Theorem pridl 38234

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 2736	. . . . . . 7 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		pridl.1	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
3		eqid 2736	. . . . . . 7 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
4	1, 2, 3	ispridl 38231	. . . . . 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 1088	. . . . . 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	bitrdi 287	. . . . 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 499	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
8		raleq 3293	. . . . . 6 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9		sseq1 3959	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝑃 ↔ 𝐴 ⊆ 𝑃))
10	9	orbi1d 916	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
11	8, 10	imbi12d 344	. . . . 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 3293	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
13	12	ralbidv 3159	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14		sseq1 3959	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝑃 ↔ 𝐵 ⊆ 𝑃))
15	14	orbi2d 915	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃)))
16	13, 15	imbi12d 344	. . . . 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 3587	. . . 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 415	. 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 1350	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 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051   ⊆ wss 3901  ran crn 5625  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  RingOpscrngo 38091  Idlcidl 38204  PrIdlcpridl 38205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-pridl 38208

This theorem is referenced by: (None)