Theorem mxidlmax 33485

Description: A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)

Hypothesis

Ref	Expression
mxidlval.1	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
mxidlmax	⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))

Proof of Theorem mxidlmax

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3990	. . . 4 ⊢ (𝑗 = 𝐼 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝐼))
2		eqeq1 2740	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑀 ↔ 𝐼 = 𝑀))
3		eqeq1 2740	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝐵 ↔ 𝐼 = 𝐵))
4	2, 3	orbi12d 918	. . . 4 ⊢ (𝑗 = 𝐼 → ((𝑗 = 𝑀 ∨ 𝑗 = 𝐵) ↔ (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)))
5	1, 4	imbi12d 344	. . 3 ⊢ (𝑗 = 𝐼 → ((𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)) ↔ (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))))
6		mxidlval.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
7	6	ismxidl 33482	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
8	7	biimpa 476	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))
9	8	simp3d 1144	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))
10	9	adantr 480	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))
11		simpr 484	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
12	5, 10, 11	rspcdva 3607	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)))
13	12	impr 454	1 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052   ⊆ wss 3931  ‘cfv 6536  Basecbs 17233  Ringcrg 20198  LIdealclidl 21172  MaxIdealcmxidl 33479

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-mxidl 33480

This theorem is referenced by:  mxidlmaxv  33488  mxidlprm  33490  opprmxidlabs  33507  zarclssn  33909