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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 20881 | . . . . 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 20883 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘𝑌) |
9 | eqid 2798 | . . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌) | |
10 | eqid 2798 | . . . . 5 ⊢ (1r‘( I ‘𝑅)) = (1r‘( I ‘𝑅)) | |
11 | eqid 2798 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
12 | 11 | grpinvfvi 18138 | . . . . 5 ⊢ (invg‘𝑅) = (invg‘( I ‘𝑅)) |
13 | 6, 7, 8, 9, 10, 12 | lmodvneg1 19670 | . . . 4 ⊢ ((𝑌 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
14 | 4, 5, 13 | syl2anc 587 | . . 3 ⊢ (𝜑 → (((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹) = (𝑁‘𝐹)) |
15 | 14 | fveq2d 6649 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘(𝑁‘𝐹))) |
16 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
17 | eqid 2798 | . . 3 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
18 | fvi 6715 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → ( I ‘𝑅) = 𝑅) | |
19 | 1, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
20 | 19 | fveq2d 6649 | . . . . 5 ⊢ (𝜑 → (1r‘( I ‘𝑅)) = (1r‘𝑅)) |
21 | 20 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) = ((invg‘𝑅)‘(1r‘𝑅))) |
22 | eqid 2798 | . . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
23 | 17, 22 | unitrrg 20059 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
24 | 1, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
25 | eqid 2798 | . . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
26 | 22, 25 | 1unit 19404 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅)) |
27 | 22, 11 | unitnegcl 19427 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Unit‘𝑅)) → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
28 | 1, 26, 27 | syl2anc2 588 | . . . . 5 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (Unit‘𝑅)) |
29 | 24, 28 | sseldd 3916 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘𝑅)) ∈ (RLReg‘𝑅)) |
30 | 21, 29 | eqeltrd 2890 | . . 3 ⊢ (𝜑 → ((invg‘𝑅)‘(1r‘( I ‘𝑅))) ∈ (RLReg‘𝑅)) |
31 | 2, 16, 1, 6, 17, 9, 30, 5 | deg1vsca 24706 | . 2 ⊢ (𝜑 → (𝐷‘(((invg‘𝑅)‘(1r‘( I ‘𝑅)))( ·𝑠 ‘𝑌)𝐹)) = (𝐷‘𝐹)) |
32 | 15, 31 | eqtr3d 2835 | 1 ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 I cid 5424 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ·𝑠 cvsca 16561 invgcminusg 18096 1rcur 19244 Ringcrg 19290 Unitcui 19385 LModclmod 19627 RLRegcrlreg 20045 Poly1cpl1 20806 deg1 cdg1 24655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-lmod 19629 df-lss 19697 df-rlreg 20049 df-psr 20594 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-ply1 20811 df-mdeg 24656 df-deg1 24657 |
This theorem is referenced by: deg1suble 24708 deg1sub 24709 |
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