Theorem opprmxidlabs 33567

Description: The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)

Hypotheses

Ref	Expression
oppreqg.o	⊢ 𝑂 = (oppr‘𝑅)
oppr2idl.2	⊢ (𝜑 → 𝑅 ∈ Ring)
opprmxidl.3	⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))

Assertion

Ref	Expression
opprmxidlabs	⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))

Proof of Theorem opprmxidlabs

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppr2idl.2	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		oppreqg.o	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
3	2	opprring 20316	. . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
4		eqid 2737	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
5	4	opprring 20316	. . 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 33543	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
10	1, 7, 9	syl2anc 585	. . 3 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
11	2, 1	opprlidlabs 33565	. . 3 ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
12	10, 11	eleqtrd 2839	. 2 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘(oppr‘𝑂)))
13	8	mxidlnr 33544	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
14	1, 7, 13	syl2anc 585	. 2 ⊢ (𝜑 → 𝑀 ≠ (Base‘𝑅))
15	1	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑅 ∈ Ring)
16	7	ad2antrr 727	. . . . 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 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
19	17, 18	eleqtrrd 2840	. . . . 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 33545	. . . . 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 3130	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (LIdeal‘(oppr‘𝑂))(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
25	2, 8	opprbas 20312	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂)
26	4, 25	opprbas 20312	. . . 4 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑂))
27	26	ismxidl 33542	. . 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 1375	1 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3890  ‘cfv 6490  Basecbs 17168  Ringcrg 20203  opp_rcoppr 20305  LIdealclidl 21194  MaxIdealcmxidl 33539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-sets 17123  df-slot 17141  df-ndx 17153  df-base 17169  df-ress 17190  df-plusg 17222  df-mulr 17223  df-sca 17225  df-vsca 17226  df-ip 17227  df-0g 17393  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18901  df-minusg 18902  df-cmn 19746  df-abl 19747  df-mgp 20111  df-rng 20123  df-ur 20152  df-ring 20205  df-oppr 20306  df-lss 20916  df-sra 21158  df-rgmod 21159  df-lidl 21196  df-mxidl 33540

This theorem is referenced by:  qsdrngi  33575