Theorem issubdrg 17328

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 760	. . . . . 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 17306	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
4	1, 3	syl 16	. . . . 5 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> S e. Ring )
5		simpr 461	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. ( A \ { .0. } ) )
6		eldifsn 4140	. . . . . . . . 9 |- ( x e. ( A \ { .0. } ) <-> ( x e. A /\ x =/= .0. ) )
7	5, 6	sylib 196	. . . . . . . 8 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( x e. A /\ x =/= .0. ) )
8	7	simpld 459	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. A )
9	2	subrgbas 17312	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
10	1, 9	syl 16	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> A = ( Base ` S ) )
11	8, 10	eleqtrd 2533	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. ( Base ` S ) )
12	7	simprd 463	. . . . . . 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 17310	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> .0. = ( 0g ` S ) )
15	1, 14	syl 16	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> .0. = ( 0g ` S ) )
16	12, 15	neeqtrd 2738	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x =/= ( 0g ` S ) )
17		eqid 2443	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
18		eqid 2443	. . . . . . . 8 |- (Unit ` S ) = (Unit ` S )
19		eqid 2443	. . . . . . . 8 |- ( 0g ` S ) = ( 0g ` S )
20	17, 18, 19	drngunit 17275	. . . . . . 7 |- ( S e. DivRing -> ( x e. (Unit ` S ) <-> ( x e. ( Base ` S ) /\ x =/= ( 0g ` S ) ) ) )
21	20	ad2antlr 726	. . . . . 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 922	. . . . 5 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. (Unit ` S ) )
23		eqid 2443	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
24	18, 23, 17	ringinvcl 17199	. . . . 5 |- ( ( S e. Ring /\ x e. (Unit ` S ) ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
25	4, 22, 24	syl2anc 661	. . . 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 17319	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. (Unit ` S ) ) -> ( I ` x ) = ( ( invr ` S ) ` x ) )
28	1, 22, 27	syl2anc 661	. . . 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 2546	. . 3 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( I ` x ) e. A )
30	29	ralrimiva 2857	. 2 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) -> A. x e. ( A \ { .0. } ) ( I ` x ) e. A )
31	3	ad2antlr 726	. . 3 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> S e. Ring )
32		eqid 2443	. . . . . . . . . 10 |- (Unit ` R ) = (Unit ` R )
33	2, 32, 18	subrguss 17318	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> (Unit ` S ) C_ (Unit ` R ) )
34	33	ad2antlr 726	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) C_ (Unit ` R ) )
35		eqid 2443	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
36	35, 32, 13	isdrng 17274	. . . . . . . . . 10 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( ( Base ` R ) \ { .0. } ) ) )
37	36	simprbi 464	. . . . . . . . 9 |- ( R e. DivRing -> (Unit ` R ) = ( ( Base ` R ) \ { .0. } ) )
38	37	ad2antrr 725	. . . . . . . 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 3525	. . . . . . 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 17183	. . . . . . . 8 |- (Unit ` S ) C_ ( Base ` S )
41	9	ad2antlr 726	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> A = ( Base ` S ) )
42	40, 41	syl5sseqr 3538	. . . . . . 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 3707	. . . . . 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 17304	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
45	44	ad2antlr 726	. . . . . . 7 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> A C_ ( Base ` R ) )
46		difin2 3745	. . . . . . 7 |- ( A C_ ( Base ` R ) -> ( A \ { .0. } ) = ( ( ( Base ` R ) \ { .0. } ) i^i A ) )
47	45, 46	syl 16	. . . . . 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 3526	. . . . 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 726	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> A C_ ( Base ` R ) )
50		simprl 756	. . . . . . . . . . . . . 14 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. ( A \ { .0. } ) )
51	50, 6	sylib 196	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> ( x e. A /\ x =/= .0. ) )
52	51	simpld 459	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. A )
53	49, 52	sseldd 3490	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. ( Base ` R ) )
54	51	simprd 463	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x =/= .0. )
55	35, 32, 13	drngunit 17275	. . . . . . . . . . . 12 |- ( R e. DivRing -> ( x e. (Unit ` R ) <-> ( x e. ( Base ` R ) /\ x =/= .0. ) ) )
56	55	ad2antrr 725	. . . . . . . . . . 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 922	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. (Unit ` R ) )
58		simprr 757	. . . . . . . . . 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 17321	. . . . . . . . . . 11 |- ( A e. (SubRing ` R ) -> ( x e. (Unit ` S ) <-> ( x e. (Unit ` R ) /\ x e. A /\ ( I ` x ) e. A ) ) )
60	59	ad2antlr 726	. . . . . . . . . 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 1180	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. (Unit ` S ) )
62	61	expr 615	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ x e. ( A \ { .0. } ) ) -> ( ( I ` x ) e. A -> x e. (Unit ` S ) ) )
63	62	ralimdva 2851	. . . . . . 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 429	. . . . . 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 3479	. . . . . 6 |- ( ( A \ { .0. } ) C_ (Unit ` S ) <-> A. x e. ( A \ { .0. } ) x e. (Unit ` S ) )
66	64, 65	sylibr 212	. . . . 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 3506	. . . 4 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) = ( A \ { .0. } ) )
68	14	ad2antlr 726	. . . . . 6 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> .0. = ( 0g ` S ) )
69	68	sneqd 4026	. . . . 5 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> { .0. } = { ( 0g ` S ) } )
70	41, 69	difeq12d 3608	. . . 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 2484	. . 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 17274	. . 3 |- ( S e. DivRing <-> ( S e. Ring /\ (Unit ` S ) = ( ( Base ` S ) \ { ( 0g ` S ) } ) ) )
73	31, 71, 72	sylanbrc 664	. 2 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> S e. DivRing )
74	30, 73	impbida 832	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 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   \ cdif 3458   i^i cin 3460   C_ wss 3461   {csn 4014   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾_s cress 14510   0gc0g 14714   Ringcrg 17072  Unitcui 17162   invrcinvr 17194   DivRingcdr 17270  SubRingcsubrg 17299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-subg 16072  df-mgp 17016  df-ur 17028  df-ring 17074  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-drng 17272  df-subrg 17301

This theorem is referenced by:  cnsubdrglem  18343  issdrg2  31123