Theorem ismxidl 33440

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 33439	. . 3 ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))})
3	2	eleq2d 2815	. 2 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))}))
4		neeq1 2988	. . . . 5 ⊢ (𝑖 = 𝑀 → (𝑖 ≠ 𝐵 ↔ 𝑀 ≠ 𝐵))
5		sseq1 3975	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
6		eqeq2 2742	. . . . . . . 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 3157	. . . . 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 3662	. . 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 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408   ⊆ wss 3917  ‘cfv 6514  Basecbs 17186  Ringcrg 20149  LIdealclidl 21123  MaxIdealcmxidl 33437

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-mxidl 33438

This theorem is referenced by:  mxidlidl  33441  mxidlnr  33442  mxidlmax  33443  crngmxidl  33447  mxidlirred  33450  ssmxidl  33452  drng0mxidl  33454  opprmxidlabs  33465  qsdrng  33475  zarclssn  33870