Theorem prmidl 31615

Description: The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
prmidl	⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑃,𝑦   𝑥,𝐼   𝑥,𝐽,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   · (𝑥,𝑦)   𝐼(𝑦)

Proof of Theorem prmidl

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3342	. . . . 5 ⊢ (𝑏 = 𝐽 → (∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
2	1	ralbidv 3112	. . . 4 ⊢ (𝑏 = 𝐽 → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
3		sseq1 3946	. . . . 5 ⊢ (𝑏 = 𝐽 → (𝑏 ⊆ 𝑃 ↔ 𝐽 ⊆ 𝑃))
4	3	orbi2d 913	. . . 4 ⊢ (𝑏 = 𝐽 → ((𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)))
5	2, 4	imbi12d 345	. . 3 ⊢ (𝑏 = 𝐽 → ((∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)) ↔ (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))))
6		raleq 3342	. . . . . 6 ⊢ (𝑎 = 𝐼 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃))
7		sseq1 3946	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝑃 ↔ 𝐼 ⊆ 𝑃))
8	7	orbi1d 914	. . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
9	6, 8	imbi12d 345	. . . . 5 ⊢ (𝑎 = 𝐼 → ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)) ↔ (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))
10	9	ralbidv 3112	. . . 4 ⊢ (𝑎 = 𝐼 → (∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)) ↔ ∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))
11		prmidlval.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
12		prmidlval.2	. . . . . . . 8 ⊢ · = (.r‘𝑅)
13	11, 12	isprmidl 31613	. . . . . . 7 ⊢ (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)))))
14	13	biimpa 477	. . . . . 6 ⊢ ((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))))
15	14	simp3d 1143	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
16	15	adantr 481	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
17		simprl 768	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐼 ∈ (LIdeal‘𝑅))
18	10, 16, 17	rspcdva 3562	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → ∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
19		simprr 770	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐽 ∈ (LIdeal‘𝑅))
20	5, 18, 19	rspcdva 3562	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)))
21	20	imp 407	1 ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ⊆ wss 3887  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  ._rcmulr 16963  Ringcrg 19783  LIdealclidl 20432  PrmIdealcprmidl 31610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-prmidl 31611

This theorem is referenced by:  idlmulssprm  31617  isprmidlc  31623