mxidlirred - Mathbox for Thierry Arnoux

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Theorem mxidlirred 33479

Description: In a principal ideal domain, maximal ideals are exactly the ideals generated by irreducible elements. (Contributed by Thierry Arnoux, 22-Mar-2025.)

**Hypotheses**
Ref	Expression
mxidlirred.b	⊢ 𝐵 = (Base‘𝑅)
mxidlirred.k	⊢ 𝐾 = (RSpan‘𝑅)
mxidlirred.0	⊢ 0 = (0_g‘𝑅)
mxidlirred.m	⊢ 𝑀 = (𝐾‘{𝑋})
mxidlirred.r	⊢ (𝜑 → 𝑅 ∈ PID)
mxidlirred.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mxidlirred.y	⊢ (𝜑 → 𝑋 ≠ 0 )
mxidlirred.1	⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))

**Assertion**
Ref	Expression
mxidlirred	⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))

Proof of Theorem mxidlirred Dummy variables 𝑡 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mxidlirred.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		mxidlirred.k	. . 3 ⊢ 𝐾 = (RSpan‘𝑅)
3		mxidlirred.0	. . 3 ⊢ 0 = (0_g‘𝑅)
4		mxidlirred.m	. . 3 ⊢ 𝑀 = (𝐾‘{𝑋})
5		mxidlirred.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ PID)
6		df-pid 21364	. . . . . 6 ⊢ PID = (IDomn ∩ LPIR)
7	5, 6	eleqtrdi 2848	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (IDomn ∩ LPIR))
8	7	elin1d 4213	. . . 4 ⊢ (𝜑 → 𝑅 ∈ IDomn)
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ IDomn)
10		mxidlirred.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ 𝐵)
12		mxidlirred.y	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 )
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ≠ 0 )
14		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
15	1, 2, 3, 4, 9, 11, 13, 14	mxidlirredi 33478	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ (Irred‘𝑅))
16		eqid 2734	. . . . . . . . . . 11 ⊢ (∥_r‘𝑅) = (∥_r‘𝑅)
17		simplr 769	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑥 ∈ 𝐵)
18	17	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑥 ∈ 𝐵)
19	10	ad8antr 740	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑋 ∈ 𝐵)
20		eqid 2734	. . . . . . . . . . 11 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
21		eqid 2734	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (._r‘𝑅)
22	8	idomringd 20744	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ad4antr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑅 ∈ Ring)
24	23	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑅 ∈ Ring)
25	24	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑅 ∈ Ring)
26		simplr 769	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑡 ∈ 𝐵)
27		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑋 = (𝑡(._r‘𝑅)𝑥))
28		simp-8r 792	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑋 ∈ (Irred‘𝑅))
29	27, 28	eqeltrrd 2839	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → (𝑡(._r‘𝑅)𝑥) ∈ (Irred‘𝑅))
30		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
31	30, 1, 20, 21	irredmul 20445	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ (𝑡(._r‘𝑅)𝑥) ∈ (Irred‘𝑅)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
32	26, 18, 29, 31	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
33		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑘 = (𝐾‘{𝑥}))
34	33	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑘 = (𝐾‘{𝑥}))
35		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
36		annim 403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ⊆ 𝑘 ∧ ¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)) ↔ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
37	35, 36	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → (𝑀 ⊆ 𝑘 ∧ ¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
38	37	simprd 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))
39		ioran 985	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵) ↔ (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
40	38, 39	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
41	40	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ 𝑘 = 𝐵)
42	41	neqned 2944	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ≠ 𝐵)
43	42	ad4antr 732	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑘 ≠ 𝐵)
44	34, 43	eqnetrrd 3006	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → (𝐾‘{𝑥}) ≠ 𝐵)
45	44	neneqd 2942	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → ¬ (𝐾‘{𝑥}) = 𝐵)
46		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (𝐾‘{𝑥}) = (𝐾‘{𝑥})
47	8	ad8antr 740	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑅 ∈ IDomn)
48	20, 2, 46, 1, 18, 47	unitpidl1 33431	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → ((𝐾‘{𝑥}) = 𝐵 ↔ 𝑥 ∈ (Unit‘𝑅)))
49	45, 48	mtbid 324	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → ¬ 𝑥 ∈ (Unit‘𝑅))
50	32, 49	olcnd 877	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑡 ∈ (Unit‘𝑅))
51	27	eqcomd 2740	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → (𝑡(._r‘𝑅)𝑥) = 𝑋)
52	1, 2, 16, 18, 19, 20, 21, 25, 50, 51	dvdsruassoi 33391	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → (𝑥(∥_r‘𝑅)𝑋 ∧ 𝑋(∥_r‘𝑅)𝑥))
53	1, 2, 16, 18, 19, 25	rspsnasso 33395	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → ((𝑥(∥_r‘𝑅)𝑋 ∧ 𝑋(∥_r‘𝑅)𝑥) ↔ (𝐾‘{𝑋}) = (𝐾‘{𝑥})))
54	52, 53	mpbid 232	. . . . . . . . 9 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → (𝐾‘{𝑋}) = (𝐾‘{𝑥}))
55	54, 34	eqtr4d 2777	. . . . . . . 8 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → (𝐾‘{𝑋}) = 𝑘)
56	4, 55	eqtr2id 2787	. . . . . . 7 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑘 = 𝑀)
57	40	simpld 494	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ 𝑘 = 𝑀)
