Theorem prmidl 21627

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 3316	. . . . 5 ⊢ (𝑏 = 𝐽 → (∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
2	1	ralbidv 3185	. . . 4 ⊢ (𝑏 = 𝐽 → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃))
3		sseq1 3933	. . . . 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 3316	. . . . . 6 ⊢ (𝑎 = 𝐼 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃))
7		sseq1 3933	. . . . . . 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 3185	. . . 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 21625	. . . . . . 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 3577	. . 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 3577	. 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   ⊆ wss 3895  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  ._rcmulr 17435  Ringcrg 20465  LIdealclidl 21490  PrmIdealcprmidl 21622

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-prmidl 21623

This theorem is used by:  idlmulssprm  21629  isprmidlc  21634