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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 1132 | . . . 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 24729 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
6 | 5 | 3ad2ant3 1130 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
7 | r1pval.e | . . . . 5 ⊢ 𝐸 = (rem1p‘𝑅) | |
8 | eqid 2819 | . . . . 5 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅) | |
9 | eqid 2819 | . . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
10 | eqid 2819 | . . . . 5 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
11 | 7, 2, 3, 8, 9, 10 | r1pval 24742 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
12 | 1, 6, 11 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
13 | 12 | fveq2d 6667 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) = (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))) |
14 | eqid 2819 | . . . 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 24740 | . . . 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 260 | . . 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 498 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) < (𝐷‘𝐺)) |
19 | 13, 18 | eqbrtrd 5079 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 < clt 10667 Basecbs 16475 .rcmulr 16558 -gcsg 18097 Ringcrg 19289 Poly1cpl1 20337 deg1 cdg1 24640 Unic1pcuc1p 24712 quot1pcq1p 24713 rem1pcr1p 24714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 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-starv 16572 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-subrg 19525 df-lmod 19628 df-lss 19696 df-rlreg 20048 df-psr 20128 df-mvr 20129 df-mpl 20130 df-opsr 20132 df-psr1 20340 df-vr1 20341 df-ply1 20342 df-coe1 20343 df-cnfld 20538 df-mdeg 24641 df-deg1 24642 df-uc1p 24717 df-q1p 24718 df-r1p 24719 |
This theorem is referenced by: ply1rem 24749 ig1pdvds 24762 |
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