Theorem issubdrg 20665

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 775	. . . . . 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 20459	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
4	1, 3	syl 17	. . . . 5 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑆 ∈ Ring)
5		simpr 484	. . . . . . . . 9 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (𝐴 ∖ { 0 }))
6		eldifsn 4737	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
7	5, 6	sylib 218	. . . . . . . 8 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
8	7	simpld 494	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ 𝐴)
9	2	subrgbas 20466	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
10	1, 9	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝐴 = (Base‘𝑆))
11	8, 10	eleqtrd 2830	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Base‘𝑆))
12	7	simprd 495	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ 0 )
13		issubdrg.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
14	2, 13	subrg0 20464	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆))
15	1, 14	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 0 = (0g‘𝑆))
16	12, 15	neeqtrd 2994	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ (0g‘𝑆))
17		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
18		eqid 2729	. . . . . . . 8 ⊢ (Unit‘𝑆) = (Unit‘𝑆)
19		eqid 2729	. . . . . . . 8 ⊢ (0g‘𝑆) = (0g‘𝑆)
20	17, 18, 19	drngunit 20619	. . . . . . 7 ⊢ (𝑆 ∈ DivRing → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))))
21	20	ad2antlr 727	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))))
22	11, 16, 21	mpbir2and 713	. . . . 5 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑆))
23		eqid 2729	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
24	18, 23, 17	ringinvcl 20277	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝑆)) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
25	4, 22, 24	syl2anc 584	. . . 4 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
26		issubdrg.i	. . . . . 6 ⊢ 𝐼 = (invr‘𝑅)
27	2, 26, 18, 23	subrginv 20473	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Unit‘𝑆)) → (𝐼‘𝑥) = ((invr‘𝑆)‘𝑥))
28	1, 22, 27	syl2anc 584	. . . 4 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼‘𝑥) = ((invr‘𝑆)‘𝑥))
29	25, 28, 10	3eltr4d 2843	. . 3 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼‘𝑥) ∈ 𝐴)
30	29	ralrimiva 3121	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) → ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴)
31	3	ad2antlr 727	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝑆 ∈ Ring)
32		eqid 2729	. . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
33	2, 32, 18	subrguss 20472	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
34	33	ad2antlr 727	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
35		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	35, 32, 13	isdrng 20618	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 })))
37	36	simprbi 496	. . . . . . . . 9 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
38	37	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
39	34, 38	sseqtrd 3972	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
40	17, 18	unitss 20261	. . . . . . . 8 ⊢ (Unit‘𝑆) ⊆ (Base‘𝑆)
41	9	ad2antlr 727	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝐴 = (Base‘𝑆))
42	40, 41	sseqtrrid 3979	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
43	39, 42	ssind 4192	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
44	35	subrgss 20457	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
45	44	ad2antlr 727	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝐴 ⊆ (Base‘𝑅))
46		difin2 4252	. . . . . . 7 ⊢ (𝐴 ⊆ (Base‘𝑅) → (𝐴 ∖ { 0 }) = (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
47	45, 46	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
48	43, 47	sseqtrrd 3973	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (𝐴 ∖ { 0 }))
49	44	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝐴 ⊆ (Base‘𝑅))
50		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (𝐴 ∖ { 0 }))
51	50, 6	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
52	51	simpld 494	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ 𝐴)
53	49, 52	sseldd 3936	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
54	51	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ≠ 0 )
55	35, 32, 13	drngunit 20619	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ DivRing → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ 0 )))
56	55	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ 0 )))
57	53, 54, 56	mpbir2and 713	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑅))
58		simprr 772	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝐼‘𝑥) ∈ 𝐴)
59	2, 32, 18, 26	subrgunit 20475	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ (𝐼‘𝑥) ∈ 𝐴)))
60	59	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ (𝐼‘𝑥) ∈ 𝐴)))
61	57, 52, 58, 60	mpbir3and 1343	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑆))
62	61	expr 456	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((𝐼‘𝑥) ∈ 𝐴 → 𝑥 ∈ (Unit‘𝑆)))
63	62	ralimdva 3141	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆)))
64	63	imp 406	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
65		dfss3 3924	. . . . . 6 ⊢ ((𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆) ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
66	64, 65	sylibr 234	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆))
67	48, 66	eqssd 3953	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) = (𝐴 ∖ { 0 }))
68	14	ad2antlr 727	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 0 = (0g‘𝑆))
69	68	sneqd 4589	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → { 0 } = {(0g‘𝑆)})
70	41, 69	difeq12d 4078	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
71	67, 70	eqtrd 2764	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
72	17, 18, 19	isdrng 20618	. . 3 ⊢ (𝑆 ∈ DivRing ↔ (𝑆 ∈ Ring ∧ (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g‘𝑆)})))
73	31, 71, 72	sylanbrc 583	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝑆 ∈ DivRing)
74	30, 73	impbida 800	1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  {csn 4577  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_s cress 17141  0_gc0g 17343  Ringcrg 20118  Unitcui 20240  inv_rcinvr 20272  SubRingcsubrg 20454  DivRingcdr 20614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-grp 18815  df-minusg 18816  df-subg 19002  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-subrg 20455  df-drng 20616

This theorem is referenced by:  issdrg2  20680  cnsubdrglem  21325  extdg1id  33633