mxidlirred - Mathbox for Thierry Arnoux

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Theorem mxidlirred 33821

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 21557	. . . . . 6 ⊢ PID = (IDomn ∩ LPIR)
7	5, 6	eleqtrdi 2875	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (IDomn ∩ LPIR))
8	7	elin1d 4157	. . . 4 ⊢ (𝜑 → 𝑅 ∈ IDomn)
9	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ IDomn)
10		mxidlirred.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ 𝐵)
12		mxidlirred.y	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 )
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ≠ 0 )
14		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
15	1, 2, 3, 4, 9, 11, 13, 14	mxidlirredi 33820	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ (Irred‘𝑅))
16		eqid 2765	. . . . . . . . . . 11 ⊢ (∥_r‘𝑅) = (∥_r‘𝑅)
17		simplr 781	. . . . . . . . . . . 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 739	. . . . . . . . . . 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 753	. . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
21		eqid 2765	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (._r‘𝑅)
22	8	idomringd 20878	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑅 ∈ Ring)
24	23	ad2antrr 739	. . . . . . . . . . . 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 739	. . . . . . . . . . 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 781	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 804	. . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
31	30, 1, 20, 21	irredmul 20559	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ (𝑡(._r‘𝑅)𝑥) ∈ (Irred‘𝑅)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
32	26, 18, 29, 31	syl3anc 1398	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ⊆ 𝑘 ∧ ¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)) ↔ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
37	35, 36	sylibr 237	. . . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))
39		ioran 999	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵) ↔ (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
40	38, 39	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
41	40	simprd 501	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ 𝑘 = 𝐵)
42	41	neqned 2967	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ≠ 𝐵)
43	42	ad4antr 745	. . . . . . . . . . . . . . 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 3028	. . . . . . . . . . . . . 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 2965	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ (𝐾‘{𝑥}) = (𝐾‘{𝑥})
47	8	ad8antr 753	. . . . . . . . . . . . . . 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)) → 𝑅 ∈ IDomn)
48	47	idomcringd 20877	. . . . . . . . . . . . . 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 ∈ CRing)
49	20, 2, 46, 1, 18, 48	unitpidl1 33798	. . . . . . . . . . . . 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)))
50	45, 49	mtbid 327	. . . . . . . . . . . 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))
51	32, 50	olcnd 891	. . . . . . . . . . 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))
52	27	eqcomd 2771	. . . . . . . . . . 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)
53	1, 2, 16, 18, 19, 20, 21, 25, 51, 52	dvdsruassoi 33763	. . . . . . . . . 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))
54	1, 2, 16, 18, 19, 25	rspsnasso 33767	. . . . . . . . . 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})))
55	53, 54	mpbid 235	. . . . . . . . 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}))
56	55, 34	eqtr4d 2803	. . . . . . . 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)
57	4, 56	eqtr2id 2813	. . . . . . 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)
58	40	simpld 500	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ 𝑘 = 𝑀)
59	58	ad4antr 745	. . . . . . 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)
60	57, 59	pm2.21dd 198	. . . . . 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))
61	37	simpld 500	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑀 ⊆ 𝑘)
62	61	ad2antrr 739	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑀 ⊆ 𝑘)
63	10	snssd 4754	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
64	2, 1	rspssid 21417	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → {𝑋} ⊆ (𝐾‘{𝑋}))
65	22, 63, 64	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
66	65, 4	sseqtrrdi 3979	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑋} ⊆ 𝑀)
67		snssg 4751	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝑀 ↔ {𝑋} ⊆ 𝑀))
68	67	biimpar 483	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ {𝑋} ⊆ 𝑀) → 𝑋 ∈ 𝑀)
69	10, 66, 68	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
70	69	ad6antr 749	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋 ∈ 𝑀)
