Theorem opprmxidlabs 33549

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 20287	. . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
4		eqid 2737	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
5	4	opprring 20287	. . 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 33525	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
10	1, 7, 9	syl2anc 585	. . 3 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
11	2, 1	opprlidlabs 33547	. . 3 ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
12	10, 11	eleqtrd 2839	. 2 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘(oppr‘𝑂)))
13	8	mxidlnr 33526	. . 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 33527	. . . . 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 3129	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (LIdeal‘(oppr‘𝑂))(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
25	2, 8	opprbas 20283	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂)
26	4, 25	opprbas 20283	. . . 4 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑂))
27	26	ismxidl 33524	. . 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 3902  ‘cfv 6493  Basecbs 17140  Ringcrg 20172  opp_rcoppr 20276  LIdealclidl 21165  MaxIdealcmxidl 33521

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-sca 17197  df-vsca 17198  df-ip 17199  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18870  df-minusg 18871  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-oppr 20277  df-lss 20887  df-sra 21129  df-rgmod 21130  df-lidl 21167  df-mxidl 33522

This theorem is referenced by:  qsdrngi  33557