Theorem pridl 34444

Description: The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypothesis

Ref	Expression
pridl.1	⊢ 𝐻 = (2nd ‘𝑅)

Assertion

Ref	Expression
pridl	⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃))

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑃,𝑦   𝑥,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem pridl

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

Step	Hyp	Ref	Expression
1		eqid 2777	. . . . . . 7 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		pridl.1	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
3		eqid 2777	. . . . . . 7 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
4	1, 2, 3	ispridl 34441	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st ‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
5		df-3an 1073	. . . . . 6 ⊢ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st ‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st ‘𝑅)) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))
6	4, 5	syl6bb 279	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st ‘𝑅)) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
7	6	simplbda 495	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
8		raleq 3329	. . . . . 6 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9		sseq1 3844	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝑃 ↔ 𝐴 ⊆ 𝑃))
10	9	orbi1d 903	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
11	8, 10	imbi12d 336	. . . . 5 ⊢ (𝑎 = 𝐴 → ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))
12		raleq 3329	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
13	12	ralbidv 3167	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14		sseq1 3844	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝑃 ↔ 𝐵 ⊆ 𝑃))
15	14	orbi2d 902	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃)))
16	13, 15	imbi12d 336	. . . . 5 ⊢ (𝑏 = 𝐵 → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃))))
17	11, 16	rspc2v 3523	. . . 4 ⊢ ((𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅)) → (∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃))))
18	7, 17	syl5com 31	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ((𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃))))
19	18	expd 406	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (Idl‘𝑅) → (𝐵 ∈ (Idl‘𝑅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃)))))
20	19	3imp2 1411	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃))

Syntax hints:   → wi 4   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  ∀wral 3089   ⊆ wss 3791  ran crn 5356  ‘cfv 6135  (class class class)co 6922  1^st c1st 7443  2^nd c2nd 7444  RingOpscrngo 34301  Idlcidl 34414  PrIdlcpridl 34415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-pridl 34418

This theorem is referenced by: (None)