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Mirrors > Home > MPE Home > Th. List > q1pcl | Structured version Visualization version GIF version |
Description: Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
q1pcl.q | ⊢ 𝑄 = (quot1p‘𝑅) |
q1pcl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
q1pcl.b | ⊢ 𝐵 = (Base‘𝑃) |
q1pcl.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
Ref | Expression |
---|---|
q1pcl | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 ⊢ (𝐹𝑄𝐺) = (𝐹𝑄𝐺) | |
2 | q1pcl.q | . . . 4 ⊢ 𝑄 = (quot1p‘𝑅) | |
3 | q1pcl.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | q1pcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
5 | eqid 2725 | . . . 4 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
6 | eqid 2725 | . . . 4 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
7 | eqid 2725 | . . . 4 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
8 | q1pcl.c | . . . 4 ⊢ 𝐶 = (Unic1p‘𝑅) | |
9 | 2, 3, 4, 5, 6, 7, 8 | q1peqb 26141 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (((𝐹𝑄𝐺) ∈ 𝐵 ∧ (( deg1 ‘𝑅)‘(𝐹(-g‘𝑃)((𝐹𝑄𝐺)(.r‘𝑃)𝐺))) < (( deg1 ‘𝑅)‘𝐺)) ↔ (𝐹𝑄𝐺) = (𝐹𝑄𝐺))) |
10 | 1, 9 | mpbiri 257 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹𝑄𝐺) ∈ 𝐵 ∧ (( deg1 ‘𝑅)‘(𝐹(-g‘𝑃)((𝐹𝑄𝐺)(.r‘𝑃)𝐺))) < (( deg1 ‘𝑅)‘𝐺))) |
11 | 10 | simpld 493 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 < clt 11285 Basecbs 17188 .rcmulr 17242 -gcsg 18905 Ringcrg 20190 Poly1cpl1 22124 deg1 cdg1 26036 Unic1pcuc1p 26112 quot1pcq1p 26113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-sup 9472 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14008 df-hash 14331 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-starv 17256 df-sca 17257 df-vsca 17258 df-ip 17259 df-tset 17260 df-ple 17261 df-ds 17263 df-unif 17264 df-hom 17265 df-cco 17266 df-0g 17431 df-gsum 17432 df-prds 17437 df-pws 17439 df-mre 17574 df-mrc 17575 df-acs 17577 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18748 df-submnd 18749 df-grp 18906 df-minusg 18907 df-sbg 18908 df-mulg 19037 df-subg 19091 df-ghm 19181 df-cntz 19285 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-cring 20193 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-subrng 20500 df-subrg 20525 df-lmod 20762 df-lss 20833 df-rlreg 21252 df-cnfld 21302 df-psr 21864 df-mvr 21865 df-mpl 21866 df-opsr 21868 df-psr1 22127 df-vr1 22128 df-ply1 22129 df-coe1 22130 df-mdeg 26037 df-deg1 26038 df-uc1p 26117 df-q1p 26118 |
This theorem is referenced by: r1pcl 26144 r1pid 26146 dvdsq1p 26147 dvdsr1p 26148 ply1rem 26150 fta1glem1 26152 fta1glem2 26153 ig1pdvds 26164 q1pdir 33406 q1pvsca 33407 r1pvsca 33408 r1pcyc 33410 r1padd1 33411 r1pid2 33412 |
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