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Mirrors > Home > MPE Home > Th. List > r1pdeglt | Structured version Visualization version GIF version |
Description: The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
r1pval.e | ⊢ 𝐸 = (rem1p‘𝑅) |
r1pval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
r1pval.b | ⊢ 𝐵 = (Base‘𝑃) |
r1pcl.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
r1pdeglt.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
Ref | Expression |
---|---|
r1pdeglt | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) < (𝐷‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1130 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 ∈ 𝐵) | |
2 | r1pval.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | r1pval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
4 | r1pcl.c | . . . . . 6 ⊢ 𝐶 = (Unic1p‘𝑅) | |
5 | 2, 3, 4 | uc1pcl 24122 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
6 | 5 | 3ad2ant3 1128 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
7 | r1pval.e | . . . . 5 ⊢ 𝐸 = (rem1p‘𝑅) | |
8 | eqid 2770 | . . . . 5 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅) | |
9 | eqid 2770 | . . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
10 | eqid 2770 | . . . . 5 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
11 | 7, 2, 3, 8, 9, 10 | r1pval 24135 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
12 | 1, 6, 11 | syl2anc 565 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
13 | 12 | fveq2d 6336 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) = (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))) |
14 | eqid 2770 | . . . 4 ⊢ (𝐹(quot1p‘𝑅)𝐺) = (𝐹(quot1p‘𝑅)𝐺) | |
15 | r1pdeglt.d | . . . . 5 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
16 | 8, 2, 3, 15, 10, 9, 4 | q1peqb 24133 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (((𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵 ∧ (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) < (𝐷‘𝐺)) ↔ (𝐹(quot1p‘𝑅)𝐺) = (𝐹(quot1p‘𝑅)𝐺))) |
17 | 14, 16 | mpbiri 248 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵 ∧ (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) < (𝐷‘𝐺))) |
18 | 17 | simprd 477 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) < (𝐷‘𝐺)) |
19 | 13, 18 | eqbrtrd 4806 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1070 = wceq 1630 ∈ wcel 2144 class class class wbr 4784 ‘cfv 6031 (class class class)co 6792 < clt 10275 Basecbs 16063 .rcmulr 16149 -gcsg 17631 Ringcrg 18754 Poly1cpl1 19761 deg1 cdg1 24033 Unic1pcuc1p 24105 quot1pcq1p 24106 rem1pcr1p 24107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7043 df-ofr 7044 df-om 7212 df-1st 7314 df-2nd 7315 df-supp 7446 df-tpos 7503 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-ixp 8062 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fsupp 8431 df-sup 8503 df-oi 8570 df-card 8964 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-fz 12533 df-fzo 12673 df-seq 13008 df-hash 13321 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-0g 16309 df-gsum 16310 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-mhm 17542 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-mulg 17748 df-subg 17798 df-ghm 17865 df-cntz 17956 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-cring 18757 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-subrg 18987 df-lmod 19074 df-lss 19142 df-rlreg 19497 df-psr 19570 df-mvr 19571 df-mpl 19572 df-opsr 19574 df-psr1 19764 df-vr1 19765 df-ply1 19766 df-coe1 19767 df-cnfld 19961 df-mdeg 24034 df-deg1 24035 df-uc1p 24110 df-q1p 24111 df-r1p 24112 |
This theorem is referenced by: ply1rem 24142 ig1pdvds 24155 |
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