Theorem ispridl2 37998

Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38030 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 37976	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟 ⊆ 𝑋)
4		ssralv 4077	. . . . . . . . . . . . 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 716	. . . . . . . . . . 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 37976	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠 ⊆ 𝑋)
8		ssralv 4077	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑋 → (∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
9	8	ralimdv 3175	. . . . . . . . . . . . 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 715	. . . . . . . . . . 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 714	. . . . . . . . 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 3144	. . . . . . . . . . 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 3129	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) → ∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
17		2ralor 3237	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ 𝑟 ∀𝑏 ∈ 𝑠 (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃 ∨ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃))
18		dfss3 3997	. . . . . . . . . . . . . 14 ⊢ (𝑟 ⊆ 𝑃 ↔ ∀𝑎 ∈ 𝑟 𝑎 ∈ 𝑃)
19		dfss3 3997	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝑃 ↔ ∀𝑏 ∈ 𝑠 𝑏 ∈ 𝑃)
20	18, 19	orbi12i 913	. . . . . . . . . . . . 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 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 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 37994	. . 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 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067   ⊆ wss 3976  ran crn 5701  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  RingOpscrngo 37854  Idlcidl 37967  PrIdlcpridl 37968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-idl 37970  df-pridl 37971

This theorem is referenced by:  ispridlc  38030