Theorem isprmidl 33409

Description: The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)

Hypotheses

Ref	Expression
prmidlval.1	⊢ 𝐵 = (Base‘𝑅)
prmidlval.2	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
isprmidl	⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))

Distinct variable groups:   𝑅,𝑎,𝑏,𝑥,𝑦   𝑃,𝑎,𝑏,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑎,𝑏)   · (𝑥,𝑦,𝑎,𝑏)

Proof of Theorem isprmidl

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmidlval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
2		prmidlval.2	. . . . 5 ⊢ · = (.r‘𝑅)
3	1, 2	prmidlval 33408	. . . 4 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
4	3	eleq2d 2814	. . 3 ⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ 𝑃 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))}))
5		neeq1 2987	. . . . 5 ⊢ (𝑖 = 𝑃 → (𝑖 ≠ 𝐵 ↔ 𝑃 ≠ 𝐵))
6		eleq2 2817	. . . . . . . 8 ⊢ (𝑖 = 𝑃 → ((𝑥 · 𝑦) ∈ 𝑖 ↔ (𝑥 · 𝑦) ∈ 𝑃))
7	6	2ralbidv 3201	. . . . . . 7 ⊢ (𝑖 = 𝑃 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 ↔ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃))
8		sseq2 3973	. . . . . . . 8 ⊢ (𝑖 = 𝑃 → (𝑎 ⊆ 𝑖 ↔ 𝑎 ⊆ 𝑃))
9		sseq2 3973	. . . . . . . 8 ⊢ (𝑖 = 𝑃 → (𝑏 ⊆ 𝑖 ↔ 𝑏 ⊆ 𝑃))
10	8, 9	orbi12d 918	. . . . . . 7 ⊢ (𝑖 = 𝑃 → ((𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖) ↔ (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
11	7, 10	imbi12d 344	. . . . . 6 ⊢ (𝑖 = 𝑃 → ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)) ↔ (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))
12	11	2ralbidv 3201	. . . . 5 ⊢ (𝑖 = 𝑃 → (∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)) ↔ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))
13	5, 12	anbi12d 632	. . . 4 ⊢ (𝑖 = 𝑃 → ((𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖))) ↔ (𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
14	13	elrab 3659	. . 3 ⊢ (𝑃 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))} ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
15	4, 14	bitrdi 287	. 2 ⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))))
16		3anass 1094	. 2 ⊢ ((𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
17	15, 16	bitr4di 289	1 ⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3405   ⊆ wss 3914  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  ._rcmulr 17221  Ringcrg 20142  LIdealclidl 21116  PrmIdealcprmidl 33406

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-prmidl 33407

This theorem is referenced by:  prmidlnr  33410  prmidl  33411  prmidl2  33412  prmidlidl  33415