mxidlirred - Mathbox for Thierry Arnoux

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Theorem mxidlirred 33443

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 21247	. . . . . 6 ⊢ PID = (IDomn ∩ LPIR)
7	5, 6	eleqtrdi 2838	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (IDomn ∩ LPIR))
8	7	elin1d 4167	. . . 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 33442	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ (Irred‘𝑅))
16		eqid 2729	. . . . . . . . . . 11 ⊢ (∥_r‘𝑅) = (∥_r‘𝑅)
17		simplr 768	. . . . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
21		eqid 2729	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (._r‘𝑅)
22	8	idomringd 20637	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . 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 791	. . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
31	30, 1, 20, 21	irredmul 20338	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ (𝑡(._r‘𝑅)𝑥) ∈ (Irred‘𝑅)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
32	26, 18, 29, 31	syl3anc 1373	. . . . . . . . . . . 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 2932	. . . . . . . . . . . . . . . 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 2993	. . . . . . . . . . . . . 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 2930	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 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 33395	. . . . . . . . . . . . 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 2735	. . . . . . . . . . 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 33355	. . . . . . . . . 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 33359	. . . . . . . . . 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 2767	. . . . . . . 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 2777	. . . . . . 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 4773	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
63	2, 1	rspssid 21146	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → {𝑋} ⊆ (𝐾‘{𝑋}))
64	22, 62, 63	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
65	64, 4	sseqtrrdi 3988	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑋} ⊆ 𝑀)
66		snssg 4747	. . . . . . . . . . . 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 3947	. . . . . . . 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 2830	. . . . . . 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 21150	. . . . . . . 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 837	. . . . . 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 3141	. . . . 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 768	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ∈ (LIdeal‘𝑅))
77	7	elin2d 4168	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ LPIR)
78		eqid 2729	. . . . . . . . . . 11 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
79		eqid 2729	. . . . . . . . . . 11 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
80	78, 79	islpir 21238	. . . . . . . . . 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 2830	. . . . . 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 21235	. . . . . . 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 3141	. . . 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 20334	. . . . . . . . . 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 33395	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
94	93	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
95	94	necon3abid 2961	. . . . . . . . 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 33433	. . . . . . . . . . 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 3082	. . . . 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 3141	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
110	109	pm2.18da 799	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
111	15, 110	impbida 800	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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 ⊆ wss 3914 {csn 4589 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 ._rcmulr 17221 0_gc0g 17402 Ringcrg 20142 ∥_rcdsr 20263 Unitcui 20264 Irredcir 20265 IDomncidom 20602 LIdealclidl 21116 RSpancrsp 21117 LPIdealclpidl 21230 LPIRclpir 21231 PIDcpid 21246 MaxIdealcmxidl 33430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-irred 20268 df-invr 20297 df-nzr 20422 df-subrg 20479 df-domn 20604 df-idom 20605 df-lmod 20768 df-lss 20838 df-lsp 20878 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-lpidl 21232 df-lpir 21233 df-pid 21247 df-mxidl 33431

This theorem is referenced by: rprmirredb 33503 algextdeglem4 33710

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