Theorem issubdrg 20883

Description: Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)

Hypotheses

Ref	Expression
issubdrg.s	⊢ 𝑆 = (𝑅 ↾s 𝐴)
issubdrg.z	⊢ 0 = (0g‘𝑅)
issubdrg.i	⊢ 𝐼 = (invr‘𝑅)

Assertion

Ref	Expression
issubdrg	⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝑆   𝑥, 0

Allowed substitution hint:   𝐼(𝑥)

Proof of Theorem issubdrg

Step	Hyp	Ref	Expression
1		simpllr 787	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝐴 ∈ (SubRing‘𝑅))
2		issubdrg.s	. . . . . . 7 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
3	2	subrgring 20673	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
4	1, 3	syl 18	. . . . 5 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑆 ∈ Ring)
5		eldifsn 4753	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
6	5	bilani 509	. . . . . . . 8 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
7	6	simpld 499	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ 𝐴)
8	2	subrgbas 20680	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
9	1, 8	syl 18	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝐴 = (Base‘𝑆))
10	7, 9	eleqtrd 2865	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Base‘𝑆))
11	6	simprd 500	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ 0 )
12		issubdrg.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
13	2, 12	subrg0 20678	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆))
14	1, 13	syl 18	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 0 = (0g‘𝑆))
15	11, 14	neeqtrd 3027	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ (0g‘𝑆))
16		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
17		eqid 2763	. . . . . . . 8 ⊢ (Unit‘𝑆) = (Unit‘𝑆)
18		eqid 2763	. . . . . . . 8 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	16, 17, 18	drngunit 20832	. . . . . . 7 ⊢ (𝑆 ∈ DivRing → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))))
20	19	ad2antlr 739	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))))
21	10, 15, 20	mpbir2and 725	. . . . 5 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑆))
22		eqid 2763	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
23	17, 22, 16	ringinvcl 20470	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝑆)) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
24	4, 21, 23	syl2anc 595	. . . 4 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
25		issubdrg.i	. . . . . 6 ⊢ 𝐼 = (invr‘𝑅)
26	2, 25, 17, 22	subrginv 20687	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Unit‘𝑆)) → (𝐼‘𝑥) = ((invr‘𝑆)‘𝑥))
27	1, 21, 26	syl2anc 595	. . . 4 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼‘𝑥) = ((invr‘𝑆)‘𝑥))
28	24, 27, 9	3eltr4d 2878	. . 3 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼‘𝑥) ∈ 𝐴)
29	28	ralrimiva 3157	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) → ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴)
30	3	ad2antlr 739	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝑆 ∈ Ring)
31		eqid 2763	. . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
32	2, 31, 17	subrguss 20686	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
33	32	ad2antlr 739	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
34		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
35	34, 31, 12	isdrng 20831	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 })))
36	35	simprbi 502	. . . . . . . . 9 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
37	36	ad2antrr 738	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
38	33, 37	sseqtrd 3973	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
39	16, 17	unitss 20454	. . . . . . . 8 ⊢ (Unit‘𝑆) ⊆ (Base‘𝑆)
40	8	ad2antlr 739	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝐴 = (Base‘𝑆))
41	39, 40	sseqtrrid 3980	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
42	38, 41	ssind 4193	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
43	34	subrgss 20671	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
44	43	ad2antlr 739	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝐴 ⊆ (Base‘𝑅))
45		difin2 4254	. . . . . . 7 ⊢ (𝐴 ⊆ (Base‘𝑅) → (𝐴 ∖ { 0 }) = (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
46	44, 45	syl 18	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
47	42, 46	sseqtrrd 3974	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (𝐴 ∖ { 0 }))
48	43	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝐴 ⊆ (Base‘𝑅))
49		simprl 782	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (𝐴 ∖ { 0 }))
50	49, 5	sylib 221	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
51	50	simpld 499	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ 𝐴)
52	48, 51	sseldd 3938	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
53	50	simprd 500	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ≠ 0 )
54	34, 31, 12	drngunit 20832	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ DivRing → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ 0 )))
55	54	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ 0 )))
56	52, 53, 55	mpbir2and 725	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑅))
57		simprr 784	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝐼‘𝑥) ∈ 𝐴)
58	2, 31, 17, 25	subrgunit 20689	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ (𝐼‘𝑥) ∈ 𝐴)))
59	58	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ (𝐼‘𝑥) ∈ 𝐴)))
60	56, 51, 57, 59	mpbir3and 1361	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑆))
61	60	expr 461	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((𝐼‘𝑥) ∈ 𝐴 → 𝑥 ∈ (Unit‘𝑆)))
62	61	ralimdva 3177	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆)))
63	62	imp 411	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
64		dfss3 3926	. . . . . 6 ⊢ ((𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆) ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
65	63, 64	sylibr 237	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆))
66	47, 65	eqssd 3954	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) = (𝐴 ∖ { 0 }))
67	13	ad2antlr 739	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 0 = (0g‘𝑆))
68	67	sneqd 4601	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → { 0 } = {(0g‘𝑆)})
69	40, 68	difeq12d 4082	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
70	66, 69	eqtrd 2798	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
71	16, 17, 18	isdrng 20831	. . 3 ⊢ (𝑆 ∈ DivRing ↔ (𝑆 ∈ Ring ∧ (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g‘𝑆)})))
72	30, 70, 71	sylanbrc 594	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝑆 ∈ DivRing)
73	29, 72	impbida 812	1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  {csn 4589  ‘cfv 6536  (class class class)co 7410  Basecbs 17264   ↾_s cress 17285  0_gc0g 17487  Ringcrg 20310  Unitcui 20433  inv_rcinvr 20465  SubRingcsubrg 20668  DivRingcdr 20827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-minusg 18999  df-subg 19184  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-subrg 20669  df-drng 20829

This theorem is referenced by:  issdrg2  20898  cnsubdrglem  21568  extdg1id  34056