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Mirrors > Home > MPE Home > Th. List > deg1sub | Structured version Visualization version GIF version |
Description: Exact degree of a difference of two polynomials of unequal degree. (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) |
deg1suble.b | ⊢ 𝐵 = (Base‘𝑌) |
deg1suble.m | ⊢ − = (-g‘𝑌) |
deg1suble.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
deg1suble.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
deg1sub.l | ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹)) |
Ref | Expression |
---|---|
deg1sub | ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1suble.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
2 | deg1suble.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
3 | deg1suble.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
4 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
5 | eqid 2737 | . . . . 5 ⊢ (invg‘𝑌) = (invg‘𝑌) | |
6 | deg1suble.m | . . . . 5 ⊢ − = (-g‘𝑌) | |
7 | 3, 4, 5, 6 | grpsubval 18756 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) = (𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) |
8 | 1, 2, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐹 − 𝐺) = (𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) |
9 | 8 | fveq2d 6843 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘(𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺)))) |
10 | deg1addle.y | . . 3 ⊢ 𝑌 = (Poly1‘𝑅) | |
11 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
12 | deg1addle.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | 10 | ply1ring 21571 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
14 | ringgrp 19923 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
15 | 12, 13, 14 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ Grp) |
16 | 3, 5 | grpinvcl 18758 | . . . 4 ⊢ ((𝑌 ∈ Grp ∧ 𝐺 ∈ 𝐵) → ((invg‘𝑌)‘𝐺) ∈ 𝐵) |
17 | 15, 2, 16 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((invg‘𝑌)‘𝐺) ∈ 𝐵) |
18 | 10, 11, 12, 3, 5, 2 | deg1invg 25423 | . . . 4 ⊢ (𝜑 → (𝐷‘((invg‘𝑌)‘𝐺)) = (𝐷‘𝐺)) |
19 | deg1sub.l | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹)) | |
20 | 18, 19 | eqbrtrd 5125 | . . 3 ⊢ (𝜑 → (𝐷‘((invg‘𝑌)‘𝐺)) < (𝐷‘𝐹)) |
21 | 10, 11, 12, 3, 4, 1, 17, 20 | deg1add 25420 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) = (𝐷‘𝐹)) |
22 | 9, 21 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 < clt 11147 Basecbs 17043 +gcplusg 17093 Grpcgrp 18708 invgcminusg 18709 -gcsg 18710 Ringcrg 19918 Poly1cpl1 21500 deg1 cdg1 25368 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14185 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-starv 17108 df-sca 17109 df-vsca 17110 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-0g 17283 df-gsum 17284 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-mulg 18832 df-subg 18884 df-ghm 18965 df-cntz 19056 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-cring 19921 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-subrg 20173 df-lmod 20277 df-lss 20346 df-rlreg 20706 df-cnfld 20750 df-psr 21264 df-mpl 21266 df-opsr 21268 df-psr1 21503 df-ply1 21505 df-coe1 21506 df-mdeg 25369 df-deg1 25370 |
This theorem is referenced by: ply1remlem 25479 lgsqrlem4 26649 idomrootle 41431 |
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