 1 โข ((๐ โ Ring โง ๐น โ ๐ต โง ๐บ โ ๐ต) โ (coe1โ(๐น โ ๐บ)) = (๐ โ โ0 โฆ (๐ ฮฃg (๐ฅ โ (0...๐) โฆ (((coe1โ๐น)โ๐ฅ) ยท ((coe1โ๐บ)โ(๐ โ ๐ฅ))))))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1087 = wceq 1541 โ wcel 2106 โฆ cmpt 5231 โcfv 6543 (class class class)co 7411 1oc1o 8461 0cc0 11112 โ cmin 11448 โ0cn0 12476 ...cfz 13488 Basecbs 17148 .rcmulr 17202 ฮฃg cgsu 17390 Ringcrg 20127 mPwSer cmps 21676 mPoly cmpl 21678 PwSer1cps1 21918 Poly1cpl1 21920 coe1cco1 21921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-mulg 18987 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-psr 21681 df-mpl 21683 df-opsr 21685 df-psr1 21923 df-ply1 21925 df-coe1 21926 |
This theorem is referenced by: coe1tmmul2 22018 coe1tmmul 22019 cply1mul 22038 decpmatmullem 22493 pm2mpmhmlem2 22541 coe1mul3 25841 ply1mulgsum 47159 |
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