Theorem issubdrg 18805

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 e. DivRing /\ A e. (SubRing ` R ) ) -> ( S e. DivRing <-> A. x e. ( A \ { .0. } ) ( I ` x ) e. 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 799	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> A e. (SubRing ` R ) )
2		issubdrg.s	. . . . . . 7 |- S = ( R ↾s A )
3	2	subrgring 18783	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
4	1, 3	syl 17	. . . . 5 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> S e. Ring )
5		simpr 477	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. ( A \ { .0. } ) )
6		eldifsn 4317	. . . . . . . . 9 |- ( x e. ( A \ { .0. } ) <-> ( x e. A /\ x =/= .0. ) )
7	5, 6	sylib 208	. . . . . . . 8 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( x e. A /\ x =/= .0. ) )
8	7	simpld 475	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. A )
9	2	subrgbas 18789	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
10	1, 9	syl 17	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> A = ( Base ` S ) )
11	8, 10	eleqtrd 2703	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. ( Base ` S ) )
12	7	simprd 479	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x =/= .0. )
13		issubdrg.z	. . . . . . . . 9 |- .0. = ( 0g ` R )
14	2, 13	subrg0 18787	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> .0. = ( 0g ` S ) )
15	1, 14	syl 17	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> .0. = ( 0g ` S ) )
16	12, 15	neeqtrd 2863	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x =/= ( 0g ` S ) )
17		eqid 2622	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
18		eqid 2622	. . . . . . . 8 |- (Unit ` S ) = (Unit ` S )
19		eqid 2622	. . . . . . . 8 |- ( 0g ` S ) = ( 0g ` S )
20	17, 18, 19	drngunit 18752	. . . . . . 7 |- ( S e. DivRing -> ( x e. (Unit ` S ) <-> ( x e. ( Base ` S ) /\ x =/= ( 0g ` S ) ) ) )
21	20	ad2antlr 763	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( x e. (Unit ` S ) <-> ( x e. ( Base ` S ) /\ x =/= ( 0g ` S ) ) ) )
22	11, 16, 21	mpbir2and 957	. . . . 5 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. (Unit ` S ) )
23		eqid 2622	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
24	18, 23, 17	ringinvcl 18676	. . . . 5 |- ( ( S e. Ring /\ x e. (Unit ` S ) ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
25	4, 22, 24	syl2anc 693	. . . 4 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
26		issubdrg.i	. . . . . 6 |- I = ( invr ` R )
27	2, 26, 18, 23	subrginv 18796	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. (Unit ` S ) ) -> ( I ` x ) = ( ( invr ` S ) ` x ) )
28	1, 22, 27	syl2anc 693	. . . 4 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( I ` x ) = ( ( invr ` S ) ` x ) )
29	25, 28, 10	3eltr4d 2716	. . 3 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( I ` x ) e. A )
30	29	ralrimiva 2966	. 2 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) -> A. x e. ( A \ { .0. } ) ( I ` x ) e. A )
31	3	ad2antlr 763	. . 3 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> S e. Ring )
32		eqid 2622	. . . . . . . . . 10 |- (Unit ` R ) = (Unit ` R )
33	2, 32, 18	subrguss 18795	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> (Unit ` S ) C_ (Unit ` R ) )
34	33	ad2antlr 763	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) C_ (Unit ` R ) )
35		eqid 2622	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
36	35, 32, 13	isdrng 18751	. . . . . . . . . 10 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( ( Base ` R ) \ { .0. } ) ) )
37	36	simprbi 480	. . . . . . . . 9 |- ( R e. DivRing -> (Unit ` R ) = ( ( Base ` R ) \ { .0. } ) )
38	37	ad2antrr 762	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` R ) = ( ( Base ` R ) \ { .0. } ) )
39	34, 38	sseqtrd 3641	. . . . . . 7 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) C_ ( ( Base ` R ) \ { .0. } ) )
40	17, 18	unitss 18660	. . . . . . . 8 |- (Unit ` S ) C_ ( Base ` S )
41	9	ad2antlr 763	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> A = ( Base ` S ) )
