Theorem imadrhmcl 20964

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 20958	. . . . 5 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑆 ∈ (SubRing‘𝑀))
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))
5		rhmima 20767	. . . 4 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
6	1, 4, 5	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
7		imadrhmcl.r	. . . 4 ⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆))
8	7	subrgring 20737	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝑅 ∈ Ring)
9	6, 8	syl 18	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
10		eqid 2762	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2762	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12	10, 11	unitss 20518	. . . . 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 2762	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
16		eqid 2762	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	15, 16	rhmf 20627	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
18	1, 17	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
19	18	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
20		rhmrcl2 20620	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
21	1, 20	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ Ring)
22		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (1r‘𝑅) = (0g‘𝑅))
23		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑁) = (1r‘𝑁)
24	7, 23	subrg1 20745	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → (1r‘𝑁) = (1r‘𝑅))
25	6, 24	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1r‘𝑁) = (1r‘𝑅))
26	25	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (1r‘𝑁) = (1r‘𝑅))
27		imadrhmcl.0	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑁)
28	7, 27	subrg0 20742	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 0 = (0g‘𝑅))
29	6, 28	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 = (0g‘𝑅))
30	29	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 0 = (0g‘𝑅))
31	22, 26, 30	3eqtr4rd 2808	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 0 = (1r‘𝑁))
32	16, 27, 23	01eq0ring 20692	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Ring ∧ 0 = (1r‘𝑁)) → (Base‘𝑁) = { 0 })
33	21, 31, 32	syl2an2r 698	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (Base‘𝑁) = { 0 })
34	33	feq3d 6691	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (𝐹:(Base‘𝑀)⟶(Base‘𝑁) ↔ 𝐹:(Base‘𝑀)⟶{ 0 }))
35	19, 34	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹:(Base‘𝑀)⟶{ 0 })
36	27	fvexi 6896	. . . . . . . . 9 ⊢ 0 ∈ V
37	36	fconst2 7207	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑀)⟶{ 0 } ↔ 𝐹 = ((Base‘𝑀) × { 0 }))
38	35, 37	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → 𝐹 = ((Base‘𝑀) × { 0 }))
39	18	ffnd 6707	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑀))
40		sdrgrcl 20956	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → 𝑀 ∈ DivRing)
41	2, 40	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ DivRing)
42	41	drngringd 20899	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Ring)
43		eqid 2762	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
44	15, 43	ring0cl 20409	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Ring → (0g‘𝑀) ∈ (Base‘𝑀))
45	42, 44	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑀) ∈ (Base‘𝑀))
46	45	ne0d 4291	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑀) ≠ ∅)
47		fconst5 7208	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ (Base‘𝑀) ≠ ∅) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
48	39, 46, 47	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
49	48	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
50	38, 49	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (1r‘𝑅) = (0g‘𝑅)) → ran 𝐹 = { 0 })
51	14, 50	mteqand 3048	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))
52		eqid 2762	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
53		eqid 2762	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
54	11, 52, 53	0unit 20538	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) = (0g‘𝑅)))
55	9, 54	syl 18	. . . . . 6 ⊢ (𝜑 → ((0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) = (0g‘𝑅)))
56	55	necon3bbid 2994	. . . . 5 ⊢ (𝜑 → (¬ (0g‘𝑅) ∈ (Unit‘𝑅) ↔ (1r‘𝑅) ≠ (0g‘𝑅)))
57	51, 56	mpbird 260	. . . 4 ⊢ (𝜑 → ¬ (0g‘𝑅) ∈ (Unit‘𝑅))
58		ssdifsn 4754	. . . 4 ⊢ ((Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ↔ ((Unit‘𝑅) ⊆ (Base‘𝑅) ∧ ¬ (0g‘𝑅) ∈ (Unit‘𝑅)))
59	13, 57, 58	sylanbrc 595	. . 3 ⊢ (𝜑 → (Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
60	39	fnfund 6637	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
61	7	ressbasss2 17337	. . . . . 6 ⊢ (Base‘𝑅) ⊆ (𝐹 “ 𝑆)
62		eldifi 4081	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (Base‘𝑅))
63	61, 62	sselid 3932	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ∈ (𝐹 “ 𝑆))
64		fvelima 6947	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
