Theorem ismxidl 33425

Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)

Hypothesis

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

Assertion

Ref	Expression
ismxidl	⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))

Distinct variable groups:   𝑅,𝑗   𝑗,𝑀

Allowed substitution hint:   𝐵(𝑗)

Proof of Theorem ismxidl

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mxidlval.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
2	1	mxidlval 33424	. . 3 ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))})
3	2	eleq2d 2817	. 2 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))}))
4		neeq1 2990	. . . . 5 ⊢ (𝑖 = 𝑀 → (𝑖 ≠ 𝐵 ↔ 𝑀 ≠ 𝐵))
5		sseq1 3960	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
6		eqeq2 2743	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑗 = 𝑖 ↔ 𝑗 = 𝑀))
7	6	orbi1d 916	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝑗 = 𝑖 ∨ 𝑗 = 𝐵) ↔ (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))
8	5, 7	imbi12d 344	. . . . . 6 ⊢ (𝑖 = 𝑀 → ((𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)) ↔ (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))
9	8	ralbidv 3155	. . . . 5 ⊢ (𝑖 = 𝑀 → (∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)) ↔ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))
10	4, 9	anbi12d 632	. . . 4 ⊢ (𝑖 = 𝑀 → ((𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵))) ↔ (𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
11	10	elrab 3647	. . 3 ⊢ (𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))} ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ (𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
12		3anass 1094	. . 3 ⊢ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ (𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
13	11, 12	bitr4i 278	. 2 ⊢ (𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))} ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))
14	3, 13	bitrdi 287	1 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  {crab 3395   ⊆ wss 3902  ‘cfv 6481  Basecbs 17120  Ringcrg 20152  LIdealclidl 21144  MaxIdealcmxidl 33422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-mxidl 33423

This theorem is referenced by:  mxidlidl  33426  mxidlnr  33427  mxidlmax  33428  crngmxidl  33432  mxidlirred  33435  ssmxidl  33437  drng0mxidl  33439  opprmxidlabs  33450  qsdrng  33460  zarclssn  33884