Theorem imadrhmcl 20757

Description: The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.)

Hypotheses

Ref	Expression
imadrhmcl.r	⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆))
imadrhmcl.0	⊢ 0 = (0g‘𝑁)
imadrhmcl.h	⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))
imadrhmcl.s	⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑀))
imadrhmcl.1	⊢ (𝜑 → ran 𝐹 ≠ { 0 })

Assertion

Ref	Expression
imadrhmcl	⊢ (𝜑 → 𝑅 ∈ DivRing)

Proof of Theorem imadrhmcl

Dummy variables 𝑎 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imadrhmcl.h	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))
2		imadrhmcl.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑀))
3		sdrgsubrg 20751	. . . . 5 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑆 ∈ (SubRing‘𝑀))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))
5		rhmima 20564	. . . 4 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
6	1, 4, 5	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
7		imadrhmcl.r	. . . 4 ⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆))
8	7	subrgring 20534	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝑅 ∈ Ring)
9	6, 8	syl 17	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
10		eqid 2735	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2735	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12	10, 11	unitss 20336	. . . . 5 ⊢ (Unit‘𝑅) ⊆ (Base‘𝑅)
13	12	a1i 11	. . . 4 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
14		imadrhmcl.1	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ≠ { 0 })
15		eqid 2735	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
16		eqid 2735	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	15, 16	rhmf 20445	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
18	1, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
20		rhmrcl2 20437	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
21	1, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ Ring)
22		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (1r‘𝑅) = (0g‘𝑅))
23		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑁) = (1r‘𝑁)
24	7, 23	subrg1 20542	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → (1r‘𝑁) = (1r‘𝑅))
25	6, 24	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1r‘𝑁) = (1r‘𝑅))
26	25	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (1r‘𝑁) = (1r‘𝑅))
27		imadrhmcl.0	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑁)
28	7, 27	subrg0 20539	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 0 = (0g‘𝑅))
29	6, 28	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 = (0g‘𝑅))
30	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 0 = (0g‘𝑅))
31	22, 26, 30	3eqtr4rd 2781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 0 = (1r‘𝑁))
32	16, 27, 23	01eq0ring 20490	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Ring ∧ 0 = (1r‘𝑁)) → (Base‘𝑁) = { 0 })
33	21, 31, 32	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (Base‘𝑁) = { 0 })
34	33	feq3d 6693	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (𝐹:(Base‘𝑀)⟶(Base‘𝑁) ↔ 𝐹:(Base‘𝑀)⟶{ 0 }))
35	19, 34	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹:(Base‘𝑀)⟶{ 0 })
36	27	fvexi 6890	. . . . . . . . 9 ⊢ 0 ∈ V
37	36	fconst2 7197	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑀)⟶{ 0 } ↔ 𝐹 = ((Base‘𝑀) × { 0 }))
38	35, 37	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹 = ((Base‘𝑀) × { 0 }))
39	18	ffnd 6707	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑀))
40		sdrgrcl 20749	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑀 ∈ DivRing)
41	2, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ DivRing)
42	41	drngringd 20697	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Ring)
43		eqid 2735	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
44	15, 43	ring0cl 20227	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Ring → (0g‘𝑀) ∈ (Base‘𝑀))
45	42, 44	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑀) ∈ (Base‘𝑀))
46	45	ne0d 4317	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑀) ≠ ∅)
47		fconst5 7198	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ (Base‘𝑀) ≠ ∅) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
48	39, 46, 47	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
50	38, 49	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → ran 𝐹 = { 0 })
51	14, 50	mteqand 3023	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))
52		eqid 2735	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
53		eqid 2735	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
54	11, 52, 53	0unit 20356	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) = (0g‘𝑅)))
55	9, 54	syl 17	. . . . . 6 ⊢ (𝜑 → ((0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) = (0g‘𝑅)))
56	55	necon3bbid 2969	. . . . 5 ⊢ (𝜑 → (¬ (0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) ≠ (0g‘𝑅)))
57	51, 56	mpbird 257	. . . 4 ⊢ (𝜑 → ¬ (0g‘𝑅) ∈ (Unit‘𝑅))
58		ssdifsn 4764	. . . 4 ⊢ ((Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ↔ ((Unit‘𝑅) ⊆ (Base‘𝑅) ∧ ¬ (0g‘𝑅) ∈ (Unit‘𝑅)))
59	13, 57, 58	sylanbrc 583	. . 3 ⊢ (𝜑 → (Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
60	39	fnfund 6639	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
61	7	ressbasss2 17262	. . . . . 6 ⊢ (Base‘𝑅) ⊆ (𝐹 “ 𝑆)
