Theorem issubdrg 17302

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 758	. . . . . 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 17280	. . . . . 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 4157	. . . . . . . . 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 17286	. . . . . . . 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 2557	. . . . . 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 17284	. . . . . . . 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 2762	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x =/= ( 0g ` S ) )
17		eqid 2467	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
18		eqid 2467	. . . . . . . 8 |- (Unit ` S ) = (Unit ` S )
19		eqid 2467	. . . . . . . 8 |- ( 0g ` S ) = ( 0g ` S )
20	17, 18, 19	drngunit 17249	. . . . . . 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 920	. . . . 5 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. (Unit ` S ) )
23		eqid 2467	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
24	18, 23, 17	ringinvcl 17174	. . . . 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 17293	. . . . 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 2570	. . 3 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( I ` x ) e. A )
30	29	ralrimiva 2881	. 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 2467	. . . . . . . . . 10 |- (Unit ` R ) = (Unit ` R )
33	2, 32, 18	subrguss 17292	. . . . . . . . 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 2467	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
36	35, 32, 13	isdrng 17248	. . . . . . . . . 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 3545	. . . . . . 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 17158	. . . . . . . 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 3558	. . . . . . 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 3727	. . . . . 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 17278	. . . . . . . 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 3765	. . . . . . 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 3546	. . . . 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 755	. . . . . . . . . . . . . 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 3510	. . . . . . . . . . 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 17249	. . . . . . . . . . . 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 920	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. (Unit ` R ) )
58		simprr 756	. . . . . . . . . 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 17295	. . . . . . . . . . 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 1179	. . . . . . . . 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 2875	. . . . . . 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 3499	. . . . . 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 3526	. . . 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 4044	. . . . 5 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> { .0. } = { ( 0g ` S ) } )
70	41, 69	difeq12d 3628	. . . 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 2508	. . 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 17248	. . 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 830	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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   \ cdif 3478   i^i cin 3480   C_ wss 3481   {csn 4032   ` cfv 5593  (class class class)co 6294   Basecbs 14502   ↾_s cress 14503   0gc0g 14707   Ringcrg 17047  Unitcui 17137   invrcinvr 17169   DivRingcdr 17244  SubRingcsubrg 17273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-tpos 6965  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907  df-subg 16047  df-mgp 16991  df-ur 17003  df-ring 17049  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-drng 17246  df-subrg 17275

This theorem is referenced by:  cnsubdrglem  18316  issdrg2  31044