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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 | ⊢ 𝑌 = (Poly1‘𝑅) |
deg1addle.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1addle.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
deg1invg.b | ⊢ 𝐵 = (Base‘𝑌) |
deg1invg.n | ⊢ 𝑁 = (invg‘𝑌) |
deg1invg.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
deg1invg | ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1addle.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | deg1addle.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
3 | 2 | ply1lmod 20022 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ LMod) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ LMod) |
5 | deg1invg.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
6 | deg1invg.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
7 | deg1invg.n | . . . . 5 ⊢ 𝑁 = (invg‘𝑌) | |
8 | 2 | ply1sca2 20024 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘𝑌) |
9 | eqid 2778 | . . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌) | |
10 | eqid 2778 | . . . . 5 ⊢ (1r‘( I ‘𝑅)) = (1r‘( I ‘𝑅)) | |
11 | eqid 2778 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
12 | 11 | grpinvfvi 17854 | . . . . 5 ⊢ (invg‘𝑅) = (invg‘( I ‘𝑅)) |
13 | 6, 7, 8, 9, 10, 12 | lmodvneg1 19302 | . . . 4 ⊢ ((𝑌 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
14 | 4, 5, 13 | syl2anc 579 | . . 3 ⊢ (𝜑 → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
15 | 14 | fveq2d 6452 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘(𝑁‘𝐹))) |
16 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
17 | eqid 2778 | . . 3 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
18 | fvi 6517 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → ( I ‘𝑅) = 𝑅) | |
19 | 1, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
20 | 19 | fveq2d 6452 | . . . . 5 ⊢ (𝜑 → (1r‘( I ‘𝑅)) = (1r‘𝑅)) |
21 | 20 | fveq2d 6452 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) = ((invg‘𝑅)‘(1r‘𝑅))) |
22 | eqid 2778 | . . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
23 | 17, 22 | unitrrg 19694 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
24 | 1, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
25 | eqid 2778 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
26 | 22, 25 | 1unit 19049 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅)) |
27 | 1, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ∈ (Unit‘𝑅)) |
28 | 22, 11 | unitnegcl 19072 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Unit‘𝑅)) → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
29 | 1, 27, 28 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
30 | 24, 29 | sseldd 3822 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (RLReg‘𝑅)) |
31 | 21, 30 | eqeltrd 2859 | . . 3 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) ∈ (RLReg‘𝑅)) |
32 | 2, 16, 1, 6, 17, 9, 31, 5 | deg1vsca 24306 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘𝐹)) |
33 | 15, 32 | eqtr3d 2816 | 1 ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 I cid 5262 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 ·𝑠 cvsca 16346 invgcminusg 17814 1rcur 18892 Ringcrg 18938 Unitcui 19030 LModclmod 19259 RLRegcrlreg 19680 Poly1cpl1 19947 deg1 cdg1 24255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-fz 12648 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-sca 16358 df-vsca 16359 df-tset 16361 df-ple 16362 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-minusg 17817 df-sbg 17818 df-subg 17979 df-mgp 18881 df-ur 18893 df-ring 18940 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 df-lmod 19261 df-lss 19329 df-rlreg 19684 df-psr 19757 df-mpl 19759 df-opsr 19761 df-psr1 19950 df-ply1 19952 df-mdeg 24256 df-deg1 24257 |
This theorem is referenced by: deg1suble 24308 deg1sub 24309 |
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