Theorem opprmxidlabs 33686

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 20420	. . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
4		eqid 2765	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
5	4	opprring 20420	. . 3 ⊢ (𝑂 ∈ Ring → (oppr‘𝑂) ∈ Ring)
6	1, 3, 5	3syl 19	. 2 ⊢ (𝜑 → (oppr‘𝑂) ∈ Ring)
7		opprmxidl.3	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
8		eqid 2765	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9	8	mxidlidl 33663	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
10	1, 7, 9	syl2anc 595	. . 3 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
11	2, 1	opprlidlabs 33684	. . 3 ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
12	10, 11	eleqtrd 2867	. 2 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘(oppr‘𝑂)))
13	8	mxidlnr 33664	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
14	1, 7, 13	syl2anc 595	. 2 ⊢ (𝜑 → 𝑀 ≠ (Base‘𝑅))
15	1	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑅 ∈ Ring)
16	7	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑀 ∈ (MaxIdeal‘𝑅))
17		simplr 780	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘(oppr‘𝑂)))
18	11	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
19	17, 18	eleqtrrd 2868	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘𝑅))
20		simpr 489	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → 𝑀 ⊆ 𝑗)
21	8	mxidlmax 33665	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
22	15, 16, 19, 20, 21	syl22anc 851	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
23	22	ex 417	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (LIdeal‘(oppr‘𝑂))) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
24	23	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (LIdeal‘(oppr‘𝑂))(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
25	2, 8	opprbas 20416	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂)
26	4, 25	opprbas 20416	. . . 4 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑂))
27	26	ismxidl 33662	. . 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 482	. 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 1395	1 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079   ⊆ wss 3907  ‘cfv 6525  Basecbs 17259  Ringcrg 20306  opp_rcoppr 20409  LIdealclidl 21299  MaxIdealcmxidl 33659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-0g 17484  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-grp 18993  df-minusg 18994  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-oppr 20410  df-lss 21022  df-sra 21263  df-rgmod 21264  df-lidl 21301  df-mxidl 33660

This theorem is referenced by:  qsdrngi  33694