71	62, 70	sseldd 3939	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋 ∈ 𝑘)
72	71, 33	eleqtrd 2867	. . . . . . 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}))
73	1, 21, 2	elrspsn 21423	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑋 ∈ (𝐾‘{𝑥}) ↔ ∃𝑡 ∈ 𝐵 𝑋 = (𝑡(._r‘𝑅)𝑥)))
74	73	biimpa 482	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑋 ∈ (𝐾‘{𝑥})) → ∃𝑡 ∈ 𝐵 𝑋 = (𝑡(._r‘𝑅)𝑥))
75	24, 17, 72, 74	syl21anc 851	. . . . . 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))
76	60, 75	r19.29a 3175	. . . . 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))
77		simplr 781	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ∈ (LIdeal‘𝑅))
78	7	elin2d 4158	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ LPIR)
79		eqid 2765	. . . . . . . . . . 11 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
80		eqid 2765	. . . . . . . . . . 11 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
81	79, 80	islpir 21548	. . . . . . . . . 10 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
82	81	simprbi 503	. . . . . . . . 9 ⊢ (𝑅 ∈ LPIR → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
83	78, 82	syl 18	. . . . . . . 8 ⊢ (𝜑 → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
84	83	ad4antr 745	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
85	77, 84	eleqtrd 2867	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ∈ (LPIdeal‘𝑅))
86	79, 2, 1	islpidl 21545	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑘 ∈ (LPIdeal‘𝑅) ↔ ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥})))
87	86	biimpa 482	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LPIdeal‘𝑅)) → ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥}))
88	23, 85, 87	syl2anc 596	. . . . 5 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥}))
89	76, 88	r19.29a 3175	. . . 4 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑀 ∈ (MaxIdeal‘𝑅))
90		mxidlirred.1	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
91	90	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
92	30, 20	irrednu 20555	. . . . . . . . . 10 ⊢ (𝑋 ∈ (Irred‘𝑅) → ¬ 𝑋 ∈ (Unit‘𝑅))
93	92	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → ¬ 𝑋 ∈ (Unit‘𝑅))
94	8	idomcringd 20877	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ CRing)
95	20, 2, 4, 1, 10, 94	unitpidl1 33798	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
96	95	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
97	96	necon3abid 2996	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
98	93, 97	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → 𝑀 ≠ 𝐵)
99	98	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
100	91, 99	jca 521	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵))
101	1	ismxidl 33811	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
102	22, 101	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
103		df-3an 1105	. . . . . . . . . 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))))
104	102, 103	bitrdi 290	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
105	104	notbid 321	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑀 ∈ (MaxIdeal‘𝑅) ↔ ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
106	105	biimpa 482	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))))
107	106	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))))
108	100, 107	mpnanrd 415	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
109		rexnal 3119	. . . . 5 ⊢ (∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)) ↔ ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
110	108, 109	sylibr 237	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
111	89, 110	r19.29a 3175	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
112	111	pm2.18da 812	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
113	15, 112	impbida 813	1 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 ∩ cin 3905 ⊆ wss 3906 {csn 4591 class class class wbr 5111 ‘cfv 6540 (class class class)co 7419 Basecbs 17293 ._rcmulr 17335 0_gc0g 17516 Ringcrg 20361 ∥_rcdsr 20484 Unitcui 20485 Irredcir 20486 IDomncidom 20844 LIdealclidl 21382 RSpancrsp 21383 LPIdealclpidl 21540 LPIRclpir 21541 PIDcpid 21556 MaxIdealcmxidl 33808

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-sca 17350 df-vsca 17351 df-ip 17352 df-0g 17518 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-grp 19049 df-minusg 19050 df-sbg 19051 df-subg 19235 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-cring 20364 df-oppr 20467 df-dvdsr 20487 df-unit 20488 df-irred 20489 df-invr 20518 df-nzr 20662 df-subrg 20721 df-domn 20846 df-idom 20847 df-lmod 21035 df-lss 21105 df-lsp 21145 df-sra 21346 df-rgmod 21347 df-lidl 21384 df-rsp 21385 df-lpidl 21542 df-lpir 21543 df-pid 21557 df-mxidl 33809

This theorem is used by: rprmirredb 33888 algextdeglem4 34176

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