Theorem ispridl2 39048

Description: Obsolete theorem, use prmidl2 21623 instead. A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 39080 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 39026	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟 ⊆ 𝑋)
4		ssralv 3977	. . . . . . . . . . . . 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 39026	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠 ⊆ 𝑋)
8		ssralv 3977	. . . . . . . . . . . . . 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 3897	. . . . . . . . . . . . . 14 ⊢ (𝑟 ⊆ 𝑃 ↔ ∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃)
19		dfss3 3897	. . . . . . . . . . . . . 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 39044	. . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   ⊆ wss 3895  ran crn 5651  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  RingOpscrngo 38904  Idlcidl 39017  PrIdlcpridl 39018

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-idl 39020  df-pridl 39021

This theorem is used by:  ispridlc  39080