Theorem mxidlmax 33607

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 3957	. . . 4 ⊢ (𝑗 = 𝐼 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝐼))
2		eqeq1 2760	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑀 ↔ 𝐼 = 𝑀))
3		eqeq1 2760	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝐵 ↔ 𝐼 = 𝐵))
4	2, 3	orbi12d 927	. . . 4 ⊢ (𝑗 = 𝐼 → ((𝑗 = 𝑀 ∨ 𝑗 = 𝐵) ↔ (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)))
5	1, 4	imbi12d 346	. . 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 33604	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
8	7	biimpa 479	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))
9	8	simp3d 1153	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))
10	9	adantr 483	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))
11		simpr 487	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
12	5, 10, 11	rspcdva 3577	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)))
13	12	impr 457	1 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 856   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  ∀wral 3070   ⊆ wss 3899  ‘cfv 6510  Basecbs 17221  Ringcrg 20255  LIdealclidl 21249  MaxIdealcmxidl 33601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6512  df-fv 6518  df-mxidl 33602

This theorem is referenced by:  mxidlmaxv  33610  mxidlprm  33612  opprmxidlabs  33629  dflring3  33647  dflring4  33648  zarclssn  34124