Theorem ispridl2 38210

Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38242 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 38188	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟 ⊆ 𝑋)
4		ssralv 4003	. . . . . . . . . . . . 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 718	. . . . . . . . . . 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 38188	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠 ⊆ 𝑋)
8		ssralv 4003	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑋 → (∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
9	8	ralimdv 3151	. . . . . . . . . . . . 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 717	. . . . . . . . . . 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 716	. . . . . . . . 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 3122	. . . . . . . . . . 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		pm3.35 803	. . . . . . . . . . . . 13 ⊢ (((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
16	15	2ralimi 3107	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
17		2ralor 3211	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃 ∨ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃))
18		dfss3 3923	. . . . . . . . . . . . . 14 ⊢ (𝑟 ⊆ 𝑃 ↔ ∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃)
19		dfss3 3923	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑃 ↔ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃)
20	18, 19	orbi12i 915	. . . . . . . . . . . . 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 3192	. . . . . . 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 1089	. . . 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 1089	. . . 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 38206	. . 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3902  ran crn 5626  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  RingOpscrngo 38066  Idlcidl 38179  PrIdlcpridl 38180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-idl 38182  df-pridl 38183

This theorem is referenced by:  ispridlc  38242