Theorem issubdrg 20861

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 20659	. . . . . 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 4758	. . . . . . . . 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 20666	. . . . . . . 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 2871	. . . . . 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 20664	. . . . . . . 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 3033	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ (0g‘𝑆))
16		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
17		eqid 2769	. . . . . . . 8 ⊢ (Unit‘𝑆) = (Unit‘𝑆)
18		eqid 2769	. . . . . . . 8 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	16, 17, 18	drngunit 20818	. . . . . . 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 2769	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
23	17, 22, 16	ringinvcl 20474	. . . . 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 20673	. . . . 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 2884	. . 3 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼‘𝑥) ∈ 𝐴)
29	28	ralrimiva 3163	. 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 2769	. . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
32	2, 31, 17	subrguss 20672	. . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
35	34, 31, 12	isdrng 20817	. . . . . . . . . 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 3981	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
39	16, 17	unitss 20458	. . . . . . . 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 3988	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
42	38, 41	ssind 4201	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
43	34	subrgss 20657	. . . . . . . 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 4262	. . . . . . 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 3982	. . . . 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 3946	. . . . . . . . . . 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 20818	. . . . . . . . . . . 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 20675	. . . . . . . . . . 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 1359	. . . . . . . . 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 3183	. . . . . . 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 3934	. . . . . 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 3962	. . . 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 4606	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → { 0 } = {(0g‘𝑆)})
69	40, 68	difeq12d 4090	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
70	66, 69	eqtrd 2804	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
71	16, 17, 18	isdrng 20817	. . 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 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  {csn 4594  ‘cfv 6537  (class class class)co 7411  Basecbs 17269   ↾_s cress 17290  0_gc0g 17492  Ringcrg 20315  Unitcui 20437  inv_rcinvr 20469  SubRingcsubrg 20654  DivRingcdr 20813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-0g 17494  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-minusg 19004  df-subg 19189  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-ring 20317  df-oppr 20419  df-dvdsr 20439  df-unit 20440  df-invr 20470  df-subrg 20655  df-drng 20815

This theorem is referenced by:  issdrg2  20876  cnsubdrglem  21537  extdg1id  34001