mxidlirredi - Mathbox for Thierry Arnoux

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Theorem mxidlirredi 33486

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 20688	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		mxidlirredi.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
5		mxidlirredi.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6	5	mxidlnr 33479	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
7	3, 4, 6	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐵)
8		eqid 2735	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
9		mxidlirredi.k	. . . . . 6 ⊢ 𝐾 = (RSpan‘𝑅)
10		mxidlirredi.m	. . . . . 6 ⊢ 𝑀 = (𝐾‘{𝑋})
11	8, 9, 10, 5, 1, 2	unitpidl1 33439	. . . . 5 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
12	11	necon3abid 2968	. . . 4 ⊢ (𝜑 → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
13	7, 12	mpbid 232	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅))
14	1, 13	eldifd 3937	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)))
15	3	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ Ring)
16	4	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ∈ (MaxIdeal‘𝑅))
17		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
18	17	eldifad 3938	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ 𝐵)
19	18	snssd 4785	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵)
20		eqid 2735	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
21	9, 5, 20	rspcl 21196	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
22	15, 19, 21	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
23	3	ad4antr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring)
24	23	ad2antrr 726	. . . . . . . . . . . 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)
25		simp-5r 785	. . . . . . . . . . . . 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)))
26	25	eldifad 3938	. . . . . . . . . . . 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)
27		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑅) = (._r‘𝑅)
28		simplr 768	. . . . . . . . . . . . . 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)
29		simp-6r 787	. . . . . . . . . . . . . . 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)))
30	29	eldifad 3938	. . . . . . . . . . . . . 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)
31	5, 27, 24, 28, 30	ringcld 20220	. . . . . . . . . . . . 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)
32		oveq1 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑦(._r‘𝑅)𝑔) = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔))
33	32	eqeq2d 2746	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑥 = (𝑦(._r‘𝑅)𝑔) ↔ 𝑥 = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔)))
34	33	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(._r‘𝑅)𝑋)) ∧ 𝑦 = (𝑞(._r‘𝑅)𝑓)) → (𝑥 = (𝑦(._r‘𝑅)𝑔) ↔ 𝑥 = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔)))
35		simp-4r 783	. . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . 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, 27, 24, 28, 30, 26	ringassd 20217	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . 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	31, 34, 39	rspcedvd 3603	. . . . . . . . . . . 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, 27, 9	elrspsn 21201	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑔}) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)))
42	41	biimpar 477	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) ∧ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)) → 𝑥 ∈ (𝐾‘{𝑔}))
43	24, 26, 40, 42	syl21anc 837	. . . . . . . . . . 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 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵)
45		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
46	45, 10	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑋}))
47	5, 27, 9	elrspsn 21201	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑋}) ↔ ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋)))
48	47	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ (𝐾‘{𝑋})) → ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋))
49	23, 44, 46, 48	syl21anc 837	. . . . . . . . . . 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 3148	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑔}))
51	50	ex 412	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝑥 ∈ 𝑀 → 𝑥 ∈ (𝐾‘{𝑔})))
52	51	ssrdv 3964	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔}))
53	9, 5	rspssid 21197	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔}))
54		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
55	54	snss 4761	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐾‘{𝑔}) ↔ {𝑔} ⊆ (𝐾‘{𝑔}))
56	53, 55	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → 𝑔 ∈ (𝐾‘{𝑔}))
57	15, 19, 56	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐾‘{𝑔}))
58		df-idom 20656	. . . . . . . . . . . . . . 15 ⊢ IDomn = (CRing ∩ Domn)
59	2, 58	eleqtrdi 2844	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
60	59	elin1d 4179	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ CRing)
61	60	ad6antr 736	. . . . . . . . . . . 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)
62		simplr 768	. . . . . . . . . . . 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)
63		simp-6r 787	. . . . . . . . . . . . 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)))
64	63	eldifad 3938	. . . . . . . . . . . 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)
65		mxidlirredi.0	. . . . . . . . . . . . . 14 ⊢ 0 = (0_g‘𝑅)
66	15	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑅 ∈ Ring)
67	66	ad2antrr 726	. . . . . . . . . . . . . . 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)
68	5, 27, 67, 62, 64	ringcld 20220	. . . . . . . . . . . . . 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)
69		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
70	5, 69	ringidcl 20225	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
71	3, 70	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝑅) ∈ 𝐵)
72	71	ad6antr 736	. . . . . . . . . . . . . 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)
73	18	ad3antrrr 730	. . . . . . . . . . . . . . 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)