65	60, 63, 64	syl2an 608	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
66		simprr 785	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) = 𝑥)
67		simprl 783	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ∈ 𝑆)
68	67	fvresd 6902	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹 ↾ 𝑆)‘𝑎) = (𝐹‘𝑎))
69		eqid 2762	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝑆) = (𝑀 ↾s 𝑆)
70	69	resrhm 20764	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁))
71	1, 4, 70	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁))
72		df-ima 5672	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑆) = ran (𝐹 ↾ 𝑆)
73		eqimss2 3993	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑆) = ran (𝐹 ↾ 𝑆) → ran (𝐹 ↾ 𝑆) ⊆ (𝐹 “ 𝑆))
74	72, 73	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝐹 ↾ 𝑆) ⊆ (𝐹 “ 𝑆))
75	7	resrhm2b 20765	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑁) ↔ (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅)))
77	71, 76	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅))
78	77	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅))
79		eldifsni 4756	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) → 𝑥 ≠ (0g‘𝑅))
80	79	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑥 ≠ (0g‘𝑅))
81	68	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘𝑎) = (𝐹‘𝑎))
82		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → 𝑎 = (0g‘𝑀))
83	82	fveq2d 6886	. . . . . . . . . . 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 20742	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubRing‘𝑀) → (0g‘𝑀) = (0g‘(𝑀 ↾s 𝑆)))
85	4, 84	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑀) = (0g‘(𝑀 ↾s 𝑆)))
86	85	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))))
87		rhmghm 20626	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅) → (𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) GrpHom 𝑅))
88		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑀 ↾s 𝑆)) = (0g‘(𝑀 ↾s 𝑆))
89	88, 52	ghmid 19350	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) GrpHom 𝑅) → ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))) = (0g‘𝑅))
90	77, 87, 89	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘(𝑀 ↾s 𝑆))) = (0g‘𝑅))
91	86, 90	eqtrd 2797	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘(0g‘𝑀)) = (0g‘𝑅))
92	91	ad3antrrr 743	. . . . . . . . . . 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 2797	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → ((𝐹 ↾ 𝑆)‘𝑎) = (0g‘𝑅))
94		simplrr 790	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → (𝐹‘𝑎) = 𝑥)
95	81, 93, 94	3eqtr3rd 2806	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ 𝑎 = (0g‘𝑀)) → 𝑥 = (0g‘𝑅))
96	80, 95	mteqand 3048	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ≠ (0g‘𝑀))
97	2	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑆 ∈ (SubDRing‘𝑀))
98		eqid 2762	. . . . . . . . . 10 ⊢ (Unit‘(𝑀 ↾s 𝑆)) = (Unit‘(𝑀 ↾s 𝑆))
99	69, 43, 98	sdrgunit 20963	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubDRing‘𝑀) → (𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆)) ↔ (𝑎 ∈ 𝑆 ∧ 𝑎 ≠ (0g‘𝑀))))
100	97, 99	syl 18	. . . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆)))
102		elrhmunit 20671	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑆) ∈ ((𝑀 ↾s 𝑆) RingHom 𝑅) ∧ 𝑎 ∈ (Unit‘(𝑀 ↾s 𝑆))) → ((𝐹 ↾ 𝑆)‘𝑎) ∈ (Unit‘𝑅))
103	78, 101, 102	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹 ↾ 𝑆)‘𝑎) ∈ (Unit‘𝑅))
104	68, 103	eqeltrrd 2863	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝐹‘𝑎) ∈ (Unit‘𝑅))
105	66, 104	eqeltrrd 2863	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → 𝑥 ∈ (Unit‘𝑅))
106	65, 105	rexlimddv 3171	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → 𝑥 ∈ (Unit‘𝑅))
107	59, 106	eqelssd 3955	. 2 ⊢ (𝜑 → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)}))
108	10, 11, 52	isdrng 20895	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)})))
109	9, 107, 108	sylanbrc 595	1 ⊢ (𝜑 → 𝑅 ∈ DivRing)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4282  {csn 4587   × cxp 5657  ran crn 5660   ↾ cres 5661   “ cima 5662  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  Basecbs 17305   ↾_s cress 17326  0_gc0g 17528   GrpHom cghm 19341  1_rcur 20321  Ringcrg 20373  Unitcui 20497   RingHom crh 20611  SubRingcsubrg 20732  DivRingcdr 20891  SubDRingcsdrg 20953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-tpos 8227  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-0g 17530  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-submnd 18893  df-grp 19061  df-minusg 19062  df-subg 19247  df-ghm 19342  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-oppr 20479  df-dvdsr 20499  df-unit 20500  df-invr 20530  df-rhm 20614  df-subrng 20709  df-subrg 20733  df-drng 20893  df-sdrg 20954

This theorem is used by:  rndrhmcl  33724  ricdrng1  43397