|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > opprmxidlabs | Structured version Visualization version GIF version | ||
| Description: The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| oppreqg.o | ⊢ 𝑂 = (oppr‘𝑅) | 
| oppr2idl.2 | ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| opprmxidl.3 | ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) | 
| Ref | Expression | 
|---|---|
| opprmxidlabs | ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oppr2idl.2 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 2 | oppreqg.o | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
| 3 | 2 | opprring 20347 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) | 
| 4 | eqid 2737 | . . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂) | |
| 5 | 4 | opprring 20347 | . . 3 ⊢ (𝑂 ∈ Ring → (oppr‘𝑂) ∈ Ring) | 
| 6 | 1, 3, 5 | 3syl 18 | . 2 ⊢ (𝜑 → (oppr‘𝑂) ∈ Ring) | 
| 7 | opprmxidl.3 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) | |
| 8 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 9 | 8 | mxidlidl 33491 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅)) | 
| 10 | 1, 7, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) | 
| 11 | 2, 1 | opprlidlabs 33513 | . . 3 ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂))) | 
| 12 | 10, 11 | eleqtrd 2843 | . 2 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘(oppr‘𝑂))) | 
| 13 | 8 | mxidlnr 33492 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅)) | 
| 14 | 1, 7, 13 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝑀 ≠ (Base‘𝑅)) | 
| 15 | 1 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑅 ∈ Ring) | 
| 16 | 7 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑀 ∈ (MaxIdeal‘𝑅)) | 
| 17 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) | |
| 18 | 11 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂))) | 
| 19 | 17, 18 | eleqtrrd 2844 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘𝑅)) | 
| 20 | simpr 484 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑀 ⊆ 𝑗) | |
| 21 | 8 | mxidlmax 33493 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))) | 
| 22 | 15, 16, 19, 20, 21 | syl22anc 839 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))) | 
| 23 | 22 | ex 412 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))) | 
| 24 | 23 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑗 ∈ (LIdeal‘(oppr‘𝑂))(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))) | 
| 25 | 2, 8 | opprbas 20341 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) | 
| 26 | 4, 25 | opprbas 20341 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑂)) | 
| 27 | 26 | ismxidl 33490 | . . 3 ⊢ ((oppr‘𝑂) ∈ Ring → (𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)) ↔ (𝑀 ∈ (LIdeal‘(oppr‘𝑂)) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘(oppr‘𝑂))(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))))) | 
| 28 | 27 | biimpar 477 | . 2 ⊢ (((oppr‘𝑂) ∈ Ring ∧ (𝑀 ∈ (LIdeal‘(oppr‘𝑂)) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘(oppr‘𝑂))(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂))) | 
| 29 | 6, 12, 14, 24, 28 | syl13anc 1374 | 1 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3951 ‘cfv 6561 Basecbs 17247 Ringcrg 20230 opprcoppr 20333 LIdealclidl 21216 MaxIdealcmxidl 33487 | 
| 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 | 
| 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-op 4633 df-uni 4908 df-iun 4993 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 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-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-mxidl 33488 | 
| This theorem is referenced by: qsdrngi 33523 | 
| Copyright terms: Public domain | W3C validator |