74		simpr 484	. . . . . . . . . . . . . . . . . . 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 )
75	74	oveq2d 7421	. . . . . . . . . . . . . . . . . 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 ))
76		simp-5r 785	. . . . . . . . . . . . . . . . . 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)
77	66	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 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)
78	64	adantr 480	. . . . . . . . . . . . . . . . . . 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)
79	5, 27, 65	ringrz 20254	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → (𝑓(._r‘𝑅) 0 ) = 0 )
80	77, 78, 79	syl2anc 584	. . . . . . . . . . . . . . . . . 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 )
81	75, 76, 80	3eqtr3d 2778	. . . . . . . . . . . . . . . . 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 )
82		mxidlirredi.y	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ≠ 0 )
83	82	neneqd 2937	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ 𝑋 = 0 )
84	83	ad7antr 738	. . . . . . . . . . . . . . . . 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 )
85	81, 84	pm2.65da 816	. . . . . . . . . . . . . . . 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 )
86	85	neqned 2939	. . . . . . . . . . . . . . 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 )
87		eldifsn 4762	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝐵 ∖ { 0 }) ↔ (𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ))
88	73, 86, 87	sylanbrc 583	. . . . . . . . . . . . . 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 }))
89	2	ad6antr 736	. . . . . . . . . . . . . 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)
90	5, 27, 69, 67, 73	ringlidmd 20232	. . . . . . . . . . . . . . 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)
91		simpr 484	. . . . . . . . . . . . . . 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))
92	5, 27, 67, 62, 64, 73	ringassd 20217	. . . . . . . . . . . . . . . 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)))
93		simp-4r 783	. . . . . . . . . . . . . . . . 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)
94	93	oveq2d 7421	. . . . . . . . . . . . . . . 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))
95	92, 94	eqtr2d 2771	. . . . . . . . . . . . . . 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))
96	90, 91, 95	3eqtrrd 2775	. . . . . . . . . . . . . 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))
97	5, 65, 27, 68, 72, 88, 89, 96	idomrcan 33273	. . . . . . . . . . . . 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))
98	8, 69	1unit 20334	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Unit‘𝑅))
99	3, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Unit‘𝑅))
100	99	ad6antr 736	. . . . . . . . . . . . 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))
101	97, 100	eqeltrd 2834	. . . . . . . . . . . 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))
102	8, 27, 5	unitmulclb 20341	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → ((𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅) ↔ (𝑟 ∈ (Unit‘𝑅) ∧ 𝑓 ∈ (Unit‘𝑅))))
103	102	simplbda 499	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) ∧ (𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅)) → 𝑓 ∈ (Unit‘𝑅))
104	61, 62, 64, 101, 103	syl31anc 1375	. . . . . . . . . . 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))
105	1	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑋 ∈ 𝐵)
106		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ 𝑀)
107	106, 10	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ (𝐾‘{𝑋}))
108	5, 27, 9	elrspsn 21201	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑔 ∈ (𝐾‘{𝑋}) ↔ ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋)))
109	108	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑔 ∈ (𝐾‘{𝑋})) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋))
110	66, 105, 107, 109	syl21anc 837	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋))
111	104, 110	r19.29a 3148	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (Unit‘𝑅))
112		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
113	112	eldifbd 3939	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → ¬ 𝑓 ∈ (Unit‘𝑅))
114	111, 113	pm2.65da 816	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ 𝑀)
115	57, 114	eldifd 3937	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ ((𝐾‘{𝑔}) ∖ 𝑀))
116	5, 15, 16, 22, 52, 115	mxidlmaxv 33483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵)
117		eqid 2735	. . . . . . . 8 ⊢ (𝐾‘{𝑔}) = (𝐾‘{𝑔})
118	2	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ IDomn)
119	8, 9, 117, 5, 18, 118	unitpidl1 33439	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ((𝐾‘{𝑔}) = 𝐵 ↔ 𝑔 ∈ (Unit‘𝑅)))
120	116, 119	mpbid 232	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (Unit‘𝑅))
121	17	eldifbd 3939	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ (Unit‘𝑅))
122	120, 121	pm2.65da 816	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
123	122	anasss 466	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
124	123	neqned 2939	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
125	124	ralrimivva 3187	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
126		eqid 2735	. . 3 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
127		eqid 2735	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅))
128	5, 8, 126, 127, 27	isirred 20379	. 2 ⊢ (𝑋 ∈ (Irred‘𝑅) ↔ (𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(._r‘𝑅)𝑔) ≠ 𝑋))
129	14, 125, 128	sylanbrc 583	1 ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ∖ cdif 3923 ∩ cin 3925 ⊆ wss 3926 {csn 4601 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 ._rcmulr 17272 0_gc0g 17453 1_rcur 20141 Ringcrg 20193 CRingccrg 20194 Unitcui 20315 Irredcir 20316 Domncdomn 20652 IDomncidom 20653 LIdealclidl 21167 RSpancrsp 21168 MaxIdealcmxidl 33474

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-ip 17289 df-0g 17455 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-irred 20319 df-invr 20348 df-nzr 20473 df-subrg 20530 df-domn 20655 df-idom 20656 df-lmod 20819 df-lss 20889 df-lsp 20929 df-sra 21131 df-rgmod 21132 df-lidl 21169 df-rsp 21170 df-mxidl 33475

This theorem is referenced by: mxidlirred 33487

W3C validator