mxidlirredi - Mathbox for Thierry Arnoux

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Theorem mxidlirredi 33701

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 20814	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		mxidlirredi.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
5		mxidlirredi.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6	5	mxidlnr 33694	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
7	3, 4, 6	syl2anc 595	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐵)
8		eqid 2769	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
9		mxidlirredi.k	. . . . . 6 ⊢ 𝐾 = (RSpan‘𝑅)
10		mxidlirredi.m	. . . . . 6 ⊢ 𝑀 = (𝐾‘{𝑋})
11	2	idomcringd 20813	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
12	8, 9, 10, 5, 1, 11	unitpidl1 33678	. . . . 5 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
13	12	necon3abid 3000	. . . 4 ⊢ (𝜑 → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
14	7, 13	mpbid 235	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅))
15	1, 14	eldifd 3924	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)))
16	3	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ Ring)
17	4	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ∈ (MaxIdeal‘𝑅))
18		simplr 780	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
19	18	eldifad 3925	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ 𝐵)
20	19	snssd 4757	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵)
21		eqid 2769	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
22	9, 5, 21	rspcl 21344	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
23	16, 20, 22	syl2anc 595	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
24	3	ad4antr 744	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring)
25	24	ad2antrr 738	. . . . . . . . . . . 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 797	. . . . . . . . . . . . 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 3925	. . . . . . . . . . . 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 7420	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑦(._r‘𝑅)𝑔) = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔))
29	28	eqeq2d 2780	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑥 = (𝑦(._r‘𝑅)𝑔) ↔ 𝑥 = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔)))
30		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑅) = (._r‘𝑅)
31		simplr 780	. . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . . 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 20344	. . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . 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 20341	. . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . 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 3593	. . . . . . . . . . . 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 21349	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑔}) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)))
42	41	biimpar 482	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) ∧ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)) → 𝑥 ∈ (𝐾‘{𝑔}))
43	25, 27, 40, 42	syl21anc 850	. . . . . . . . . . 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 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵)
45		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
46	45, 10	eleqtrdi 2879	. . . . . . . . . . . 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 21349	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑋}) ↔ ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋)))
48	47	biimpa 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ (𝐾‘{𝑋})) → ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋))
49	24, 44, 46, 48	syl21anc 850	. . . . . . . . . . 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 3179	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑔}))
51	50	ex 417	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝑥 ∈ 𝑀 → 𝑥 ∈ (𝐾‘{𝑔})))
52	51	ssrdv 3951	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔}))
53	9, 5	rspssid 21345	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔}))
54		vex 3467	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
55	54	snss 4755	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐾‘{𝑔}) ↔ {𝑔} ⊆ (𝐾‘{𝑔}))
56	53, 55	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → 𝑔 ∈ (𝐾‘{𝑔}))
57	16, 20, 56	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐾‘{𝑔}))
58	11	ad6antr 748	. . . . . . . . . . . 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 780	. . . . . . . . . . . 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 799	. . . . . . . . . . . . 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 3925	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑅 ∈ Ring)
64	63	ad2antrr 738	. . . . . . . . . . . . . . 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 20344	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
67	5, 66, 3	ringidcld 20351	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝑅) ∈ 𝐵)
68	67	ad6antr 748	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 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 20381	. . . . . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . . 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 2969	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ 𝑋 = 0 )
79	78	ad7antr 750	. . . . . . . . . . . . . . . . 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 828	. . . . . . . . . . . . . . . 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 2971	. . . . . . . . . . . . . . 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 4759	. . . . . . . . . . . . . 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 748	. . . . . . . . . . . . . 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 20357	. . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . 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 20341	. . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . 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 33545	. . . . . . . . . . . . 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 20458	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Unit‘𝑅))
93	3, 92	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Unit‘𝑅))
94	93	ad6antr 748	. . . . . . . . . . . . 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 2869	. . . . . . . . . . . 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 20465	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → ((𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅) ↔ (𝑟 ∈ (Unit‘𝑅) ∧ 𝑓 ∈ (Unit‘𝑅))))
97	96	simplbda 504	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) ∧ (𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅)) → 𝑓 ∈ (Unit‘𝑅))
98	58, 59, 61, 95, 97	syl31anc 1398	. . . . . . . . . . 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 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑋 ∈ 𝐵)
100		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ 𝑀)
101	100, 10	eleqtrdi 2879	. . . . . . . . . . . 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 21349	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑔 ∈ (𝐾‘{𝑋}) ↔ ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋)))
103	102	biimpa 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑔 ∈ (𝐾‘{𝑋})) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋))
104	63, 99, 101, 103	syl21anc 850	. . . . . . . . . . 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 3179	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (Unit‘𝑅))
106		simp-4r 795	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
107	106	eldifbd 3926	. . . . . . . . . 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 828	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ 𝑀)
109	57, 108	eldifd 3924	. . . . . . . 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 33698	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵)
111		eqid 2769	. . . . . . . 8 ⊢ (𝐾‘{𝑔}) = (𝐾‘{𝑔})
112	11	ad3antrrr 742	. . . . . . . 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 33678	. . . . . . 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 235	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (Unit‘𝑅))
115	18	eldifbd 3926	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ (Unit‘𝑅))
116	114, 115	pm2.65da 828	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
117	116	anasss 471	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
118	117	neqned 2971	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
119	118	ralrimivva 3214	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
120		eqid 2769	. . 3 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
121		eqid 2769	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅))
122	5, 8, 120, 121, 30	isirred 20503	. 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 594	1 ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ∖ cdif 3910 ⊆ wss 3913 {csn 4594 ‘cfv 6539 (class class class)co 7413 Basecbs 17271 ._rcmulr 17313 0_gc0g 17494 1_rcur 20265 Ringcrg 20317 CRingccrg 20318 Unitcui 20439 Irredcir 20440 IDomncidom 20780 LIdealclidl 21310 RSpancrsp 21311 MaxIdealcmxidl 33689

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-cnex 11158 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-mulcom 11166 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7865 df-1st 7988 df-2nd 7989 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11445 df-neg 11446 df-nn 12236 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-sets 17226 df-slot 17244 df-ndx 17256 df-base 17272 df-ress 17293 df-plusg 17325 df-mulr 17326 df-sca 17328 df-vsca 17329 df-ip 17330 df-0g 17496 df-mgm 18700 df-sgrp 18779 df-mnd 18795 df-grp 19005 df-minusg 19006 df-sbg 19007 df-subg 19191 df-cmn 19854 df-abl 19855 df-mgp 20219 df-rng 20233 df-ur 20266 df-ring 20319 df-cring 20320 df-oppr 20421 df-dvdsr 20441 df-unit 20442 df-irred 20443 df-invr 20472 df-nzr 20598 df-subrg 20657 df-domn 20782 df-idom 20783 df-lmod 20963 df-lss 21033 df-lsp 21073 df-sra 21274 df-rgmod 21275 df-lidl 21312 df-rsp 21313 df-mxidl 33690

This theorem is referenced by: mxidlirred 33702

W3C validator