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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 21472 | . 2 โข (๐น โ ๐ต โ ๐น โ (Baseโ(PwSer1โ๐ ))) |
5 | 2, 3 | ply1bascl 21472 | . 2 โข (๐บ โ ๐ต โ ๐บ โ (Baseโ(PwSer1โ๐ ))) |
6 | eqid 2736 | . . 3 โข (PwSer1โ๐ ) = (PwSer1โ๐ ) | |
7 | eqid 2736 | . . . . 5 โข (1o mPoly ๐ ) = (1o mPoly ๐ ) | |
8 | eqid 2736 | . . . . 5 โข (1o mPwSer ๐ ) = (1o mPwSer ๐ ) | |
9 | coe1mul.t | . . . . . 6 โข โ = (.rโ๐) | |
10 | 2, 7, 9 | ply1mulr 21496 | . . . . 5 โข โ = (.rโ(1o mPoly ๐ )) |
11 | 7, 8, 10 | mplmulr 21490 | . . . 4 โข โ = (.rโ(1o mPwSer ๐ )) |
12 | eqid 2736 | . . . . 5 โข (.rโ(PwSer1โ๐ )) = (.rโ(PwSer1โ๐ )) | |
13 | 6, 8, 12 | psr1mulr 21493 | . . . 4 โข (.rโ(PwSer1โ๐ )) = (.rโ(1o mPwSer ๐ )) |
14 | 11, 13 | eqtr4i 2767 | . . 3 โข โ = (.rโ(PwSer1โ๐ )) |
15 | coe1mul.u | . . 3 โข ยท = (.rโ๐ ) | |
16 | eqid 2736 | . . 3 โข (Baseโ(PwSer1โ๐ )) = (Baseโ(PwSer1โ๐ )) | |
17 | 6, 14, 15, 16 | coe1mul2 21538 | . 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 1159 | 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 1086 = wceq 1540 โ wcel 2105 โฆ cmpt 5172 โcfv 6473 (class class class)co 7329 1oc1o 8352 0cc0 10964 โ cmin 11298 โ0cn0 12326 ...cfz 13332 Basecbs 17001 .rcmulr 17052 ฮฃg cgsu 17240 Ringcrg 19870 mPwSer cmps 21205 mPoly cmpl 21207 PwSer1cps1 21444 Poly1cpl1 21446 coe1cco1 21447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-ofr 7588 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-pm 8681 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-fz 13333 df-fzo 13476 df-seq 13815 df-hash 14138 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-tset 17070 df-ple 17071 df-0g 17241 df-gsum 17242 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-submnd 18520 df-grp 18668 df-minusg 18669 df-mulg 18789 df-ghm 18920 df-cntz 19011 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-psr 21210 df-mpl 21212 df-opsr 21214 df-psr1 21449 df-ply1 21451 df-coe1 21452 |
This theorem is referenced by: coe1tmmul2 21545 coe1tmmul 21546 cply1mul 21563 decpmatmullem 22018 pm2mpmhmlem2 22066 coe1mul3 25362 ply1mulgsum 46071 |
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