Theorem prmidl 33515

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 3293	. . . . 5 ⊢ (𝑏 = 𝐽 → (∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
2	1	ralbidv 3161	. . . 4 ⊢ (𝑏 = 𝐽 → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
3		sseq1 3948	. . . . 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 3293	. . . . . 6 ⊢ (𝑎 = 𝐼 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃))
7		sseq1 3948	. . . . . . 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 3161	. . . 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 33513	. . . . . . 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 3566	. . 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 3566	. 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 2933  ∀wral 3052   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  ._rcmulr 17212  Ringcrg 20205  LIdealclidl 21196  PrmIdealcprmidl 33510

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 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-prmidl 33511

This theorem is referenced by:  idlmulssprm  33517  isprmidlc  33522