Theorem ispridl2 38235

Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38267 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
ispridl2.1	⊢ 𝐺 = (1st ‘𝑅)
ispridl2.2	⊢ 𝐻 = (2nd ‘𝑅)
ispridl2.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
ispridl2	⊢ ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))

Distinct variable groups:   𝑅,𝑎,𝑏   𝑃,𝑎,𝑏   𝑋,𝑎,𝑏

Allowed substitution hints:   𝐺(𝑎,𝑏)   𝐻(𝑎,𝑏)

Proof of Theorem ispridl2

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispridl2.1	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (1st ‘𝑅)
2		ispridl2.3	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran 𝐺
3	1, 2	idlss 38213	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟 ⊆ 𝑋)
4		ssralv 4002	. . . . . . . . . . . . 13 ⊢ (𝑟 ⊆ 𝑋 → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
5	3, 4	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
6	5	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
7	1, 2	idlss 38213	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠 ⊆ 𝑋)
8		ssralv 4002	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑋 → (∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
9	8	ralimdv 3150	. . . . . . . . . . . . 13 ⊢ (𝑠 ⊆ 𝑋 → (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
10	7, 9	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
11	10	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
12	6, 11	syld 47	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
13	12	adantlr 715	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
14		r19.26-2 3121	. . . . . . . . . . 11 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) ↔ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 ∧ ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
15		pm3.35 802	. . . . . . . . . . . . 13 ⊢ (((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
16	15	2ralimi 3106	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
17		2ralor 3210	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃 ∨ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃))
18		dfss3 3922	. . . . . . . . . . . . . 14 ⊢ (𝑟 ⊆ 𝑃 ↔ ∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃)
19		dfss3 3922	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑃 ↔ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃)
20	18, 19	orbi12i 914	. . . . . . . . . . . . 13 ⊢ ((𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃) ↔ (∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃 ∨ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃))
21	17, 20	sylbb2 238	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))
22	16, 21	syl 17	. . . . . . . . . . 11 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))
23	14, 22	sylbir 235	. . . . . . . . . 10 ⊢ ((∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 ∧ ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))
24	23	expcom 413	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃)))
25	13, 24	syl6 35	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))))
26	25	ralrimdvva 3191	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))))
27	26	ex 412	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (Idl‘𝑅) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃)))))
28	27	adantrd 491	. . . . 5 ⊢ (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃)))))
29	28	imdistand 570	. . . 4 ⊢ (𝑅 ∈ RingOps → (((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋) ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃)))))
30		df-3an 1088	. . . 4 ⊢ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
31		df-3an 1088	. . . 4 ⊢ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋) ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))))
32	29, 30, 31	3imtr4g 296	. . 3 ⊢ (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃)))))
33		ispridl2.2	. . . 4 ⊢ 𝐻 = (2nd ‘𝑅)
34	1, 33, 2	ispridl 38231	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃)))))
35	32, 34	sylibrd 259	. 2 ⊢ (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → 𝑃 ∈ (PrIdl‘𝑅)))
36	35	imp 406	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))

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-pow 5310  ax-pr 5377  ax-un 7680

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-idl 38207  df-pridl 38208

This theorem is referenced by:  ispridlc  38267