Theorem opprmxidlabs 33577

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 20325	. . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
4		eqid 2740	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
5	4	opprring 20325	. . 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 2740	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9	8	mxidlidl 33553	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
10	1, 7, 9	syl2anc 590	. . 3 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
11	2, 1	opprlidlabs 33575	. . 3 ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
12	10, 11	eleqtrd 2842	. 2 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘(oppr‘𝑂)))
13	8	mxidlnr 33554	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
14	1, 7, 13	syl2anc 590	. 2 ⊢ (𝜑 → 𝑀 ≠ (Base‘𝑅))
15	1	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑅 ∈ Ring)
16	7	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑀 ∈ (MaxIdeal‘𝑅))
17		simplr 774	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘(oppr‘𝑂)))
18	11	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
19	17, 18	eleqtrrd 2843	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘𝑅))
20		simpr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑀 ⊆ 𝑗)
21	8	mxidlmax 33555	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
22	15, 16, 19, 20, 21	syl22anc 844	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
23	22	ex 413	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
24	23	ralrimiva 3132	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (LIdeal‘(oppr‘𝑂))(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
25	2, 8	opprbas 20321	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂)
26	4, 25	opprbas 20321	. . . 4 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑂))
27	26	ismxidl 33552	. . 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 478	. 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 1380	1 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054   ⊆ wss 3890  ‘cfv 6492  Basecbs 17177  Ringcrg 20212  opp_rcoppr 20314  LIdealclidl 21206  MaxIdealcmxidl 33549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  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 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-tpos 8173  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-ip 17236  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-oppr 20315  df-lss 20929  df-sra 21170  df-rgmod 21171  df-lidl 21208  df-mxidl 33550

This theorem is referenced by:  qsdrngi  33585