Theorem mxidlmax 31539

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 3943	. . . 4 ⊢ (𝑗 = 𝐼 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝐼))
2		eqeq1 2742	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑀 ↔ 𝐼 = 𝑀))
3		eqeq1 2742	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝐵 ↔ 𝐼 = 𝐵))
4	2, 3	orbi12d 915	. . . 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 31536	. . . . . 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 1142	. . . 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 3554	. 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 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   ⊆ wss 3883  ‘cfv 6418  Basecbs 16840  Ringcrg 19698  LIdealclidl 20347  MaxIdealcmxidl 31533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-mxidl 31534

This theorem is referenced by:  mxidlprm  31542  zarclssn  31725