mxidlirredi - Mathbox for Thierry Arnoux

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Theorem mxidlirredi 33232

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 21259	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		mxidlirredi.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
5		mxidlirredi.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6	5	mxidlnr 33225	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
7	3, 4, 6	syl2anc 582	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐵)
8		eqid 2725	. . . . . 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 33187	. . . . 5 ⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅)))
12	11	necon3abid 2967	. . . 4 ⊢ (𝜑 → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
13	7, 12	mpbid 231	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅))
14	1, 13	eldifd 3951	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)))
15	3	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ Ring)
16	4	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ∈ (MaxIdeal‘𝑅))
17		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
18	17	eldifad 3952	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ 𝐵)
19	18	snssd 4808	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵)
20		eqid 2725	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
21	9, 5, 20	rspcl 21133	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
22	15, 19, 21	syl2anc 582	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
23	3	ad4antr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring)
24	23	ad2antrr 724	. . . . . . . . . . . 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 784	. . . . . . . . . . . . 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 3952	. . . . . . . . . . . 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 2725	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑅) = (._r‘𝑅)
28		simplr 767	. . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . 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 3952	. . . . . . . . . . . . . 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 20201	. . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑦(._r‘𝑅)𝑔) = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔))
33	32	eqeq2d 2736	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑞(._r‘𝑅)𝑓) → (𝑥 = (𝑦(._r‘𝑅)𝑔) ↔ 𝑥 = ((𝑞(._r‘𝑅)𝑓)(._r‘𝑅)𝑔)))
34	33	adantl 480	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . 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 20199	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 3604	. . . . . . . . . . . 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	rspsnel 33129	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑔}) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)))
42	41	biimpar 476	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) ∧ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(._r‘𝑅)𝑔)) → 𝑥 ∈ (𝐾‘{𝑔}))
43	24, 26, 40, 42	syl21anc 836	. . . . . . . . . . 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 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵)
45		simpr 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
46	45, 10	eleqtrdi 2835	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑋}))
47	5, 27, 9	rspsnel 33129	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑋}) ↔ ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋)))
48	47	biimpa 475	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ (𝐾‘{𝑋})) → ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(._r‘𝑅)𝑋))
49	23, 44, 46, 48	syl21anc 836	. . . . . . . . . . 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 3152	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑔}))
51	50	ex 411	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝑥 ∈ 𝑀 → 𝑥 ∈ (𝐾‘{𝑔})))
52	51	ssrdv 3978	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔}))
53	9, 5	rspssid 21134	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔}))
54		vex 3467	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
55	54	snss 4785	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐾‘{𝑔}) ↔ {𝑔} ⊆ (𝐾‘{𝑔}))
56	53, 55	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → 𝑔 ∈ (𝐾‘{𝑔}))
57	15, 19, 56	syl2anc 582	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐾‘{𝑔}))
58		df-idom 21234	. . . . . . . . . . . . . . 15 ⊢ IDomn = (CRing ∩ Domn)
59	2, 58	eleqtrdi 2835	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
60	59	elin1d 4192	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ CRing)
61	60	ad6antr 734	. . . . . . . . . . . 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 767	. . . . . . . . . . . 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 786	. . . . . . . . . . . . 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 3952	. . . . . . . . . . . 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	18	ad3antrrr 728	. . . . . . . . . . . . . . 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)
67		simpr 483	. . . . . . . . . . . . . . . . . . 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 )
68	67	oveq2d 7431	. . . . . . . . . . . . . . . . . 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 ))
69		simp-5r 784	. . . . . . . . . . . . . . . . . 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)
70	15	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑅 ∈ Ring)
71	70	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 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)
72	64	adantr 479	. . . . . . . . . . . . . . . . . . 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)
