mxidlirredi - Mathbox for Thierry Arnoux

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Theorem mxidlirredi 33558

Description: In an integral domain, the generator of a maximal ideal is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)

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

**Assertion**
Ref	Expression
mxidlirredi	⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅))

Proof of Theorem mxidlirredi Dummy variables 𝑓 𝑔 𝑞 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mxidlirredi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		mxidlirredi.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn)
3	2	idomringd 20704	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		mxidlirredi.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
5		mxidlirredi.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6	5	mxidlnr 33551	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
7	3, 4, 6	syl2anc 591	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐵)
8		eqid 2741	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
9		mxidlirredi.k	. . . . . 6 ⊢ 𝐾 = (RSpan‘𝑅)
10		mxidlirredi.m	. . . . . 6 ⊢ 𝑀 = (𝐾‘{𝑋})
11	2	idomcringd 20703	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
12	8, 9, 10, 5, 1, 11	unitpidl1 33511	. . . . 5 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
13	12	necon3abid 2972	. . . 4 ⊢ (𝜑 → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
14	7, 13	mpbid 234	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅))
15	1, 14	eldifd 3896	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)))
16	3	ad3antrrr 737	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ Ring)
17	4	ad3antrrr 737	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ∈ (MaxIdeal‘𝑅))
18		simplr 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
19	18	eldifad 3897	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ 𝐵)
20	19	snssd 4721	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵)
21		eqid 2741	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
22	9, 5, 21	rspcl 21232	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
23	16, 20, 22	syl2anc 591	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
24	3	ad4antr 739	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring)
25	24	ad2antrr 733	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑅 ∈ Ring)
26		simp-5r 792	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
27	26	eldifad 3897	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑔 ∈ 𝐵)
28		oveq1 7367	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑦(._r‘𝑅)𝑔) = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔))
29	28	eqeq2d 2752	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑥 = (𝑦(._r‘𝑅)𝑔) ↔ 𝑥 = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔)))
30		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑅) = (._r‘𝑅)
31		simplr 775	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑞 ∈ 𝐵)
32		simp-6r 794	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
33	32	eldifad 3897	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑓 ∈ 𝐵)
34	5, 30, 25, 31, 33	ringcld 20236	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → (𝑞(._r‘𝑅)𝑓) ∈ 𝐵)
35		simp-4r 790	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → (𝑓(._r‘𝑅)𝑔) = 𝑋)
36	35	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → (𝑞(._r‘𝑅)(𝑓(._r‘𝑅)𝑔)) = (𝑞(._r‘𝑅)𝑋))
37	5, 30, 25, 31, 33, 27	ringassd 20233	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔) = (𝑞(._r‘𝑅)(𝑓(._r‘𝑅)𝑔)))
38		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑥 = (𝑞(._r‘𝑅)𝑋))
39	36, 37, 38	3eqtr4rd 2787	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑥 = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔))
40	29, 34, 39	rspcedvdw 3565	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔))
41	5, 30, 9	elrspsn 21237	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑔}) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)))
42	41	biimpar 479	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) ∧ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)) → 𝑥 ∈ (𝐾‘{𝑔}))
43	25, 27, 40, 42	syl21anc 844	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) → 𝑥 ∈ (𝐾‘{𝑔}))
44	1	ad4antr 739	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵)
45		simpr 486	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
46	45, 10	eleqtrdi 2851	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑋}))
47	5, 30, 9	elrspsn 21237	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑋}) ↔ ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋)))
48	47	biimpa 478	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ (𝐾‘{𝑋})) → ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋))
49	24, 44, 46, 48	syl21anc 844	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋))
50	43, 49	r19.29a 3149	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑔}))
51	50	ex 414	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝑥 ∈ 𝑀 → 𝑥 ∈ (𝐾‘{𝑔})))
52	51	ssrdv 3923	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔}))
53	9, 5	rspssid 21233	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔}))
54		vex 3437	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
55	54	snss 4719	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐾‘{𝑔}) ↔ {𝑔} ⊆ (𝐾‘{𝑔}))
56	53, 55	sylibr 236	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → 𝑔 ∈ (𝐾‘{𝑔}))
57	16, 20, 56	syl2anc 591	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐾‘{𝑔}))
58	11	ad6antr 743	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑅 ∈ CRing)
59		simplr 775	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑟 ∈ 𝐵)
60		simp-6r 794	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
61	60	eldifad 3897	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑓 ∈ 𝐵)
62		mxidlirredi.0	. . . . . . . . . . . . . 14 ⊢ 0 = (0_g‘𝑅)
63	16	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑅 ∈ Ring)
64	63	ad2antrr 733	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑅 ∈ Ring)
65	5, 30, 64, 59, 61	ringcld 20236	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (𝑟(._r‘𝑅)𝑓) ∈ 𝐵)
