isprmidlc - Mathbox for Thierry Arnoux

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Theorem isprmidlc 33001

Description: The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)

**Hypotheses**
Ref	Expression
isprmidlc.1	⊢ 𝐵 = (Base‘𝑅)
isprmidlc.2	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
isprmidlc	⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))

Distinct variable groups: 𝑥,𝐵,𝑦 𝑥,𝑃,𝑦 𝑥,𝑅,𝑦 Allowed substitution hints: · (𝑥,𝑦)

Proof of Theorem isprmidlc Dummy variables 𝑚 𝑛 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20139	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		prmidlidl 32997	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
3	1, 2	sylan 579	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
4		isprmidlc.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
5		isprmidlc.2	. . . . 5 ⊢ · = (._r‘𝑅)
6	4, 5	prmidlnr 32992	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ≠ 𝐵)
7	1, 6	sylan 579	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ≠ 𝐵)
8	1	ad4antr 729	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → 𝑅 ∈ Ring)
9		simp-4r 781	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → 𝑃 ∈ (PrmIdeal‘𝑅))
10		simpllr 773	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → 𝑥 ∈ 𝐵)
11	10	snssd 4804	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → {𝑥} ⊆ 𝐵)
12		eqid 2724	. . . . . . . . . . 11 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
13		eqid 2724	. . . . . . . . . . 11 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	12, 4, 13	rspcl 21083	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ {𝑥} ⊆ 𝐵) → ((RSpan‘𝑅)‘{𝑥}) ∈ (LIdeal‘𝑅))
15	8, 11, 14	syl2anc 583	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → ((RSpan‘𝑅)‘{𝑥}) ∈ (LIdeal‘𝑅))
16		simplr 766	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → 𝑦 ∈ 𝐵)
17	16	snssd 4804	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → {𝑦} ⊆ 𝐵)
18	12, 4, 13	rspcl 21083	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ {𝑦} ⊆ 𝐵) → ((RSpan‘𝑅)‘{𝑦}) ∈ (LIdeal‘𝑅))
19	8, 17, 18	syl2anc 583	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → ((RSpan‘𝑅)‘{𝑦}) ∈ (LIdeal‘𝑅))
20	15, 19	jca 511	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → (((RSpan‘𝑅)‘{𝑥}) ∈ (LIdeal‘𝑅) ∧ ((RSpan‘𝑅)‘{𝑦}) ∈ (LIdeal‘𝑅)))
21		simpllr 773	. . . . . . . . . . . . . 14 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑟 = (𝑚 · 𝑥))
22		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑠 = (𝑛 · 𝑦))
23	21, 22	oveq12d 7419	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → (𝑟 · 𝑠) = ((𝑚 · 𝑥) · (𝑛 · 𝑦)))
24		simp-10l 792	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑅 ∈ CRing)
25		simp-4r 781	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑚 ∈ 𝐵)
26	10	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) → 𝑥 ∈ 𝐵)
27	26	ad4antr 729	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑥 ∈ 𝐵)
28		simplr 766	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑛 ∈ 𝐵)
29	16	ad4antr 729	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) → 𝑦 ∈ 𝐵)
30	29	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑦 ∈ 𝐵)
31	4, 5	cringm4 33000	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ (𝑚 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑛 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑚 · 𝑥) · (𝑛 · 𝑦)) = ((𝑚 · 𝑛) · (𝑥 · 𝑦)))
32	24, 25, 27, 28, 30, 31	syl122anc 1376	. . . . . . . . . . . . . 14 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → ((𝑚 · 𝑥) · (𝑛 · 𝑦)) = ((𝑚 · 𝑛) · (𝑥 · 𝑦)))
33	24, 1	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑅 ∈ Ring)
34	3	ad9antr 739	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → 𝑃 ∈ (LIdeal‘𝑅))
35	4, 5	ringcl 20144	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝑚 · 𝑛) ∈ 𝐵)
36	33, 25, 28, 35	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → (𝑚 · 𝑛) ∈ 𝐵)
37		simp-7r 787	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → (𝑥 · 𝑦) ∈ 𝑃)
38	13, 4, 5	lidlmcl 21073	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ ((𝑚 · 𝑛) ∈ 𝐵 ∧ (𝑥 · 𝑦) ∈ 𝑃)) → ((𝑚 · 𝑛) · (𝑥 · 𝑦)) ∈ 𝑃)
39	33, 34, 36, 37, 38	syl22anc 836	. . . . . . . . . . . . . 14 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → ((𝑚 · 𝑛) · (𝑥 · 𝑦)) ∈ 𝑃)
40	32, 39	eqeltrd 2825	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → ((𝑚 · 𝑥) · (𝑛 · 𝑦)) ∈ 𝑃)
41	23, 40	eqeltrd 2825	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) ∧ 𝑛 ∈ 𝐵) ∧ 𝑠 = (𝑛 · 𝑦)) → (𝑟 · 𝑠) ∈ 𝑃)
42	8	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) → 𝑅 ∈ Ring)
43	42	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) → 𝑅 ∈ Ring)
44		simpllr 773	. . . . . . . . . . . . 13 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) → 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦}))
