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Mirrors > Home > MPE Home > Th. List > deg1suble | Structured version Visualization version GIF version |
Description: The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-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) |
Ref | Expression |
---|---|
deg1suble | ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1addle.y | . . 3 ⊢ 𝑌 = (Poly1‘𝑅) | |
2 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | deg1addle.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
4 | deg1suble.b | . . 3 ⊢ 𝐵 = (Base‘𝑌) | |
5 | eqid 2824 | . . 3 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
6 | deg1suble.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
7 | 1 | ply1ring 19977 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
8 | ringgrp 18905 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
9 | 3, 7, 8 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ Grp) |
10 | deg1suble.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
11 | eqid 2824 | . . . . 5 ⊢ (invg‘𝑌) = (invg‘𝑌) | |
12 | 4, 11 | grpinvcl 17820 | . . . 4 ⊢ ((𝑌 ∈ Grp ∧ 𝐺 ∈ 𝐵) → ((invg‘𝑌)‘𝐺) ∈ 𝐵) |
13 | 9, 10, 12 | syl2anc 581 | . . 3 ⊢ (𝜑 → ((invg‘𝑌)‘𝐺) ∈ 𝐵) |
14 | 1, 2, 3, 4, 5, 6, 13 | deg1addle 24259 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) ≤ if((𝐷‘𝐹) ≤ (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘𝐹))) |
15 | deg1suble.m | . . . . 5 ⊢ − = (-g‘𝑌) | |
16 | 4, 5, 11, 15 | grpsubval 17818 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) = (𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) |
17 | 6, 10, 16 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝐹 − 𝐺) = (𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) |
18 | 17 | fveq2d 6436 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘(𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺)))) |
19 | 1, 2, 3, 4, 11, 10 | deg1invg 24264 | . . . . 5 ⊢ (𝜑 → (𝐷‘((invg‘𝑌)‘𝐺)) = (𝐷‘𝐺)) |
20 | 19 | eqcomd 2830 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) = (𝐷‘((invg‘𝑌)‘𝐺))) |
21 | 20 | breq2d 4884 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ (𝐷‘𝐺) ↔ (𝐷‘𝐹) ≤ (𝐷‘((invg‘𝑌)‘𝐺)))) |
22 | 21, 20 | ifbieq1d 4328 | . 2 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) = if((𝐷‘𝐹) ≤ (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘𝐹))) |
23 | 14, 18, 22 | 3brtr4d 4904 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ifcif 4305 class class class wbr 4872 ‘cfv 6122 (class class class)co 6904 ≤ cle 10391 Basecbs 16221 +gcplusg 16304 Grpcgrp 17775 invgcminusg 17776 -gcsg 17777 Ringcrg 18900 Poly1cpl1 19906 deg1 cdg1 24212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 ax-addf 10330 ax-mulf 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-ofr 7157 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-tpos 7616 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-ixp 8175 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-sup 8616 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-fz 12619 df-fzo 12760 df-seq 13095 df-hash 13410 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-starv 16319 df-sca 16320 df-vsca 16321 df-tset 16323 df-ple 16324 df-ds 16326 df-unif 16327 df-0g 16454 df-gsum 16455 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-mhm 17687 df-submnd 17688 df-grp 17778 df-minusg 17779 df-sbg 17780 df-mulg 17894 df-subg 17941 df-ghm 18008 df-cntz 18099 df-cmn 18547 df-abl 18548 df-mgp 18843 df-ur 18855 df-ring 18902 df-cring 18903 df-oppr 18976 df-dvdsr 18994 df-unit 18995 df-invr 19025 df-subrg 19133 df-lmod 19220 df-lss 19288 df-rlreg 19643 df-psr 19716 df-mpl 19718 df-opsr 19720 df-psr1 19909 df-ply1 19911 df-cnfld 20106 df-mdeg 24213 df-deg1 24214 |
This theorem is referenced by: deg1sublt 24268 ply1divmo 24293 |
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