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Mirrors > Home > MPE Home > Th. List > ply1opprmul | Structured version Visualization version GIF version |
Description: Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
ply1opprmul.y | โข ๐ = (Poly1โ๐ ) |
ply1opprmul.s | โข ๐ = (opprโ๐ ) |
ply1opprmul.z | โข ๐ = (Poly1โ๐) |
ply1opprmul.t | โข ยท = (.rโ๐) |
ply1opprmul.u | โข โ = (.rโ๐) |
ply1opprmul.b | โข ๐ต = (Baseโ๐) |
Ref | Expression |
---|---|
ply1opprmul | โข ((๐ โ Ring โง ๐น โ ๐ต โง ๐บ โ ๐ต) โ (๐น โ ๐บ) = (๐บ ยท ๐น)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 โข (๐ โ Ring โ ๐ โ Ring) | |
2 | ply1opprmul.y | . . . 4 โข ๐ = (Poly1โ๐ ) | |
3 | ply1opprmul.b | . . . 4 โข ๐ต = (Baseโ๐) | |
4 | 2, 3 | ply1bascl 22096 | . . 3 โข (๐น โ ๐ต โ ๐น โ (Baseโ(PwSer1โ๐ ))) |
5 | eqid 2727 | . . . 4 โข (PwSer1โ๐ ) = (PwSer1โ๐ ) | |
6 | eqid 2727 | . . . 4 โข (Baseโ(PwSer1โ๐ )) = (Baseโ(PwSer1โ๐ )) | |
7 | 5, 6 | psr1bascl 22093 | . . 3 โข (๐น โ (Baseโ(PwSer1โ๐ )) โ ๐น โ (Baseโ(1o mPwSer ๐ ))) |
8 | 4, 7 | syl 17 | . 2 โข (๐น โ ๐ต โ ๐น โ (Baseโ(1o mPwSer ๐ ))) |
9 | 2, 3 | ply1bascl 22096 | . . 3 โข (๐บ โ ๐ต โ ๐บ โ (Baseโ(PwSer1โ๐ ))) |
10 | 5, 6 | psr1bascl 22093 | . . 3 โข (๐บ โ (Baseโ(PwSer1โ๐ )) โ ๐บ โ (Baseโ(1o mPwSer ๐ ))) |
11 | 9, 10 | syl 17 | . 2 โข (๐บ โ ๐ต โ ๐บ โ (Baseโ(1o mPwSer ๐ ))) |
12 | eqid 2727 | . . 3 โข (1o mPwSer ๐ ) = (1o mPwSer ๐ ) | |
13 | ply1opprmul.s | . . 3 โข ๐ = (opprโ๐ ) | |
14 | eqid 2727 | . . 3 โข (1o mPwSer ๐) = (1o mPwSer ๐) | |
15 | eqid 2727 | . . . 4 โข (1o mPoly ๐ ) = (1o mPoly ๐ ) | |
16 | ply1opprmul.t | . . . . 5 โข ยท = (.rโ๐) | |
17 | 2, 15, 16 | ply1mulr 22118 | . . . 4 โข ยท = (.rโ(1o mPoly ๐ )) |
18 | 15, 12, 17 | mplmulr 21928 | . . 3 โข ยท = (.rโ(1o mPwSer ๐ )) |
19 | eqid 2727 | . . . 4 โข (1o mPoly ๐) = (1o mPoly ๐) | |
20 | ply1opprmul.z | . . . . 5 โข ๐ = (Poly1โ๐) | |
21 | ply1opprmul.u | . . . . 5 โข โ = (.rโ๐) | |
22 | 20, 19, 21 | ply1mulr 22118 | . . . 4 โข โ = (.rโ(1o mPoly ๐)) |
23 | 19, 14, 22 | mplmulr 21928 | . . 3 โข โ = (.rโ(1o mPwSer ๐)) |
24 | eqid 2727 | . . 3 โข (Baseโ(1o mPwSer ๐ )) = (Baseโ(1o mPwSer ๐ )) | |
25 | 12, 13, 14, 18, 23, 24 | psropprmul 22130 | . 2 โข ((๐ โ Ring โง ๐น โ (Baseโ(1o mPwSer ๐ )) โง ๐บ โ (Baseโ(1o mPwSer ๐ ))) โ (๐น โ ๐บ) = (๐บ ยท ๐น)) |
26 | 1, 8, 11, 25 | syl3an 1158 | 1 โข ((๐ โ Ring โง ๐น โ ๐ต โง ๐บ โ ๐ต) โ (๐น โ ๐บ) = (๐บ ยท ๐น)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1085 = wceq 1534 โ wcel 2099 โcfv 6542 (class class class)co 7414 1oc1o 8471 Basecbs 17165 .rcmulr 17219 Ringcrg 20157 opprcoppr 20254 mPwSer cmps 21817 mPoly cmpl 21819 PwSer1cps1 22068 Poly1cpl1 22070 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-tset 17237 df-ple 17238 df-0g 17408 df-gsum 17409 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-grp 18878 df-minusg 18879 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-ur 20106 df-ring 20159 df-oppr 20255 df-psr 21822 df-mpl 21824 df-opsr 21826 df-psr1 22073 df-ply1 22075 |
This theorem is referenced by: ply1divalg2 26048 |
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