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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 6901 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → (𝐺‘𝐼) = (𝐺‘{(0g‘𝑃)})) | |
2 | id 22 | . . 3 ⊢ (𝐼 = {(0g‘𝑃)} → 𝐼 = {(0g‘𝑃)}) | |
3 | 1, 2 | eleq12d 2820 | . 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 2726 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
7 | ig1pcl.u | . . . . 5 ⊢ 𝑈 = (LIdeal‘𝑃) | |
8 | eqid 2726 | . . . . 5 ⊢ (deg1‘𝑅) = (deg1‘𝑅) | |
9 | eqid 2726 | . . . . 5 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
10 | 4, 5, 6, 7, 8, 9 | ig1pval3 26205 | . . . 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 1139 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
12 | 11 | 3expa 1115 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) ∧ 𝐼 ≠ {(0g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼) |
13 | drngring 20714 | . . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
14 | 4, 5, 6 | ig1pval2 26204 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
16 | fvex 6914 | . . . . 5 ⊢ (𝐺‘{(0g‘𝑃)}) ∈ V | |
17 | 16 | elsn 4648 | . . . 4 ⊢ ((𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)} ↔ (𝐺‘{(0g‘𝑃)}) = (0g‘𝑃)) |
18 | 15, 17 | sylibr 233 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝐺‘{(0g‘𝑃)}) ∈ {(0g‘𝑃)}) |
19 | 18 | adantr 479 | . 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 {csn 4633 “ cima 5685 ‘cfv 6554 infcinf 9484 ℝcr 11157 < clt 11298 0gc0g 17454 Ringcrg 20216 DivRingcdr 20707 LIdealclidl 21195 Poly1cpl1 22166 deg1cdg1 26078 Monic1pcmn1 26153 idlGen1pcig1p 26157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-ofr 7691 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-subrng 20528 df-subrg 20553 df-rlreg 20672 df-drng 20709 df-lmod 20838 df-lss 20909 df-sra 21151 df-rgmod 21152 df-lidl 21197 df-cnfld 21344 df-ascl 21853 df-psr 21906 df-mvr 21907 df-mpl 21908 df-opsr 21910 df-psr1 22169 df-vr1 22170 df-ply1 22171 df-coe1 22172 df-mdeg 26079 df-deg1 26080 df-mon1 26158 df-uc1p 26159 df-ig1p 26162 |
This theorem is referenced by: ig1pdvds 26207 ig1prsp 26208 ply1lpir 26209 ig1pnunit 33468 minplycl 33575 minplyann 33578 minplyirred 33580 irngnminplynz 33581 |
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