Theorem ismxidl 33455

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 33454	. . 3 ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))})
3	2	eleq2d 2830	. 2 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))}))
4		neeq1 3009	. . . . 5 ⊢ (𝑖 = 𝑀 → (𝑖 ≠ 𝐵 ↔ 𝑀 ≠ 𝐵))
5		sseq1 4034	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
6		eqeq2 2752	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑗 = 𝑖 ↔ 𝑗 = 𝑀))
7	6	orbi1d 915	. . . . . . 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 3184	. . . . 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 631	. . . 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 3708	. . 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 1095	. . 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 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443   ⊆ wss 3976  ‘cfv 6573  Basecbs 17258  Ringcrg 20260  LIdealclidl 21239  MaxIdealcmxidl 33452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-mxidl 33453

This theorem is referenced by:  mxidlidl  33456  mxidlnr  33457  mxidlmax  33458  crngmxidl  33462  mxidlirred  33465  ssmxidl  33467  drng0mxidl  33469  opprmxidlabs  33480  qsdrng  33490  zarclssn  33819