42	40, 41	syl5sseqr 3654	. . . . . . 7 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) C_ A )
43	39, 42	ssind 3837	. . . . . 6 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) C_ ( ( ( Base ` R ) \ { .0. } ) i^i A ) )
44	35	subrgss 18781	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
45	44	ad2antlr 763	. . . . . . 7 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> A C_ ( Base ` R ) )
46		difin2 3890	. . . . . . 7 |- ( A C_ ( Base ` R ) -> ( A \ { .0. } ) = ( ( ( Base ` R ) \ { .0. } ) i^i A ) )
47	45, 46	syl 17	. . . . . 6 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> ( A \ { .0. } ) = ( ( ( Base ` R ) \ { .0. } ) i^i A ) )
48	43, 47	sseqtr4d 3642	. . . . 5 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) C_ ( A \ { .0. } ) )
49	44	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> A C_ ( Base ` R ) )
50		simprl 794	. . . . . . . . . . . . . 14 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. ( A \ { .0. } ) )
51	50, 6	sylib 208	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> ( x e. A /\ x =/= .0. ) )
52	51	simpld 475	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. A )
53	49, 52	sseldd 3604	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. ( Base ` R ) )
54	51	simprd 479	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x =/= .0. )
55	35, 32, 13	drngunit 18752	. . . . . . . . . . . 12 |- ( R e. DivRing -> ( x e. (Unit ` R ) <-> ( x e. ( Base ` R ) /\ x =/= .0. ) ) )
56	55	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> ( x e. (Unit ` R ) <-> ( x e. ( Base ` R ) /\ x =/= .0. ) ) )
57	53, 54, 56	mpbir2and 957	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. (Unit ` R ) )
58		simprr 796	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> ( I ` x ) e. A )
59	2, 32, 18, 26	subrgunit 18798	. . . . . . . . . . 11 |- ( A e. (SubRing ` R ) -> ( x e. (Unit ` S ) <-> ( x e. (Unit ` R ) /\ x e. A /\ ( I ` x ) e. A ) ) )
60	59	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> ( x e. (Unit ` S ) <-> ( x e. (Unit ` R ) /\ x e. A /\ ( I ` x ) e. A ) ) )
61	57, 52, 58, 60	mpbir3and 1245	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. (Unit ` S ) )
62	61	expr 643	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ x e. ( A \ { .0. } ) ) -> ( ( I ` x ) e. A -> x e. (Unit ` S ) ) )
63	62	ralimdva 2962	. . . . . . 7 |- ( ( R e. DivRing /\ A e. (SubRing ` R ) ) -> ( A. x e. ( A \ { .0. } ) ( I ` x ) e. A -> A. x e. ( A \ { .0. } ) x e. (Unit ` S ) ) )
64	63	imp 445	. . . . . 6 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> A. x e. ( A \ { .0. } ) x e. (Unit ` S ) )
65		dfss3 3592	. . . . . 6 |- ( ( A \ { .0. } ) C_ (Unit ` S ) <-> A. x e. ( A \ { .0. } ) x e. (Unit ` S ) )
66	64, 65	sylibr 224	. . . . 5 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> ( A \ { .0. } ) C_ (Unit ` S ) )
67	48, 66	eqssd 3620	. . . 4 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) = ( A \ { .0. } ) )
68	14	ad2antlr 763	. . . . . 6 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> .0. = ( 0g ` S ) )
69	68	sneqd 4189	. . . . 5 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> { .0. } = { ( 0g ` S ) } )
70	41, 69	difeq12d 3729	. . . 4 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> ( A \ { .0. } ) = ( ( Base ` S ) \ { ( 0g ` S ) } ) )
71	67, 70	eqtrd 2656	. . 3 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) = ( ( Base ` S ) \ { ( 0g ` S ) } ) )
72	17, 18, 19	isdrng 18751	. . 3 |- ( S e. DivRing <-> ( S e. Ring /\ (Unit ` S ) = ( ( Base ` S ) \ { ( 0g ` S ) } ) ) )
73	31, 71, 72	sylanbrc 698	. 2 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> S e. DivRing )
74	30, 73	impbida 877	1 |- ( ( R e. DivRing /\ A e. (SubRing ` R ) ) -> ( S e. DivRing <-> A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   Ringcrg 18547  Unitcui 18639   invrcinvr 18671   DivRingcdr 18747  SubRingcsubrg 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-subrg 18778

This theorem is referenced by:  cnsubdrglem  19797  issdrg2  37768