Theorem prmidl 21601

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 3317	. . . . 5 ⊢ (𝑏 = 𝐽 → (∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
2	1	ralbidv 3186	. . . 4 ⊢ (𝑏 = 𝐽 → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
3		sseq1 3956	. . . . 5 ⊢ (𝑏 = 𝐽 → (𝑏 ⊆ 𝑃 ↔ 𝐽 ⊆ 𝑃))
4	3	orbi2d 929	. . . 4 ⊢ (𝑏 = 𝐽 → ((𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)))
5	2, 4	imbi12d 347	. . 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 3317	. . . . . 6 ⊢ (𝑎 = 𝐼 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃))
7		sseq1 3956	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝑃 ↔ 𝐼 ⊆ 𝑃))
8	7	orbi1d 930	. . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
9	6, 8	imbi12d 347	. . . . 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 3186	. . . 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 21599	. . . . . . 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 482	. . . . . 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 1162	. . . . 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 486	. . . 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 783	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐼 ∈ (LIdeal‘𝑅))
18	10, 16, 17	rspcdva 3578	. . 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 785	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐽 ∈ (LIdeal‘𝑅))
20	5, 18, 19	rspcdva 3578	. 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 412	1 ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ⊆ wss 3899  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  ._rcmulr 17409  Ringcrg 20439  LIdealclidl 21464  PrmIdealcprmidl 21596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-prmidl 21597

This theorem is used by:  idlmulssprm  21603  isprmidlc  21608