73	5, 27, 65	ringrz 20232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → (𝑓(._r‘𝑅) 0 ) = 0 )
74	71, 72, 73	syl2anc 582	. . . . . . . . . . . . . . . . . 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 )
75	68, 69, 74	3eqtr3d 2773	. . . . . . . . . . . . . . . . 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 )
76		mxidlirredi.y	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ≠ 0 )
77	76	neneqd 2935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ 𝑋 = 0 )
78	77	ad7antr 736	. . . . . . . . . . . . . . . . 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 )
79	75, 78	pm2.65da 815	. . . . . . . . . . . . . . . 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 )
80	79	neqned 2937	. . . . . . . . . . . . . . 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 )
81		eldifsn 4786	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝐵 ∖ { 0 }) ↔ (𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ))
82	66, 80, 81	sylanbrc 581	. . . . . . . . . . . . . 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	70	ad2antrr 724	. . . . . . . . . . . . . . 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)
84	5, 27, 83, 62, 64	ringcld 20201	. . . . . . . . . . . . . 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)
85		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
86	5, 85	ringidcl 20204	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
87	3, 86	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝑅) ∈ 𝐵)
88	87	ad6antr 734	. . . . . . . . . . . . . 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)
89	2	ad6antr 734	. . . . . . . . . . . . . 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, 85, 83, 66	ringlidmd 20210	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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, 83, 62, 64, 66	ringassd 20199	. . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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, 82, 84, 88, 89, 96	idomrcan 33015	. . . . . . . . . . . . 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, 85	1unit 20315	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Unit‘𝑅))
99	3, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Unit‘𝑅))
100	99	ad6antr 734	. . . . . . . . . . . . 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 2825	. . . . . . . . . . . 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 20322	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → ((𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅) ↔ (𝑟 ∈ (Unit‘𝑅) ∧ 𝑓 ∈ (Unit‘𝑅))))
103	102	simplbda 498	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) ∧ (𝑟(._r‘𝑅)𝑓) ∈ (Unit‘𝑅)) → 𝑓 ∈ (Unit‘𝑅))
104	61, 62, 64, 101, 103	syl31anc 1370	. . . . . . . . . . 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 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑋 ∈ 𝐵)
106		simpr 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ 𝑀)
107	106, 10	eleqtrdi 2835	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ (𝐾‘{𝑋}))
108	5, 27, 9	rspsnel 33129	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑔 ∈ (𝐾‘{𝑋}) ↔ ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋)))
109	108	biimpa 475	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑔 ∈ (𝐾‘{𝑋})) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(._r‘𝑅)𝑋))
110	70, 105, 107, 109	syl21anc 836	. . . . . . . . . . 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 3152	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (Unit‘𝑅))
112		simp-4r 782	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
113	112	eldifbd 3953	. . . . . . . . . 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 815	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ 𝑀)
115	57, 114	eldifd 3951	. . . . . . . 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 33229	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵)
117		eqid 2725	. . . . . . . 8 ⊢ (𝐾‘{𝑔}) = (𝐾‘{𝑔})
118	2	ad3antrrr 728	. . . . . . . 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 33187	. . . . . . 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 231	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (Unit‘𝑅))
121	17	eldifbd 3953	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(._r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ (Unit‘𝑅))
122	120, 121	pm2.65da 815	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
123	122	anasss 465	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → ¬ (𝑓(._r‘𝑅)𝑔) = 𝑋)
124	123	neqned 2937	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
125	124	ralrimivva 3191	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(._r‘𝑅)𝑔) ≠ 𝑋)
126		eqid 2725	. . 3 ⊢ (Irred‘𝑅) = (Irred‘𝑅)
127		eqid 2725	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅))
128	5, 8, 126, 127, 27	isirred 20360	. 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 581	1 ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ∖ cdif 3937 ∩ cin 3939 ⊆ wss 3940 {csn 4624 ‘cfv 6542 (class class class)co 7415 Basecbs 17177 ._rcmulr 17231 0_gc0g 17418 1_rcur 20123 Ringcrg 20175 CRingccrg 20176 Unitcui 20296 Irredcir 20297 LIdealclidl 21104 RSpancrsp 21105 Domncdomn 21229 IDomncidom 21230 MaxIdealcmxidl 33220

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 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 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-irred 20300 df-invr 20329 df-nzr 20454 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-rsp 21107 df-domn 21233 df-idom 21234 df-mxidl 33221

This theorem is referenced by: mxidlirred 33233

W3C validator