![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22194 | . . . . 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 22196 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘𝑌) |
9 | eqid 2725 | . . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌) | |
10 | eqid 2725 | . . . . 5 ⊢ (1r‘( I ‘𝑅)) = (1r‘( I ‘𝑅)) | |
11 | eqid 2725 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
12 | 11 | grpinvfvi 18947 | . . . . 5 ⊢ (invg‘𝑅) = (invg‘( I ‘𝑅)) |
13 | 6, 7, 8, 9, 10, 12 | lmodvneg1 20800 | . . . 4 ⊢ ((𝑌 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
14 | 4, 5, 13 | syl2anc 582 | . . 3 ⊢ (𝜑 → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
15 | 14 | fveq2d 6900 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘(𝑁‘𝐹))) |
16 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
17 | eqid 2725 | . . 3 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
18 | fvi 6973 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → ( I ‘𝑅) = 𝑅) | |
19 | 1, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
20 | 19 | fveq2d 6900 | . . . . 5 ⊢ (𝜑 → (1r‘( I ‘𝑅)) = (1r‘𝑅)) |
21 | 20 | fveq2d 6900 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) = ((invg‘𝑅)‘(1r‘𝑅))) |
22 | eqid 2725 | . . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
23 | 17, 22 | unitrrg 21257 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
24 | 1, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
25 | eqid 2725 | . . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
26 | 22, 25 | 1unit 20325 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅)) |
27 | 22, 11 | unitnegcl 20348 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Unit‘𝑅)) → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
28 | 1, 26, 27 | syl2anc2 583 | . . . . 5 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
29 | 24, 28 | sseldd 3977 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (RLReg‘𝑅)) |
30 | 21, 29 | eqeltrd 2825 | . . 3 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) ∈ (RLReg‘𝑅)) |
31 | 2, 16, 1, 6, 17, 9, 30, 5 | deg1vsca 26085 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘𝐹)) |
32 | 15, 31 | eqtr3d 2767 | 1 ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 I cid 5575 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 ·𝑠 cvsca 17240 invgcminusg 18899 1rcur 20133 Ringcrg 20185 Unitcui 20306 LModclmod 20755 RLRegcrlreg 21243 Poly1cpl1 22119 deg1 cdg1 26031 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-hom 17260 df-cco 17261 df-0g 17426 df-prds 17432 df-pws 17434 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-lmod 20757 df-lss 20828 df-rlreg 21247 df-psr 21859 df-mpl 21861 df-opsr 21863 df-psr1 22122 df-ply1 22124 df-mdeg 26032 df-deg1 26033 |
This theorem is referenced by: deg1suble 26087 deg1sub 26088 |
Copyright terms: Public domain | W3C validator |