66		eqid 2741	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
67	5, 66, 3	ringidcld 20242	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝑅) ∈ 𝐵)
68	67	ad6antr 743	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (1_r‘𝑅) ∈ 𝐵)
69	19	ad3antrrr 737	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑔 ∈ 𝐵)
70		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑔 = 0 )
71	70	oveq2d 7376	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(._r‘𝑅)𝑔) = (𝑓(._r‘𝑅) 0 ))
72		simp-5r 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(._r‘𝑅)𝑔) = 𝑋)
73	63	ad3antrrr 737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑅 ∈ Ring)
74	61	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑓 ∈ 𝐵)
75	5, 30, 62, 73, 74	ringrzd 20272	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(._r‘𝑅) 0 ) = 0 )
76	71, 72, 75	3eqtr3d 2784	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑋 = 0 )
77		mxidlirredi.y	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ≠ 0 )
78	77	neneqd 2941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ 𝑋 = 0 )
79	78	ad7antr 745	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → ¬ 𝑋 = 0 )
80	76, 79	pm2.65da 823	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → ¬ 𝑔 = 0 )
81	80	neqned 2943	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑔 ≠ 0 )
82	69, 81	eldifsnd 4723	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ { 0 }))
83	2	ad6antr 743	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑅 ∈ IDomn)
84	5, 30, 66, 64, 69	ringlidmd 20248	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → ((1_r‘𝑅)(._r‘𝑅)𝑔) = 𝑔)
85		simpr 486	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑔 = (𝑟(._r‘𝑅)𝑋))
86	5, 30, 64, 59, 61, 69	ringassd 20233	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → ((𝑟(._r‘𝑅)𝑓)(._r‘𝑅)𝑔) = (𝑟(._r‘𝑅)(𝑓(._r‘𝑅)𝑔)))
87		simp-4r 790	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (𝑓(._r‘𝑅)𝑔) = 𝑋)
88	87	oveq2d 7376	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (𝑟(._r‘𝑅)(𝑓(._r‘𝑅)𝑔)) = (𝑟(._r‘𝑅)𝑋))
89	86, 88	eqtr2d 2777	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (𝑟(._r‘𝑅)𝑋) = ((𝑟(._r‘𝑅)𝑓)(._r‘𝑅)𝑔))
90	84, 85, 89	3eqtrrd 2781	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → ((𝑟(._r‘𝑅)𝑓)(._r‘𝑅)𝑔) = ((1_r‘𝑅)(._r‘𝑅)𝑔))
91	5, 62, 30, 65, 68, 82, 83, 90	idomrcan 33364	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (𝑟(._r‘𝑅)𝑓) = (1_r‘𝑅))
92	8, 66	1unit 20349	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Unit‘𝑅))
93	3, 92	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Unit‘𝑅))
94	93	ad6antr 743	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (1_r‘𝑅) ∈ (Unit‘𝑅))
95	91, 94	eqeltrd 2841	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → (𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅))
96	8, 30, 5	unitmulclb 20356	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → ((𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅) ↔ (𝑟 ∈ (Unit‘𝑅) ∧ 𝑓 ∈ (Unit‘𝑅))))
97	96	simplbda 501	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) ∧ (𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅)) → 𝑓 ∈ (Unit‘𝑅))
98	58, 59, 61, 95, 97	syl31anc 1382	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(._r‘𝑅)𝑋)) → 𝑓 ∈ (Unit‘𝑅))
99	1	ad4antr 739	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑋 ∈ 𝐵)
100		simpr 486	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ 𝑀)
101	100, 10	eleqtrdi 2851	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ (𝐾‘{𝑋}))
102	5, 30, 9	elrspsn 21237	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑔 ∈ (𝐾‘{𝑋}) ↔ ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋)))
103	102	biimpa 478	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑔 ∈ (𝐾‘{𝑋})) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋))
104	63, 99, 101, 103	syl21anc 844	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋))
105	98, 104	r19.29a 3149	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (Unit‘𝑅))
106		simp-4r 790	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
107	106	eldifbd 3898	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → ¬ 𝑓 ∈ (Unit‘𝑅))
108	105, 107	pm2.65da 823	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ 𝑀)
109	57, 108	eldifd 3896	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ ((𝐾‘{𝑔}) ∖ 𝑀))
110	5, 16, 17, 23, 52, 109	mxidlmaxv 33555	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵)
111		eqid 2741	. . . . . . . 8 ⊢ (𝐾‘{𝑔}) = (𝐾‘{𝑔})
112	11	ad3antrrr 737	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ CRing)
113	8, 9, 111, 5, 19, 112	unitpidl1 33511	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ((𝐾‘{𝑔}) = 𝐵 ↔ 𝑔 ∈ (Unit‘𝑅)))
114	110, 113	mpbid 234	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (Unit‘𝑅))
115	18	eldifbd 3898	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ (Unit‘𝑅))
116	114, 115	pm2.65da 823	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
117	116	anasss 468	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
118	117	neqned 2943	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
119	118	ralrimivva 3184	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
120		eqid 2741	. . 3 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
121		eqid 2741	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅))
122	5, 8, 120, 121, 30	isirred 20394	. 2 ⊢ (𝑋 ∈ (Irred‘𝑅) ↔ (𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(._r‘𝑅)𝑔) ≠ 𝑋))
123	15, 119, 122	sylanbrc 590	1 ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 ∃wrex 3065 ∖ cdif 3882 ⊆ wss 3885 {csn 4558 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 ._rcmulr 17216 0_gc0g 17397 1_rcur 20157 Ringcrg 20209 CRingccrg 20210 Unitcui 20330 Irredcir 20331 IDomncidom 20669 LIdealclidl 21203 RSpancrsp 21204 MaxIdealcmxidl 33546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-ip 17233 df-0g 17399 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-irred 20334 df-invr 20363 df-nzr 20489 df-subrg 20546 df-domn 20671 df-idom 20672 df-lmod 20856 df-lss 20926 df-lsp 20966 df-sra 21167 df-rgmod 21168 df-lidl 21205 df-rsp 21206 df-mxidl 33547

This theorem is referenced by: mxidlirred 33559

W3C validator