Theorem mxidlirred 33660

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 = (0g‘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 = (0g‘𝑅)
4		mxidlirred.m	. . 3 ⊢ 𝑀 = (𝐾‘{𝑋})
5		mxidlirred.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ PID)
6		df-pid 21407	. . . . . 6 ⊢ PID = (IDomn ∩ LPIR)
7	5, 6	eleqtrdi 2872	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (IDomn ∩ LPIR))
8	7	elin1d 4156	. . . 4 ⊢ (𝜑 → 𝑅 ∈ IDomn)
9	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ IDomn)
10		mxidlirred.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	10	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ 𝐵)
12		mxidlirred.y	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 )
13	12	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ≠ 0 )
14		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
15	1, 2, 3, 4, 9, 11, 13, 14	mxidlirredi 33659	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ (Irred‘𝑅))
16		eqid 2762	. . . . . . . . . . 11 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
17		simplr 778	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 750	. . . . . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
21		eqid 2762	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
22	8	idomringd 20778	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ad4antr 742	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑅 ∈ Ring)
24	23	ad2antrr 736	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 778	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 801	. . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
31	30, 1, 20, 21	irredmul 20478	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ (𝑡(.r‘𝑅)𝑥) ∈ (Irred‘𝑅)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
32	26, 18, 29, 31	syl3anc 1390	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ⊆ 𝑘 ∧ ¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)) ↔ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
37	35, 36	sylibr 236	. . . . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))
39		ioran 997	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝑘 = 𝑀 ∨ 𝑘 = 𝐵) ↔ (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
40	38, 39	sylib 220	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
41	40	simprd 499	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ 𝑘 = 𝐵)
42	41	neqned 2964	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ≠ 𝐵)
43	42	ad4antr 742	. . . . . . . . . . . . . . 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 3025	. . . . . . . . . . . . . 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 2962	. . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ (𝐾‘{𝑥}) = (𝐾‘{𝑥})
47	8	ad8antr 750	. . . . . . . . . . . . . . 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 20777	. . . . . . . . . . . . . 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 33610	. . . . . . . . . . . . 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 326	. . . . . . . . . . . 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 888	. . . . . . . . . . 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 2768	. . . . . . . . . . 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 33570	. . . . . . . . . 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 33574	. . . . . . . . . 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 234	. . . . . . . . 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 2800	. . . . . . . 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 2810	. . . . . . 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 498	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → ¬ 𝑘 = 𝑀)
59	58	ad4antr 742	. . . . . . 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 197	. . . . . 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 498	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑀 ⊆ 𝑘)
62	61	ad2antrr 736	. . . . . . . . 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 4745	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
64	2, 1	rspssid 21306	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → {𝑋} ⊆ (𝐾‘{𝑋}))
65	22, 63, 64	syl2anc 593	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
66	65, 4	sseqtrrdi 3977	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑋} ⊆ 𝑀)
67		snssg 4742	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝑀 ↔ {𝑋} ⊆ 𝑀))
68	67	biimpar 481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ {𝑋} ⊆ 𝑀) → 𝑋 ∈ 𝑀)
69	10, 66, 68	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
70	69	ad6antr 746	. . . . . . . . 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 3937	. . . . . . . 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 2864	. . . . . . 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 21310	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑋 ∈ (𝐾‘{𝑥}) ↔ ∃𝑡 ∈ 𝐵 𝑋 = (𝑡(.r‘𝑅)𝑥)))
74	73	biimpa 480	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑋 ∈ (𝐾‘{𝑥})) → ∃𝑡 ∈ 𝐵 𝑋 = (𝑡(.r‘𝑅)𝑥))
75	24, 17, 72, 74	syl21anc 848	. . . . . 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 3170	. . . . 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 778	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))) → 𝑘 ∈ (LIdeal‘𝑅))
78	7	elin2d 4157	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ LPIR)
79		eqid 2762	. . . . . . . . . . 11 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
80		eqid 2762	. . . . . . . . . . 11 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
81	79, 80	islpir 21398	. . . . . . . . . 10 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
82	81	simprbi 501	. . . . . . . . 9 ⊢ (𝑅 ∈ LPIR → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
83	78, 82	syl 17	. . . . . . . 8 ⊢ (𝜑 → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
84	83	ad4antr 742	. . . . . . 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 2864	. . . . . 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 21395	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑘 ∈ (LPIdeal‘𝑅) ↔ ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥})))
87	86	biimpa 480	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LPIdeal‘𝑅)) → ∃𝑥 ∈ 𝐵 𝑘 = (𝐾‘{𝑥}))
88	23, 85, 87	syl2anc 593	. . . . 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 3170	. . . 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 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
92	30, 20	irrednu 20474	. . . . . . . . . 10 ⊢ (𝑋 ∈ (Irred‘𝑅) → ¬ 𝑋 ∈ (Unit‘𝑅))
93	92	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → ¬ 𝑋 ∈ (Unit‘𝑅))
94	8	idomcringd 20777	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ CRing)
95	20, 2, 4, 1, 10, 94	unitpidl1 33610	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
96	95	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
97	96	necon3abid 2993	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
98	93, 97	mpbird 259	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → 𝑀 ≠ 𝐵)
99	98	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
100	91, 99	jca 519	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵))
101	1	ismxidl 33650	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
102	22, 101	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
103		df-3an 1100	. . . . . . . . . 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 289	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
105	104	notbid 320	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑀 ∈ (MaxIdeal‘𝑅) ↔ ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))))
106	105	biimpa 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵))))
107	106	adantlr 725	. . . . . 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 413	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
109		rexnal 3114	. . . . 5 ⊢ (∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)) ↔ ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
110	108, 109	sylibr 236	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀 ⊆ 𝑘 → (𝑘 = 𝑀 ∨ 𝑘 = 𝐵)))
111	89, 110	r19.29a 3170	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
112	111	pm2.18da 809	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (Irred‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
113	15, 112	impbida 810	1 ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086   ∩ cin 3903   ⊆ wss 3904  {csn 4582   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  ._rcmulr 17287  0_gc0g 17468  Ringcrg 20283  ∥_rcdsr 20403  Unitcui 20404  Irredcir 20405  IDomncidom 20743  LIdealclidl 21276  RSpancrsp 21277  LPIdealclpidl 21390  LPIRclpir 21391  PIDcpid 21406  MaxIdealcmxidl 33647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-sca 17302  df-vsca 17303  df-ip 17304  df-0g 17470  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-grp 18978  df-minusg 18979  df-sbg 18980  df-subg 19165  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20232  df-ring 20285  df-cring 20286  df-oppr 20386  df-dvdsr 20406  df-unit 20407  df-irred 20408  df-invr 20437  df-nzr 20563  df-subrg 20620  df-domn 20745  df-idom 20746  df-lmod 20929  df-lss 20999  df-lsp 21039  df-sra 21240  df-rgmod 21241  df-lidl 21278  df-rsp 21279  df-lpidl 21392  df-lpir 21393  df-pid 21407  df-mxidl 33648

This theorem is referenced by:  rprmirredb  33728  algextdeglem4  34017