Theorem issubdrg 20381

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 774	. . . . . 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 20358	. . . . . 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 485	. . . . . . . . 9 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (𝐴 ∖ { 0 }))
6		eldifsn 4789	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
7	5, 6	sylib 217	. . . . . . . 8 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
8	7	simpld 495	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ 𝐴)
9	2	subrgbas 20364	. . . . . . . 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 2835	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Base‘𝑆))
12	7	simprd 496	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ 0 )
13		issubdrg.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
14	2, 13	subrg0 20362	. . . . . . . 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 3010	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ (0g‘𝑆))
17		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
18		eqid 2732	. . . . . . . 8 ⊢ (Unit‘𝑆) = (Unit‘𝑆)
19		eqid 2732	. . . . . . . 8 ⊢ (0g‘𝑆) = (0g‘𝑆)
20	17, 18, 19	drngunit 20312	. . . . . . 7 ⊢ (𝑆 ∈ DivRing → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))))
21	20	ad2antlr 725	. . . . . 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 711	. . . . 5 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑆))
23		eqid 2732	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
24	18, 23, 17	ringinvcl 20198	. . . . 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 20371	. . . . 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 2848	. . 3 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼‘𝑥) ∈ 𝐴)
30	29	ralrimiva 3146	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) → ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴)
31	3	ad2antlr 725	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝑆 ∈ Ring)
32		eqid 2732	. . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
33	2, 32, 18	subrguss 20370	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
34	33	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
35		eqid 2732	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	35, 32, 13	isdrng 20311	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 })))
37	36	simprbi 497	. . . . . . . . 9 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
38	37	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
39	34, 38	sseqtrd 4021	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
40	17, 18	unitss 20182	. . . . . . . 8 ⊢ (Unit‘𝑆) ⊆ (Base‘𝑆)
41	9	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝐴 = (Base‘𝑆))
42	40, 41	sseqtrrid 4034	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
43	39, 42	ssind 4231	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
44	35	subrgss 20356	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
45	44	ad2antlr 725	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 𝐴 ⊆ (Base‘𝑅))
46		difin2 4290	. . . . . . 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 4022	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (𝐴 ∖ { 0 }))
49	44	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝐴 ⊆ (Base‘𝑅))
50		simprl 769	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (𝐴 ∖ { 0 }))
51	50, 6	sylib 217	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0 ))
52	51	simpld 495	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ 𝐴)
53	49, 52	sseldd 3982	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
54	51	simprd 496	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ≠ 0 )
55	35, 32, 13	drngunit 20312	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ DivRing → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ 0 )))
56	55	ad2antrr 724	. . . . . . . . . . 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 711	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑅))
58		simprr 771	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → (𝐼‘𝑥) ∈ 𝐴)
59	2, 32, 18, 26	subrgunit 20373	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ (𝐼‘𝑥) ∈ 𝐴)))
60	59	ad2antlr 725	. . . . . . . . . 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 1342	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼‘𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑆))
62	61	expr 457	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((𝐼‘𝑥) ∈ 𝐴 → 𝑥 ∈ (Unit‘𝑆)))
63	62	ralimdva 3167	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆)))
64	63	imp 407	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
65		dfss3 3969	. . . . . 6 ⊢ ((𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆) ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
66	64, 65	sylibr 233	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆))
67	48, 66	eqssd 3998	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) = (𝐴 ∖ { 0 }))
68	14	ad2antlr 725	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → 0 = (0g‘𝑆))
69	68	sneqd 4639	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → { 0 } = {(0g‘𝑆)})
70	41, 69	difeq12d 4122	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
71	67, 70	eqtrd 2772	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴) → (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g‘𝑆)}))
72	17, 18, 19	isdrng 20311	. . 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 799	1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  {csn 4627  ‘cfv 6540  (class class class)co 7405  Basecbs 17140   ↾_s cress 17169  0_gc0g 17381  Ringcrg 20049  Unitcui 20161  inv_rcinvr 20193  DivRingcdr 20307  SubRingcsubrg 20351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-subg 18997  df-mgp 19982  df-ur 19999  df-ring 20051  df-oppr 20142  df-dvdsr 20163  df-unit 20164  df-invr 20194  df-drng 20309  df-subrg 20353

This theorem is referenced by:  issdrg2  20403  cnsubdrglem  20988  extdg1id  32730