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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 | ⊢ 𝑃 = (Poly1‘𝑅) |
ig1pval.g | ⊢ 𝐺 = (idlGen1p‘𝑅) |
ig1pcl.u | ⊢ 𝑈 = (LIdeal‘𝑃) |
Ref | Expression |
---|---|
ig1pcl | ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6446 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → (𝐺‘𝐼) = (𝐺‘{(0g‘𝑃)})) | |
2 | id 22 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → 𝐼 = {(0g‘𝑃)}) | |
3 | 1, 2 | eleq12d 2852 | . 2 ⊢ (𝐼 = {(0g‘𝑃)} → ((𝐺‘𝐼) ∈ 𝐼 ↔ (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)})) |
4 | ig1pval.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | ig1pval.g | . . . . 5 ⊢ 𝐺 = (idlGen1p‘𝑅) | |
6 | eqid 2777 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
7 | ig1pcl.u | . . . . 5 ⊢ 𝑈 = (LIdeal‘𝑃) | |
8 | eqid 2777 | . . . . 5 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
9 | eqid 2777 | . . . . 5 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
10 | 4, 5, 6, 7, 8, 9 | ig1pval3 24371 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ {(0g‘𝑃)}) → ((𝐺‘𝐼) ∈ 𝐼 ∧ (𝐺‘𝐼) ∈ (Monic1p‘𝑅) ∧ (( deg1 ‘𝑅)‘(𝐺‘𝐼)) = inf((( deg1 ‘𝑅) “ (𝐼 ∖ {(0g‘𝑃)})), ℝ, < ))) |
11 | 10 | simp1d 1133 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
12 | 11 | 3expa 1108 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
13 | drngring 19146 | . . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
14 | 4, 5, 6 | ig1pval2 24370 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
16 | fvex 6459 | . . . . 5 ⊢ (𝐺‘{(0g‘𝑃)}) ∈ V | |
17 | 16 | elsn 4412 | . . . 4 ⊢ ((𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)} ↔ (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
18 | 15, 17 | sylibr 226 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)}) |
19 | 18 | adantr 474 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)}) |
20 | 3, 12, 19 | pm2.61ne 3054 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∖ cdif 3788 {csn 4397 “ cima 5358 ‘cfv 6135 infcinf 8635 ℝcr 10271 < clt 10411 0gc0g 16486 Ringcrg 18934 DivRingcdr 19139 LIdealclidl 19567 Poly1cpl1 19943 deg1 cdg1 24251 Monic1pcmn1 24322 idlGen1pcig1p 24326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-gsum 16489 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-drng 19141 df-subrg 19170 df-lmod 19257 df-lss 19325 df-sra 19569 df-rgmod 19570 df-lidl 19571 df-rlreg 19680 df-ascl 19711 df-psr 19753 df-mvr 19754 df-mpl 19755 df-opsr 19757 df-psr1 19946 df-vr1 19947 df-ply1 19948 df-coe1 19949 df-cnfld 20143 df-mdeg 24252 df-deg1 24253 df-mon1 24327 df-uc1p 24328 df-ig1p 24331 |
This theorem is referenced by: ig1pdvds 24373 ig1prsp 24374 ply1lpir 24375 |
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