Theorem prmidl 33500

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 3292	. . . . 5 ⊢ (𝑏 = 𝐽 → (∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
2	1	ralbidv 3160	. . . 4 ⊢ (𝑏 = 𝐽 → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
3		sseq1 3947	. . . . 5 ⊢ (𝑏 = 𝐽 → (𝑏 ⊆ 𝑃 ↔ 𝐽 ⊆ 𝑃))
4	3	orbi2d 916	. . . 4 ⊢ (𝑏 = 𝐽 → ((𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)))
5	2, 4	imbi12d 344	. . 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 3292	. . . . . 6 ⊢ (𝑎 = 𝐼 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃))
7		sseq1 3947	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝑃 ↔ 𝐼 ⊆ 𝑃))
8	7	orbi1d 917	. . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐼 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
9	6, 8	imbi12d 344	. . . . 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 3160	. . . 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 33498	. . . . . . 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 476	. . . . . 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 1145	. . . . 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 480	. . . 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 771	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐼 ∈ (LIdeal‘𝑅))
18	10, 16, 17	rspcdva 3565	. . 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 773	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐽 ∈ (LIdeal‘𝑅))
20	5, 18, 19	rspcdva 3565	. 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 406	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 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051   ⊆ wss 3889  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  ._rcmulr 17221  Ringcrg 20214  LIdealclidl 21204  PrmIdealcprmidl 33495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-prmidl 33496

This theorem is referenced by:  idlmulssprm  33502  isprmidlc  33507