62		eldifi 4106	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (Base‘𝑅))
63	61, 62	sselid 3956	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (𝐹 “ 𝑆))
64		fvelima 6944	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
65	60, 63, 64	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
66		simprr 772	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) = 𝑥)
67		simprl 770	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ∈ 𝑆)
68	67	fvresd 6896	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹 ↾ 𝑆)‘𝑎) = (𝐹‘𝑎))
69		eqid 2735	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝑆) = (𝑀 ↾s 𝑆)
70	69	resrhm 20561	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁))
71	1, 4, 70	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁))
72		df-ima 5667	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑆) = ran (𝐹 ↾ 𝑆)
73		eqimss2 4018	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑆) = ran (𝐹 ↾ 𝑆) → ran (𝐹 ↾ 𝑆) ⊆ (𝐹 “ 𝑆))
74	72, 73	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝐹 ↾ 𝑆) ⊆ (𝐹 “ 𝑆))
75	7	resrhm2b 20562	. . . . . . . . . 10 ⊢ (((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) ∧ ran (𝐹 ↾ 𝑆) ⊆ (𝐹 “ 𝑆)) → ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁) ↔ (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅)))
76	6, 74, 75	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁) ↔ (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅)))
77	71, 76	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅))
78	77	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅))
79		eldifsni 4766	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ≠ (0g‘𝑅))
80	79	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑥 ≠ (0g‘𝑅))
81	68	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘𝑎) = (𝐹‘𝑎))
82		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → 𝑎 = (0g‘𝑀))
83	82	fveq2d 6880	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘𝑎) = ((𝐹 ↾ 𝑆)‘(0g‘𝑀)))
84	69, 43	subrg0 20539	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubRing‘𝑀) → (0g‘𝑀) = (0g‘(𝑀 ↾s 𝑆)))
85	4, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑀) = (0g‘(𝑀 ↾s 𝑆)))
86	85	fveq2d 6880	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))))
87		rhmghm 20444	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) GrpHom 𝑅))
88		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑀 ↾s 𝑆)) = (0g‘(𝑀 ↾s 𝑆))
89	88, 52	ghmid 19205	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) GrpHom 𝑅) → ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))) = (0g‘𝑅))
90	77, 87, 89	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))) = (0g‘𝑅))
91	86, 90	eqtrd 2770	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = (0g‘𝑅))
92	91	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = (0g‘𝑅))
93	83, 92	eqtrd 2770	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘𝑎) = (0g‘𝑅))
94		simplrr 777	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → (𝐹‘𝑎) = 𝑥)
95	81, 93, 94	3eqtr3rd 2779	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → 𝑥 = (0g‘𝑅))
96	80, 95	mteqand 3023	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ≠ (0g‘𝑀))
97	2	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑆 ∈ (SubDRing‘𝑀))
98		eqid 2735	. . . . . . . . . 10 ⊢ (Unit‘(𝑀 ↾s 𝑆)) = (Unit‘(𝑀 ↾s 𝑆))
99	69, 43, 98	sdrgunit 20756	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → (𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆)) ↔ (𝑎 ∈ 𝑆 ∧ 𝑎 ≠ (0g‘𝑀))))
100	97, 99	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆)) ↔ (𝑎 ∈ 𝑆 ∧ 𝑎 ≠ (0g‘𝑀))))
101	67, 96, 100	mpbir2and 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆)))
102		elrhmunit 20470	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅) ∧ 𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆))) → ((𝐹 ↾ 𝑆)‘𝑎) ∈ (Unit‘𝑅))
103	78, 101, 102	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹 ↾ 𝑆)‘𝑎) ∈ (Unit‘𝑅))
104	68, 103	eqeltrrd 2835	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) ∈ (Unit‘𝑅))
105	66, 104	eqeltrrd 2835	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑥 ∈ (Unit‘𝑅))
106	65, 105	rexlimddv 3147	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → 𝑥 ∈ (Unit‘𝑅))
107	59, 106	eqelssd 3980	. 2 ⊢ (𝜑 → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)}))
108	10, 11, 52	isdrng 20693	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)})))
109	9, 107, 108	sylanbrc 583	1 ⊢ (𝜑 → 𝑅 ∈ DivRing)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  {csn 4601   × cxp 5652  ran crn 5655   ↾ cres 5656   “ cima 5657  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Basecbs 17228   ↾_s cress 17251  0_gc0g 17453   GrpHom cghm 19195  1_rcur 20141  Ringcrg 20193  Unitcui 20315   RingHom crh 20429  SubRingcsubrg 20529  DivRingcdr 20689  SubDRingcsdrg 20746

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-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-map 8842  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-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-subg 19106  df-ghm 19196  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-rhm 20432  df-subrng 20506  df-subrg 20530  df-drng 20691  df-sdrg 20747

This theorem is referenced by:  rndrhmcl  33290  ricdrng1  42551