45	4, 5, 12	rspsnel 32919	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑠 ∈ ((RSpan‘𝑅)‘{𝑦}) ↔ ∃𝑛 ∈ 𝐵 𝑠 = (𝑛 · 𝑦)))
46	45	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) → ∃𝑛 ∈ 𝐵 𝑠 = (𝑛 · 𝑦))
47	43, 29, 44, 46	syl21anc 835	. . . . . . . . . . . 12 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) → ∃𝑛 ∈ 𝐵 𝑠 = (𝑛 · 𝑦))
48	41, 47	r19.29a 3154	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) ∧ 𝑚 ∈ 𝐵) ∧ 𝑟 = (𝑚 · 𝑥)) → (𝑟 · 𝑠) ∈ 𝑃)
49		simplr 766	. . . . . . . . . . . 12 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) → 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥}))
50	4, 5, 12	rspsnel 32919	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑟 ∈ ((RSpan‘𝑅)‘{𝑥}) ↔ ∃𝑚 ∈ 𝐵 𝑟 = (𝑚 · 𝑥)))
51	50	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) → ∃𝑚 ∈ 𝐵 𝑟 = (𝑚 · 𝑥))
52	42, 26, 49, 51	syl21anc 835	. . . . . . . . . . 11 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) → ∃𝑚 ∈ 𝐵 𝑟 = (𝑚 · 𝑥))
53	48, 52	r19.29a 3154	. . . . . . . . . 10 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ 𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})) → (𝑟 · 𝑠) ∈ 𝑃)
54	53	anasss 466	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) ∧ (𝑟 ∈ ((RSpan‘𝑅)‘{𝑥}) ∧ 𝑠 ∈ ((RSpan‘𝑅)‘{𝑦}))) → (𝑟 · 𝑠) ∈ 𝑃)
55	54	ralrimivva 3192	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → ∀𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})∀𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})(𝑟 · 𝑠) ∈ 𝑃)
56	4, 5	prmidl 32993	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (((RSpan‘𝑅)‘{𝑥}) ∈ (LIdeal‘𝑅) ∧ ((RSpan‘𝑅)‘{𝑦}) ∈ (LIdeal‘𝑅))) ∧ ∀𝑟 ∈ ((RSpan‘𝑅)‘{𝑥})∀𝑠 ∈ ((RSpan‘𝑅)‘{𝑦})(𝑟 · 𝑠) ∈ 𝑃) → (((RSpan‘𝑅)‘{𝑥}) ⊆ 𝑃 ∨ ((RSpan‘𝑅)‘{𝑦}) ⊆ 𝑃))
57	8, 9, 20, 55, 56	syl1111anc 837	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → (((RSpan‘𝑅)‘{𝑥}) ⊆ 𝑃 ∨ ((RSpan‘𝑅)‘{𝑦}) ⊆ 𝑃))
58	4, 12	rspsnid 32920	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ ((RSpan‘𝑅)‘{𝑥}))
59	1, 58	sylan 579	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ ((RSpan‘𝑅)‘{𝑥}))
60	59	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ((RSpan‘𝑅)‘{𝑥}))
61		ssel 3967	. . . . . . . . . . 11 ⊢ (((RSpan‘𝑅)‘{𝑥}) ⊆ 𝑃 → (𝑥 ∈ ((RSpan‘𝑅)‘{𝑥}) → 𝑥 ∈ 𝑃))
62	60, 61	syl5com 31	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((RSpan‘𝑅)‘{𝑥}) ⊆ 𝑃 → 𝑥 ∈ 𝑃))
63	4, 12	rspsnid 32920	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((RSpan‘𝑅)‘{𝑦}))
64	1, 63	sylan 579	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((RSpan‘𝑅)‘{𝑦}))
65	64	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((RSpan‘𝑅)‘{𝑦}))
66		ssel 3967	. . . . . . . . . . 11 ⊢ (((RSpan‘𝑅)‘{𝑦}) ⊆ 𝑃 → (𝑦 ∈ ((RSpan‘𝑅)‘{𝑦}) → 𝑦 ∈ 𝑃))
67	65, 66	syl5com 31	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((RSpan‘𝑅)‘{𝑦}) ⊆ 𝑃 → 𝑦 ∈ 𝑃))
68	62, 67	orim12d 961	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((((RSpan‘𝑅)‘{𝑥}) ⊆ 𝑃 ∨ ((RSpan‘𝑅)‘{𝑦}) ⊆ 𝑃) → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))
69	68	adantllr 716	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((((RSpan‘𝑅)‘{𝑥}) ⊆ 𝑃 ∨ ((RSpan‘𝑅)‘{𝑦}) ⊆ 𝑃) → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))
70	69	adantr 480	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → ((((RSpan‘𝑅)‘{𝑥}) ⊆ 𝑃 ∨ ((RSpan‘𝑅)‘{𝑦}) ⊆ 𝑃) → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))
71	57, 70	mpd 15	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 · 𝑦) ∈ 𝑃) → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃))
72	71	ex 412	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))
73	72	anasss 466	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))
74	73	ralrimivva 3192	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))
75	3, 7, 74	3jca 1125	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃))))
76		3anass 1092	. . . 4 ⊢ ((𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃))) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))
77	4, 5	prmidl2 32994	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
78	77	anasss 466	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃))))) → 𝑃 ∈ (PrmIdeal‘𝑅))
79	76, 78	sylan2b 593	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
80	1, 79	sylan 579	. 2 ⊢ ((𝑅 ∈ CRing ∧ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
81	75, 80	impbida 798	1 ⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∃wrex 3062 ⊆ wss 3940 {csn 4620 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 ._rcmulr 17196 Ringcrg 20127 CRingccrg 20128 LIdealclidl 21054 RSpancrsp 21055 PrmIdealcprmidl 32988

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-ip 17213 df-0g 17385 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-subrg 20460 df-lmod 20697 df-lss 20768 df-lsp 20808 df-sra 21010 df-rgmod 21011 df-lidl 21056 df-rsp 21057 df-prmidl 32989

This theorem is referenced by: prmidlc 33002 prmidl0 33004 qsidomlem2 33007

W3C validator