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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 20348 | . . . . 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 20350 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘𝑌) |
9 | eqid 2818 | . . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌) | |
10 | eqid 2818 | . . . . 5 ⊢ (1r‘( I ‘𝑅)) = (1r‘( I ‘𝑅)) | |
11 | eqid 2818 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
12 | 11 | grpinvfvi 18084 | . . . . 5 ⊢ (invg‘𝑅) = (invg‘( I ‘𝑅)) |
13 | 6, 7, 8, 9, 10, 12 | lmodvneg1 19606 | . . . 4 ⊢ ((𝑌 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
14 | 4, 5, 13 | syl2anc 584 | . . 3 ⊢ (𝜑 → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
15 | 14 | fveq2d 6667 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘(𝑁‘𝐹))) |
16 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
17 | eqid 2818 | . . 3 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
18 | fvi 6733 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → ( I ‘𝑅) = 𝑅) | |
19 | 1, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
20 | 19 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (1r‘( I ‘𝑅)) = (1r‘𝑅)) |
21 | 20 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) = ((invg‘𝑅)‘(1r‘𝑅))) |
22 | eqid 2818 | . . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
23 | 17, 22 | unitrrg 19994 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
24 | 1, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
25 | eqid 2818 | . . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
26 | 22, 25 | 1unit 19337 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅)) |
27 | 22, 11 | unitnegcl 19360 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Unit‘𝑅)) → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
28 | 1, 26, 27 | syl2anc2 585 | . . . . 5 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
29 | 24, 28 | sseldd 3965 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (RLReg‘𝑅)) |
30 | 21, 29 | eqeltrd 2910 | . . 3 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) ∈ (RLReg‘𝑅)) |
31 | 2, 16, 1, 6, 17, 9, 30, 5 | deg1vsca 24626 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘𝐹)) |
32 | 15, 31 | eqtr3d 2855 | 1 ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 I cid 5452 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ·𝑠 cvsca 16557 invgcminusg 18042 1rcur 19180 Ringcrg 19226 Unitcui 19318 LModclmod 19563 RLRegcrlreg 19980 Poly1cpl1 20273 deg1 cdg1 24575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-lmod 19565 df-lss 19633 df-rlreg 19984 df-psr 20064 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-ply1 20278 df-mdeg 24576 df-deg1 24577 |
This theorem is referenced by: deg1suble 24628 deg1sub 24629 |
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