Theorem imadrhmcl 20648

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 20642	. . . . 5 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑆 ∈ (SubRing‘𝑀))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))
5		rhmima 20506	. . . 4 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
6	1, 4, 5	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
7		imadrhmcl.r	. . . 4 ⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆))
8	7	subrgring 20476	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝑅 ∈ Ring)
9	6, 8	syl 17	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
10		eqid 2726	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2726	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12	10, 11	unitss 20278	. . . . 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 2726	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
16		eqid 2726	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	15, 16	rhmf 20387	. . . . . . . . . . 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 20379	. . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑁) = (1r‘𝑁)
24	7, 23	subrg1 20484	. . . . . . . . . . . . . 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 20481	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 0 = (1r‘𝑁))
32	16, 27, 23	01eq0ring 20430	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Ring ∧ 0 = (1r‘𝑁)) → (Base‘𝑁) = { 0 })
33	21, 31, 32	syl2an2r 682	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (Base‘𝑁) = { 0 })
34	33	feq3d 6698	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (𝐹:(Base‘𝑀)⟶(Base‘𝑁) ↔ 𝐹:(Base‘𝑀)⟶{ 0 }))
35	19, 34	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹:(Base‘𝑀)⟶{ 0 })
36	27	fvexi 6899	. . . . . . . . 9 ⊢ 0 ∈ V
37	36	fconst2 7202	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑀)⟶{ 0 } ↔ 𝐹 = ((Base‘𝑀) × { 0 }))
38	35, 37	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹 = ((Base‘𝑀) × { 0 }))
39	18	ffnd 6712	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑀))
40		sdrgrcl 20640	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑀 ∈ DivRing)
41	2, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ DivRing)
42	41	drngringd 20595	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Ring)
43		eqid 2726	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
44	15, 43	ring0cl 20166	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Ring → (0g‘𝑀) ∈ (Base‘𝑀))
45	42, 44	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑀) ∈ (Base‘𝑀))
46	45	ne0d 4330	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑀) ≠ ∅)
47		fconst5 7203	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ (Base‘𝑀) ≠ ∅) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
48	39, 46, 47	syl2anc 583	. . . . . . . 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 231	. . . . . 6 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → ran 𝐹 = { 0 })
51	14, 50	mteqand 3027	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))
52		eqid 2726	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
53		eqid 2726	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
54	11, 52, 53	0unit 20298	. . . . . . 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 2972	. . . . 5 ⊢ (𝜑 → (¬ (0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) ≠ (0g‘𝑅)))
57	51, 56	mpbird 257	. . . 4 ⊢ (𝜑 → ¬ (0g‘𝑅) ∈ (Unit‘𝑅))
58		ssdifsn 4786	. . . 4 ⊢ ((Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ↔ ((Unit‘𝑅) ⊆ (Base‘𝑅) ∧ ¬ (0g‘𝑅) ∈ (Unit‘𝑅)))
59	13, 57, 58	sylanbrc 582	. . 3 ⊢ (𝜑 → (Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
60	39	fnfund 6644	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
61	7	ressbasss2 17194	. . . . . 6 ⊢ (Base‘𝑅) ⊆ (𝐹 “ 𝑆)
62		eldifi 4121	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (Base‘𝑅))
63	61, 62	sselid 3975	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (𝐹 “ 𝑆))
64		fvelima 6951	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
65	60, 63, 64	syl2an 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
66		simprr 770	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) = 𝑥)
67		simprl 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ∈ 𝑆)
68	67	fvresd 6905	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹 ↾ 𝑆)‘𝑎) = (𝐹‘𝑎))
69		eqid 2726	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝑆) = (𝑀 ↾s 𝑆)
70	69	resrhm 20503	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁))
71	1, 4, 70	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁))
72		df-ima 5682	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑆) = ran (𝐹 ↾ 𝑆)
73		eqimss2 4036	. . . . . . . . . . 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 20504	. . . . . . . . . 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 583	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁) ↔ (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅)))
77	71, 76	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅))
78	77	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅))
79		eldifsni 4788	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ≠ (0g‘𝑅))
80	79	ad2antlr 724	. . . . . . . . 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 6889	. . . . . . . . . . 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 20481	. . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))))
87		rhmghm 20386	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) GrpHom 𝑅))
88		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑀 ↾s 𝑆)) = (0g‘(𝑀 ↾s 𝑆))
89	88, 52	ghmid 19147	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = (0g‘𝑅))
92	91	ad3antrrr 727	. . . . . . . . . . 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 2766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘𝑎) = (0g‘𝑅))
94		simplrr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → (𝐹‘𝑎) = 𝑥)
95	81, 93, 94	3eqtr3rd 2775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → 𝑥 = (0g‘𝑅))
96	80, 95	mteqand 3027	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ≠ (0g‘𝑀))
97	2	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑆 ∈ (SubDRing‘𝑀))
98		eqid 2726	. . . . . . . . . 10 ⊢ (Unit‘(𝑀 ↾s 𝑆)) = (Unit‘(𝑀 ↾s 𝑆))
99	69, 43, 98	sdrgunit 20647	. . . . . . . . 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 710	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆)))
102		elrhmunit 20412	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅) ∧ 𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆))) → ((𝐹 ↾ 𝑆)‘𝑎) ∈ (Unit‘𝑅))
103	78, 101, 102	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹 ↾ 𝑆)‘𝑎) ∈ (Unit‘𝑅))
104	68, 103	eqeltrrd 2828	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) ∈ (Unit‘𝑅))
105	66, 104	eqeltrrd 2828	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑥 ∈ (Unit‘𝑅))
106	65, 105	rexlimddv 3155	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → 𝑥 ∈ (Unit‘𝑅))
107	59, 106	eqelssd 3998	. 2 ⊢ (𝜑 → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)}))
108	10, 11, 52	isdrng 20591	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)})))
109	9, 107, 108	sylanbrc 582	1 ⊢ (𝜑 → 𝑅 ∈ DivRing)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∃wrex 3064   ∖ cdif 3940   ⊆ wss 3943  ∅c0 4317  {csn 4623   × cxp 5667  ran crn 5670   ↾ cres 5671   “ cima 5672  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7405  Basecbs 17153   ↾_s cress 17182  0_gc0g 17394   GrpHom cghm 19138  1_rcur 20086  Ringcrg 20138  Unitcui 20257   RingHom crh 20371  SubRingcsubrg 20469  DivRingcdr 20587  SubDRingcsdrg 20637

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-grp 18866  df-minusg 18867  df-subg 19050  df-ghm 19139  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-oppr 20236  df-dvdsr 20259  df-unit 20260  df-invr 20290  df-rhm 20374  df-subrng 20446  df-subrg 20471  df-drng 20589  df-sdrg 20638

This theorem is referenced by:  rndrhmcl  32899  ricdrng1  41661