58	57	ad4antr 732	. . . . . . 7 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → ¬ 𝑘 = 𝑀)
59	56, 58	pm2.21dd 195	. . . . . 6 ⊢ (((((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡 ∈ 𝐵) ∧ 𝑋 = (𝑡(._r‘𝑅)𝑥)) → 𝑀 ∈ (MaxIdeal‘𝑅))
60	37	simpld 494	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑀 ⊆ 𝑘)
61	60	ad2antrr 726	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑀 ⊆ 𝑘)
62	10	snssd 4813	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
63	2, 1	rspssid 21263	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → {𝑋} ⊆ (𝐾‘{𝑋}))
64	22, 62, 63	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
65	64, 4	sseqtrrdi 4046	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑋} ⊆ 𝑀)
66		snssg 4787	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝑀 ↔ {𝑋} ⊆ 𝑀))
67	66	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ {𝑋} ⊆ 𝑀) → 𝑋 ∈ 𝑀)
68	10, 65, 67	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
69	68	ad6antr 736	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋 ∈ 𝑀)
70	61, 69	sseldd 3995	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋 ∈ 𝑘)
71	70, 33	eleqtrd 2840	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋 ∈ (𝐾‘{𝑥}))
72	1, 21, 2	elrspsn 21267	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑋 ∈ (𝐾‘{𝑥}) ↔ ∃𝑡 ∈ 𝐵 𝑋 = (𝑡(._r‘𝑅)𝑥)))
73	72	biimpa 476	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑋 ∈ (𝐾‘{𝑥})) → ∃𝑡 ∈ 𝐵 𝑋 = (𝑡(._r‘𝑅)𝑥))
74	24, 17, 71, 73	syl21anc 838	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → ∃𝑡 ∈ 𝐵 𝑋 = (𝑡(._r‘𝑅)𝑥))
75	59, 74	r19.29a 3159	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑀 ∈ (MaxIdeal‘𝑅))
76		simplr 769	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ∈ (LIdeal‘𝑅))
77	7	elin2d 4214	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ LPIR)
78		eqid 2734	. . . . . . . . . . 11 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
79		eqid 2734	. . . . . . . . . . 11 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
80	78, 79	islpir 21355	. . . . . . . . . 10 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
81	80	simprbi 496	. . . . . . . . 9 ⊢ (𝑅 ∈ LPIR → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
82	77, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
83	82	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
84	76, 83	eleqtrd 2840	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ∈ (LPIdeal‘𝑅))
85	78, 2, 1	islpidl 21352	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑘 ∈ (LPIdeal‘𝑅) ↔ ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥})))
86	85	biimpa 476	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LPIdeal‘𝑅)) → ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥}))
87	23, 84, 86	syl2anc 584	. . . . 5 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥}))
88	75, 87	r19.29a 3159	. . . 4 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑀 ∈ (MaxIdeal‘𝑅))
89		mxidlirred.1	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
90	89	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
91	30, 20	irrednu 20441	. . . . . . . . . 10 ⊢ (𝑋 ∈ (Irred‘𝑅) → ¬ 𝑋 ∈ (Unit‘𝑅))
92	91	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → ¬ 𝑋 ∈ (Unit‘𝑅))
93	20, 2, 4, 1, 10, 8	unitpidl1 33431	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
94	93	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
95	94	necon3abid 2974	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
96	92, 95	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → 𝑀 ≠ 𝐵)
97	96	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
98	90, 97	jca 511	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵))
99	1	ismxidl 33469	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
100	22, 99	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
101		df-3an 1088	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ↔ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))))
102	100, 101	bitrdi 287	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
103	102	notbid 318	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑀 ∈ (MaxIdeal‘𝑅) ↔ ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
104	103	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))))
105	104	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))))
106	98, 105	mpnanrd 409	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
107		rexnal 3097	. . . . 5 ⊢ (∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)) ↔ ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
108	106, 107	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
109	88, 108	r19.29a 3159	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
110	109	pm2.18da 800	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
111	15, 110	impbida 801	1 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∩ cin 3961 ⊆ wss 3962 {csn 4630 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 ._rcmulr 17298 0_gc0g 17485 Ringcrg 20250 ∥_rcdsr 20370 Unitcui 20371 Irredcir 20372 IDomncidom 20709 LIdealclidl 21233 RSpancrsp 21234 LPIdealclpidl 21347 LPIRclpir 21348 PIDcpid 21363 MaxIdealcmxidl 33466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-irred 20375 df-invr 20404 df-nzr 20529 df-subrg 20586 df-domn 20711 df-idom 20712 df-lmod 20876 df-lss 20947 df-lsp 20987 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-rsp 21236 df-lpidl 21349 df-lpir 21350 df-pid 21364 df-mxidl 33467

This theorem is referenced by: rprmirredb 33539 algextdeglem4 33725

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