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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qsfld | Structured version Visualization version GIF version |
Description: An ideal 𝑀 in the commutative ring 𝑅 is maximal if and only if the factor ring 𝑄 is a field. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
Ref | Expression |
---|---|
qsfld.1 | ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) |
qsfld.2 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
qsfld.3 | ⊢ (𝜑 → 𝑅 ∈ NzRing) |
qsfld.4 | ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) |
Ref | Expression |
---|---|
qsfld | ⊢ (𝜑 → (𝑄 ∈ Field ↔ 𝑀 ∈ (MaxIdeal‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . 3 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
2 | qsfld.1 | . . 3 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) | |
3 | qsfld.3 | . . 3 ⊢ (𝜑 → 𝑅 ∈ NzRing) | |
4 | qsfld.4 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) | |
5 | qsfld.2 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
6 | eqid 2740 | . . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
7 | 6 | crng2idl 21314 | . . . . 5 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅)) |
8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (LIdeal‘𝑅) = (2Ideal‘𝑅)) |
9 | 4, 8 | eleqtrd 2846 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅)) |
10 | 1, 2, 3, 9 | qsdrng 33490 | . 2 ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘(oppr‘𝑅))))) |
11 | isfld 20762 | . . 3 ⊢ (𝑄 ∈ Field ↔ (𝑄 ∈ DivRing ∧ 𝑄 ∈ CRing)) | |
12 | 2, 6 | quscrng 21316 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → 𝑄 ∈ CRing) |
13 | 5, 4, 12 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ CRing) |
14 | 13 | biantrud 531 | . . 3 ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑄 ∈ DivRing ∧ 𝑄 ∈ CRing))) |
15 | 11, 14 | bitr4id 290 | . 2 ⊢ (𝜑 → (𝑄 ∈ Field ↔ 𝑄 ∈ DivRing)) |
16 | eqid 2740 | . . . . . . 7 ⊢ (MaxIdeal‘𝑅) = (MaxIdeal‘𝑅) | |
17 | 16, 1 | crngmxidl 33462 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (MaxIdeal‘𝑅) = (MaxIdeal‘(oppr‘𝑅))) |
18 | 5, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (MaxIdeal‘𝑅) = (MaxIdeal‘(oppr‘𝑅))) |
19 | 18 | eleq2d 2830 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑀 ∈ (MaxIdeal‘(oppr‘𝑅)))) |
20 | 19 | biimpd 229 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑅)))) |
21 | 20 | pm4.71d 561 | . 2 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘(oppr‘𝑅))))) |
22 | 10, 15, 21 | 3bitr4d 311 | 1 ⊢ (𝜑 → (𝑄 ∈ Field ↔ 𝑀 ∈ (MaxIdeal‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ‘cfv 6573 (class class class)co 7448 /s cqus 17565 ~QG cqg 19162 CRingccrg 20261 opprcoppr 20359 NzRingcnzr 20538 DivRingcdr 20751 Fieldcfield 20752 LIdealclidl 21239 2Idealc2idl 21282 MaxIdealcmxidl 33452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-imas 17568 df-qus 17569 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-nzr 20539 df-subrg 20597 df-drng 20753 df-field 20754 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lmhm 21044 df-lbs 21097 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-dsmm 21775 df-frlm 21790 df-uvc 21826 df-mxidl 33453 |
This theorem is referenced by: mxidlprmALT 33492 |
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