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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 1136 | . . . 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 25414 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
6 | 5 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
7 | r1pval.e | . . . . 5 ⊢ 𝐸 = (rem1p‘𝑅) | |
8 | eqid 2736 | . . . . 5 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅) | |
9 | eqid 2736 | . . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
10 | eqid 2736 | . . . . 5 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
11 | 7, 2, 3, 8, 9, 10 | r1pval 25427 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
12 | 1, 6, 11 | syl2anc 584 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
13 | 12 | fveq2d 6829 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) = (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))) |
14 | eqid 2736 | . . . 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 25425 | . . . 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 257 | . . 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 496 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) < (𝐷‘𝐺)) |
19 | 13, 18 | eqbrtrd 5114 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 class class class wbr 5092 ‘cfv 6479 (class class class)co 7337 < clt 11110 Basecbs 17009 .rcmulr 17060 -gcsg 18675 Ringcrg 19878 Poly1cpl1 21454 deg1 cdg1 25322 Unic1pcuc1p 25397 quot1pcq1p 25398 rem1pcr1p 25399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 ax-addf 11051 ax-mulf 11052 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-ofr 7596 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-tpos 8112 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-pm 8689 df-ixp 8757 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-sup 9299 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-fz 13341 df-fzo 13484 df-seq 13823 df-hash 14146 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-starv 17074 df-sca 17075 df-vsca 17076 df-tset 17078 df-ple 17079 df-ds 17081 df-unif 17082 df-0g 17249 df-gsum 17250 df-mre 17392 df-mrc 17393 df-acs 17395 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-mhm 18527 df-submnd 18528 df-grp 18676 df-minusg 18677 df-sbg 18678 df-mulg 18797 df-subg 18848 df-ghm 18928 df-cntz 19019 df-cmn 19483 df-abl 19484 df-mgp 19816 df-ur 19833 df-ring 19880 df-cring 19881 df-oppr 19957 df-dvdsr 19978 df-unit 19979 df-invr 20009 df-subrg 20127 df-lmod 20231 df-lss 20300 df-rlreg 20660 df-cnfld 20704 df-psr 21218 df-mvr 21219 df-mpl 21220 df-opsr 21222 df-psr1 21457 df-vr1 21458 df-ply1 21459 df-coe1 21460 df-mdeg 25323 df-deg1 25324 df-uc1p 25402 df-q1p 25403 df-r1p 25404 |
This theorem is referenced by: ply1rem 25434 ig1pdvds 25447 |
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