Theorem imadrhmcl 20814

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 20808	. . . . 5 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑆 ∈ (SubRing‘𝑀))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))
5		rhmima 20620	. . . 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 20590	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝑅 ∈ Ring)
9	6, 8	syl 17	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
10		eqid 2734	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2734	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12	10, 11	unitss 20392	. . . . 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 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
16		eqid 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	15, 16	rhmf 20501	. . . . . . . . . . 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 20493	. . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑁) = (1r‘𝑁)
24	7, 23	subrg1 20598	. . . . . . . . . . . . . 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 20595	. . . . . . . . . . . . . 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 2785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 0 = (1r‘𝑁))
32	16, 27, 23	01eq0ring 20546	. . . . . . . . . . 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 6723	. . . . . . . . 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 6920	. . . . . . . . 9 ⊢ 0 ∈ V
37	36	fconst2 7224	. . . . . . . 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 6737	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑀))
40		sdrgrcl 20806	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑀 ∈ DivRing)
41	2, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ DivRing)
42	41	drngringd 20753	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Ring)
43		eqid 2734	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
44	15, 43	ring0cl 20280	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Ring → (0g‘𝑀) ∈ (Base‘𝑀))
45	42, 44	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑀) ∈ (Base‘𝑀))
46	45	ne0d 4347	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑀) ≠ ∅)
47		fconst5 7225	. . . . . . . . 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 3030	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))
52		eqid 2734	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
53		eqid 2734	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
54	11, 52, 53	0unit 20412	. . . . . . 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 2975	. . . . 5 ⊢ (𝜑 → (¬ (0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) ≠ (0g‘𝑅)))
57	51, 56	mpbird 257	. . . 4 ⊢ (𝜑 → ¬ (0g‘𝑅) ∈ (Unit‘𝑅))
58		ssdifsn 4792	. . . 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 6669	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
61	7	ressbasss2 17285	. . . . . 6 ⊢ (Base‘𝑅) ⊆ (𝐹 “ 𝑆)
62		eldifi 4140	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (Base‘𝑅))
63	61, 62	sselid 3992	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (𝐹 “ 𝑆))
64		fvelima 6973	. . . . 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 773	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) = 𝑥)
67		simprl 771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ∈ 𝑆)
68	67	fvresd 6926	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹 ↾ 𝑆)‘𝑎) = (𝐹‘𝑎))
69		eqid 2734	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝑆) = (𝑀 ↾s 𝑆)
70	69	resrhm 20617	. . . . . . . . . 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 5701	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑆) = ran (𝐹 ↾ 𝑆)
73		eqimss2 4054	. . . . . . . . . . 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 20618	. . . . . . . . . 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 4794	. . . . . . . . . 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 6910	. . . . . . . . . . 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 20595	. . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))))
87		rhmghm 20500	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) GrpHom 𝑅))
88		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑀 ↾s 𝑆)) = (0g‘(𝑀 ↾s 𝑆))
89	88, 52	ghmid 19252	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘𝑎) = (0g‘𝑅))
94		simplrr 778	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → (𝐹‘𝑎) = 𝑥)
95	81, 93, 94	3eqtr3rd 2783	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → 𝑥 = (0g‘𝑅))
96	80, 95	mteqand 3030	. . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (Unit‘(𝑀 ↾s 𝑆)) = (Unit‘(𝑀 ↾s 𝑆))
99	69, 43, 98	sdrgunit 20813	. . . . . . . . 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 20526	. . . . . . 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 2839	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) ∈ (Unit‘𝑅))
105	66, 104	eqeltrrd 2839	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑥 ∈ (Unit‘𝑅))
106	65, 105	rexlimddv 3158	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → 𝑥 ∈ (Unit‘𝑅))
107	59, 106	eqelssd 4016	. 2 ⊢ (𝜑 → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)}))
108	10, 11, 52	isdrng 20749	. 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 1536   ∈ wcel 2105   ≠ wne 2937  ∃wrex 3067   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338  {csn 4630   × cxp 5686  ran crn 5689   ↾ cres 5690   “ cima 5691  Fun wfun 6556   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  Basecbs 17244   ↾_s cress 17273  0_gc0g 17485   GrpHom cghm 19242  1_rcur 20198  Ringcrg 20250  Unitcui 20371   RingHom crh 20485  SubRingcsubrg 20585  DivRingcdr 20745  SubDRingcsdrg 20803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-subg 19153  df-ghm 19243  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-drng 20747  df-sdrg 20804

This theorem is referenced by:  rndrhmcl  33279  ricdrng1  42514