Theorem ispridl2 38892

Description: Obsolete theorem, use prmidl2 21584 instead. A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38924 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 38870	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟 ⊆ 𝑋)
4		ssralv 3999	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 730	. . . . . . . . . . 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 38870	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠 ⊆ 𝑋)
8		ssralv 3999	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑋 → (∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
9	8	ralimdv 3176	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 729	. . . . . . . . . . 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 48	. . . . . . . . . 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 728	. . . . . . . . 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 3147	. . . . . . . . . . 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		pm3.35 815	. . . . . . . . . . . . 13 ⊢ (((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
16	15	2ralimi 3132	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
17		2ralor 3236	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃 ∨ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃))
18		dfss3 3919	. . . . . . . . . . . . . 14 ⊢ (𝑟 ⊆ 𝑃 ↔ ∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃)
19		dfss3 3919	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑃 ↔ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃)
20	18, 19	orbi12i 928	. . . . . . . . . . . . 13 ⊢ ((𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃) ↔ (∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃 ∨ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃))
21	17, 20	sylbb2 241	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))
22	16, 21	syl 18	. . . . . . . . . . 11 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))
23	14, 22	sylbir 238	. . . . . . . . . 10 ⊢ ((∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎𝐻𝑏) ∈ 𝑃 ∧ ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑟 ⊆ 𝑃 ∨ 𝑠 ⊆ 𝑃))
24	23	expcom 419	. . . . . . . . 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 36	. . . . . . . 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 3217	. . . . . . 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 418	. . . . . 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 497	. . . . 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 581	. . . 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 1105	. . . 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 1105	. . . 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 299	. . 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 38888	. . 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 262	. 2 ⊢ (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → 𝑃 ∈ (PrIdl‘𝑅)))
36	35	imp 412	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076   ⊆ wss 3898  ran crn 5648  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  RingOpscrngo 38748  Idlcidl 38861  PrIdlcpridl 38862

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6483  df-fun 6529  df-fv 6535  df-ov 7411  df-idl 38864  df-pridl 38865

This theorem is used by:  ispridlc  38924