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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 6695 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → (𝐺‘𝐼) = (𝐺‘{(0g‘𝑃)})) | |
2 | id 22 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → 𝐼 = {(0g‘𝑃)}) | |
3 | 1, 2 | eleq12d 2825 | . 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 2736 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
7 | ig1pcl.u | . . . . 5 ⊢ 𝑈 = (LIdeal‘𝑃) | |
8 | eqid 2736 | . . . . 5 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
9 | eqid 2736 | . . . . 5 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
10 | 4, 5, 6, 7, 8, 9 | ig1pval3 25026 | . . . 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 1144 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
12 | 11 | 3expa 1120 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
13 | drngring 19728 | . . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
14 | 4, 5, 6 | ig1pval2 25025 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
16 | fvex 6708 | . . . . 5 ⊢ (𝐺‘{(0g‘𝑃)}) ∈ V | |
17 | 16 | elsn 4542 | . . . 4 ⊢ ((𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)} ↔ (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
18 | 15, 17 | sylibr 237 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)}) |
19 | 18 | adantr 484 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)}) |
20 | 3, 12, 19 | pm2.61ne 3017 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 {csn 4527 “ cima 5539 ‘cfv 6358 infcinf 9035 ℝcr 10693 < clt 10832 0gc0g 16898 Ringcrg 19516 DivRingcdr 19721 LIdealclidl 20161 Poly1cpl1 21052 deg1 cdg1 24903 Monic1pcmn1 24977 idlGen1pcig1p 24981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-drng 19723 df-subrg 19752 df-lmod 19855 df-lss 19923 df-sra 20163 df-rgmod 20164 df-lidl 20165 df-rlreg 20275 df-cnfld 20318 df-ascl 20771 df-psr 20822 df-mvr 20823 df-mpl 20824 df-opsr 20826 df-psr1 21055 df-vr1 21056 df-ply1 21057 df-coe1 21058 df-mdeg 24904 df-deg1 24905 df-mon1 24982 df-uc1p 24983 df-ig1p 24986 |
This theorem is referenced by: ig1pdvds 25028 ig1prsp 25029 ply1lpir 25030 |
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