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| Mirrors > Home > MPE Home > Th. List > ig1pcl | Structured version Visualization version GIF version | ||
| Description: The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.) |
| Ref | Expression |
|---|---|
| ig1pval.p | β’ π = (Poly1βπ ) |
| ig1pval.g | β’ πΊ = (idlGen1pβπ ) |
| ig1pcl.u | β’ π = (LIdealβπ) |
| Ref | Expression |
|---|---|
| ig1pcl | β’ ((π β DivRing β§ πΌ β π) β (πΊβπΌ) β πΌ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6891 | . . 3 β’ (πΌ = {(0gβπ)} β (πΊβπΌ) = (πΊβ{(0gβπ)})) | |
| 2 | id 22 | . . 3 β’ (πΌ = {(0gβπ)} β πΌ = {(0gβπ)}) | |
| 3 | 1, 2 | eleq12d 2827 | . 2 β’ (πΌ = {(0gβπ)} β ((πΊβπΌ) β πΌ β (πΊβ{(0gβπ)}) β {(0gβπ)})) |
| 4 | ig1pval.p | . . . . 5 β’ π = (Poly1βπ ) | |
| 5 | ig1pval.g | . . . . 5 β’ πΊ = (idlGen1pβπ ) | |
| 6 | eqid 2732 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
| 7 | ig1pcl.u | . . . . 5 β’ π = (LIdealβπ) | |
| 8 | eqid 2732 | . . . . 5 β’ ( deg1 βπ ) = ( deg1 βπ ) | |
| 9 | eqid 2732 | . . . . 5 β’ (Monic1pβπ ) = (Monic1pβπ ) | |
| 10 | 4, 5, 6, 7, 8, 9 | ig1pval3 25916 | . . . 4 β’ ((π β DivRing β§ πΌ β π β§ πΌ β {(0gβπ)}) β ((πΊβπΌ) β πΌ β§ (πΊβπΌ) β (Monic1pβπ ) β§ (( deg1 βπ )β(πΊβπΌ)) = inf((( deg1 βπ ) β (πΌ β {(0gβπ)})), β, < ))) |
| 11 | 10 | simp1d 1142 | . . 3 β’ ((π β DivRing β§ πΌ β π β§ πΌ β {(0gβπ)}) β (πΊβπΌ) β πΌ) |
| 12 | 11 | 3expa 1118 | . 2 β’ (((π β DivRing β§ πΌ β π) β§ πΌ β {(0gβπ)}) β (πΊβπΌ) β πΌ) |
| 13 | drngring 20507 | . . . . 5 β’ (π β DivRing β π β Ring) | |
| 14 | 4, 5, 6 | ig1pval2 25915 | . . . . 5 β’ (π β Ring β (πΊβ{(0gβπ)}) = (0gβπ)) |
| 15 | 13, 14 | syl 17 | . . . 4 β’ (π β DivRing β (πΊβ{(0gβπ)}) = (0gβπ)) |
| 16 | fvex 6904 | . . . . 5 β’ (πΊβ{(0gβπ)}) β V | |
| 17 | 16 | elsn 4643 | . . . 4 β’ ((πΊβ{(0gβπ)}) β {(0gβπ)} β (πΊβ{(0gβπ)}) = (0gβπ)) |
| 18 | 15, 17 | sylibr 233 | . . 3 β’ (π β DivRing β (πΊβ{(0gβπ)}) β {(0gβπ)}) |
| 19 | 18 | adantr 481 | . 2 β’ ((π β DivRing β§ πΌ β π) β (πΊβ{(0gβπ)}) β {(0gβπ)}) |
| 20 | 3, 12, 19 | pm2.61ne 3027 | 1 β’ ((π β DivRing β§ πΌ β π) β (πΊβπΌ) β πΌ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 β cdif 3945 {csn 4628 β cima 5679 βcfv 6543 infcinf 9438 βcr 11111 < clt 11252 0gc0g 17389 Ringcrg 20127 DivRingcdr 20500 LIdealclidl 20928 Poly1cpl1 21920 deg1 cdg1 25793 Monic1pcmn1 25867 idlGen1pcig1p 25871 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-subrng 20434 df-subrg 20459 df-drng 20502 df-lmod 20616 df-lss 20687 df-sra 20930 df-rgmod 20931 df-lidl 20932 df-rlreg 21099 df-cnfld 21145 df-ascl 21629 df-psr 21681 df-mvr 21682 df-mpl 21683 df-opsr 21685 df-psr1 21923 df-vr1 21924 df-ply1 21925 df-coe1 21926 df-mdeg 25794 df-deg1 25795 df-mon1 25872 df-uc1p 25873 df-ig1p 25876 |
| This theorem is referenced by: ig1pdvds 25918 ig1prsp 25919 ply1lpir 25920 ig1pnunit 32934 minplycl 33044 minplyirred 33047 irngnminplynz 33048 |
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