Theorem issubdrg 20689

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 20483	. . . . . 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 4750	. . . . . . . . 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 20490	. . . . . . . 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 20488	. . . . . . . 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 20643	. . . . . . 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 20301	. . . . 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 20497	. . . . 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 3125	. 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 20496	. . . . . . . . 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 20642	. . . . . . . . . 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 3983	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
40	17, 18	unitss 20285	. . . . . . . 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 3990	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
43	39, 42	ssind 4204	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
44	35	subrgss 20481	. . . . . . . 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 4264	. . . . . . 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 3984	. . . . 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 3947	. . . . . . . . . . 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 20643	. . . . . . . . . . . 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 20499	. . . . . . . . . . 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 3145	. . . . . . 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 3935	. . . . . 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 3964	. . . 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 4601	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → { 0 } = {(0g‘𝑆)})
70	41, 69	difeq12d 4090	. . . 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 20642	. . 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 3911   ∩ cin 3913   ⊆ wss 3914  {csn 4589  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402  Ringcrg 20142  Unitcui 20264  inv_rcinvr 20296  SubRingcsubrg 20478  DivRingcdr 20638

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-subg 19055  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-invr 20297  df-subrg 20479  df-drng 20640

This theorem is referenced by:  issdrg2  20704  cnsubdrglem  21335  extdg1id  33661