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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 6906 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → (𝐺‘𝐼) = (𝐺‘{(0g‘𝑃)})) | |
| 2 | id 22 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → 𝐼 = {(0g‘𝑃)}) | |
| 3 | 1, 2 | eleq12d 2835 | . 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 2737 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 7 | ig1pcl.u | . . . . 5 ⊢ 𝑈 = (LIdeal‘𝑃) | |
| 8 | eqid 2737 | . . . . 5 ⊢ (deg1‘𝑅) = (deg1‘𝑅) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
| 10 | 4, 5, 6, 7, 8, 9 | ig1pval3 26217 | . . . 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 1143 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
| 12 | 11 | 3expa 1119 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
| 13 | drngring 20736 | . . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
| 14 | 4, 5, 6 | ig1pval2 26216 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
| 15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
| 16 | fvex 6919 | . . . . 5 ⊢ (𝐺‘{(0g‘𝑃)}) ∈ V | |
| 17 | 16 | elsn 4641 | . . . 4 ⊢ ((𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)} ↔ (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
| 18 | 15, 17 | sylibr 234 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)}) |
| 19 | 18 | adantr 480 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)}) |
| 20 | 3, 12, 19 | pm2.61ne 3027 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 “ cima 5688 ‘cfv 6561 infcinf 9481 ℝcr 11154 < clt 11295 0gc0g 17484 Ringcrg 20230 DivRingcdr 20729 LIdealclidl 21216 Poly1cpl1 22178 deg1cdg1 26093 Monic1pcmn1 26165 idlGen1pcig1p 26169 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-drng 20731 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-cnfld 21365 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-mdeg 26094 df-deg1 26095 df-mon1 26170 df-uc1p 26171 df-ig1p 26174 |
| This theorem is referenced by: ig1pdvds 26219 ig1prsp 26220 ply1lpir 26221 ig1pnunit 33621 minplycl 33749 minplyann 33752 minplyirred 33754 irngnminplynz 33755 |
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