Theorem pridl 38548

Description: Obsolete theorem, use isprmidlc 21434 instead. The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 2765	. . . . . . 7 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		pridl.1	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
3		eqid 2765	. . . . . . 7 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
4	1, 2, 3	ispridl 38545	. . . . . 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 1103	. . . . . 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 290	. . . . 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 504	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
8		raleq 3320	. . . . . 6 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9		sseq1 3964	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝑃 ↔ 𝐴 ⊆ 𝑃))
10	9	orbi1d 929	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))
11	8, 10	imbi12d 347	. . . . 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 3320	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
13	12	ralbidv 3188	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14		sseq1 3964	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝑃 ↔ 𝐵 ⊆ 𝑃))
15	14	orbi2d 928	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃) ↔ (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃)))
16	13, 15	imbi12d 347	. . . . 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 3595	. . . 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 32	. . 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 420	. 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 1366	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 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079   ⊆ wss 3907  ran crn 5653  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  RingOpscrngo 38405  Idlcidl 38518  PrIdlcpridl 38519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-pridl 38522

This theorem is referenced by: (None)