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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 22189 | . . . . 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 22191 | . . . . 5 β’ ( I βπ ) = (Scalarβπ) |
9 | eqid 2728 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
10 | eqid 2728 | . . . . 5 β’ (1rβ( I βπ )) = (1rβ( I βπ )) | |
11 | eqid 2728 | . . . . . 6 β’ (invgβπ ) = (invgβπ ) | |
12 | 11 | grpinvfvi 18953 | . . . . 5 β’ (invgβπ ) = (invgβ( I βπ )) |
13 | 6, 7, 8, 9, 10, 12 | lmodvneg1 20802 | . . . 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 582 | . . 3 β’ (π β (((invgβπ )β(1rβ( I βπ )))( Β·π βπ)πΉ) = (πβπΉ)) |
15 | 14 | fveq2d 6906 | . 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 2728 | . . 3 β’ (RLRegβπ ) = (RLRegβπ ) | |
18 | fvi 6979 | . . . . . . 7 β’ (π β Ring β ( I βπ ) = π ) | |
19 | 1, 18 | syl 17 | . . . . . 6 β’ (π β ( I βπ ) = π ) |
20 | 19 | fveq2d 6906 | . . . . 5 β’ (π β (1rβ( I βπ )) = (1rβπ )) |
21 | 20 | fveq2d 6906 | . . . 4 β’ (π β ((invgβπ )β(1rβ( I βπ ))) = ((invgβπ )β(1rβπ ))) |
22 | eqid 2728 | . . . . . . 7 β’ (Unitβπ ) = (Unitβπ ) | |
23 | 17, 22 | unitrrg 21254 | . . . . . 6 β’ (π β Ring β (Unitβπ ) β (RLRegβπ )) |
24 | 1, 23 | syl 17 | . . . . 5 β’ (π β (Unitβπ ) β (RLRegβπ )) |
25 | eqid 2728 | . . . . . . 7 β’ (1rβπ ) = (1rβπ ) | |
26 | 22, 25 | 1unit 20327 | . . . . . 6 β’ (π β Ring β (1rβπ ) β (Unitβπ )) |
27 | 22, 11 | unitnegcl 20350 | . . . . . 6 β’ ((π β Ring β§ (1rβπ ) β (Unitβπ )) β ((invgβπ )β(1rβπ )) β (Unitβπ )) |
28 | 1, 26, 27 | syl2anc2 583 | . . . . 5 β’ (π β ((invgβπ )β(1rβπ )) β (Unitβπ )) |
29 | 24, 28 | sseldd 3983 | . . . 4 β’ (π β ((invgβπ )β(1rβπ )) β (RLRegβπ )) |
30 | 21, 29 | eqeltrd 2829 | . . 3 β’ (π β ((invgβπ )β(1rβ( I βπ ))) β (RLRegβπ )) |
31 | 2, 16, 1, 6, 17, 9, 30, 5 | deg1vsca 26069 | . 2 β’ (π β (π·β(((invgβπ )β(1rβ( I βπ )))( Β·π βπ)πΉ)) = (π·βπΉ)) |
32 | 15, 31 | eqtr3d 2770 | 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 3949 I cid 5579 βcfv 6553 (class class class)co 7426 Basecbs 17189 Β·π cvsca 17246 invgcminusg 18905 1rcur 20135 Ringcrg 20187 Unitcui 20308 LModclmod 20757 RLRegcrlreg 21240 Poly1cpl1 22114 deg1 cdg1 26015 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-hom 17266 df-cco 17267 df-0g 17432 df-prds 17438 df-pws 17440 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-lmod 20759 df-lss 20830 df-rlreg 21244 df-psr 21856 df-mpl 21858 df-opsr 21860 df-psr1 22117 df-ply1 22119 df-mdeg 26016 df-deg1 26017 |
This theorem is referenced by: deg1suble 26071 deg1sub 26072 |
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