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Mirrors > Home > MPE Home > Th. List > deg1invg | Structured version Visualization version GIF version |
Description: The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
deg1addle.y | β’ π = (Poly1βπ ) |
deg1addle.d | β’ π· = ( deg1 βπ ) |
deg1addle.r | β’ (π β π β Ring) |
deg1invg.b | β’ π΅ = (Baseβπ) |
deg1invg.n | β’ π = (invgβπ) |
deg1invg.f | β’ (π β πΉ β π΅) |
Ref | Expression |
---|---|
deg1invg | β’ (π β (π·β(πβπΉ)) = (π·βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1addle.r | . . . . 5 β’ (π β π β Ring) | |
2 | deg1addle.y | . . . . . 6 β’ π = (Poly1βπ ) | |
3 | 2 | ply1lmod 22125 | . . . . 5 β’ (π β Ring β π β LMod) |
4 | 1, 3 | syl 17 | . . . 4 β’ (π β π β LMod) |
5 | deg1invg.f | . . . 4 β’ (π β πΉ β π΅) | |
6 | deg1invg.b | . . . . 5 β’ π΅ = (Baseβπ) | |
7 | deg1invg.n | . . . . 5 β’ π = (invgβπ) | |
8 | 2 | ply1sca2 22127 | . . . . 5 β’ ( I βπ ) = (Scalarβπ) |
9 | eqid 2726 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
10 | eqid 2726 | . . . . 5 β’ (1rβ( I βπ )) = (1rβ( I βπ )) | |
11 | eqid 2726 | . . . . . 6 β’ (invgβπ ) = (invgβπ ) | |
12 | 11 | grpinvfvi 18912 | . . . . 5 β’ (invgβπ ) = (invgβ( I βπ )) |
13 | 6, 7, 8, 9, 10, 12 | lmodvneg1 20751 | . . . 4 β’ ((π β LMod β§ πΉ β π΅) β (((invgβπ )β(1rβ( I βπ )))( Β·π βπ)πΉ) = (πβπΉ)) |
14 | 4, 5, 13 | syl2anc 583 | . . 3 β’ (π β (((invgβπ )β(1rβ( I βπ )))( Β·π βπ)πΉ) = (πβπΉ)) |
15 | 14 | fveq2d 6889 | . 2 β’ (π β (π·β(((invgβπ )β(1rβ( I βπ )))( Β·π βπ)πΉ)) = (π·β(πβπΉ))) |
16 | deg1addle.d | . . 3 β’ π· = ( deg1 βπ ) | |
17 | eqid 2726 | . . 3 β’ (RLRegβπ ) = (RLRegβπ ) | |
18 | fvi 6961 | . . . . . . 7 β’ (π β Ring β ( I βπ ) = π ) | |
19 | 1, 18 | syl 17 | . . . . . 6 β’ (π β ( I βπ ) = π ) |
20 | 19 | fveq2d 6889 | . . . . 5 β’ (π β (1rβ( I βπ )) = (1rβπ )) |
21 | 20 | fveq2d 6889 | . . . 4 β’ (π β ((invgβπ )β(1rβ( I βπ ))) = ((invgβπ )β(1rβπ ))) |
22 | eqid 2726 | . . . . . . 7 β’ (Unitβπ ) = (Unitβπ ) | |
23 | 17, 22 | unitrrg 21203 | . . . . . 6 β’ (π β Ring β (Unitβπ ) β (RLRegβπ )) |
24 | 1, 23 | syl 17 | . . . . 5 β’ (π β (Unitβπ ) β (RLRegβπ )) |
25 | eqid 2726 | . . . . . . 7 β’ (1rβπ ) = (1rβπ ) | |
26 | 22, 25 | 1unit 20276 | . . . . . 6 β’ (π β Ring β (1rβπ ) β (Unitβπ )) |
27 | 22, 11 | unitnegcl 20299 | . . . . . 6 β’ ((π β Ring β§ (1rβπ ) β (Unitβπ )) β ((invgβπ )β(1rβπ )) β (Unitβπ )) |
28 | 1, 26, 27 | syl2anc2 584 | . . . . 5 β’ (π β ((invgβπ )β(1rβπ )) β (Unitβπ )) |
29 | 24, 28 | sseldd 3978 | . . . 4 β’ (π β ((invgβπ )β(1rβπ )) β (RLRegβπ )) |
30 | 21, 29 | eqeltrd 2827 | . . 3 β’ (π β ((invgβπ )β(1rβ( I βπ ))) β (RLRegβπ )) |
31 | 2, 16, 1, 6, 17, 9, 30, 5 | deg1vsca 25996 | . 2 β’ (π β (π·β(((invgβπ )β(1rβ( I βπ )))( Β·π βπ)πΉ)) = (π·βπΉ)) |
32 | 15, 31 | eqtr3d 2768 | 1 β’ (π β (π·β(πβπΉ)) = (π·βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3943 I cid 5566 βcfv 6537 (class class class)co 7405 Basecbs 17153 Β·π cvsca 17210 invgcminusg 18864 1rcur 20086 Ringcrg 20138 Unitcui 20257 LModclmod 20706 RLRegcrlreg 21189 Poly1cpl1 22051 deg1 cdg1 25942 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 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 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-lmod 20708 df-lss 20779 df-rlreg 21193 df-psr 21803 df-mpl 21805 df-opsr 21807 df-psr1 22054 df-ply1 22056 df-mdeg 25943 df-deg1 25944 |
This theorem is referenced by: deg1suble 25998 deg1sub 25999 |
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