Theorem issubdrg 18111

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 777	. . . . . 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 18089	. . . . . 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 468	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. ( A \ { .0. } ) )
6		eldifsn 4088	. . . . . . . . 9 |- ( x e. ( A \ { .0. } ) <-> ( x e. A /\ x =/= .0. ) )
7	5, 6	sylib 201	. . . . . . . 8 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( x e. A /\ x =/= .0. ) )
8	7	simpld 466	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. A )
9	2	subrgbas 18095	. . . . . . . 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 2551	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. ( Base ` S ) )
12	7	simprd 470	. . . . . . 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 18093	. . . . . . . 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 2712	. . . . . 6 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x =/= ( 0g ` S ) )
17		eqid 2471	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
18		eqid 2471	. . . . . . . 8 |- (Unit ` S ) = (Unit ` S )
19		eqid 2471	. . . . . . . 8 |- ( 0g ` S ) = ( 0g ` S )
20	17, 18, 19	drngunit 18058	. . . . . . 7 |- ( S e. DivRing -> ( x e. (Unit ` S ) <-> ( x e. ( Base ` S ) /\ x =/= ( 0g ` S ) ) ) )
21	20	ad2antlr 741	. . . . . 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 936	. . . . 5 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> x e. (Unit ` S ) )
23		eqid 2471	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
24	18, 23, 17	ringinvcl 17982	. . . . 5 |- ( ( S e. Ring /\ x e. (Unit ` S ) ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
25	4, 22, 24	syl2anc 673	. . . 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 18102	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. (Unit ` S ) ) -> ( I ` x ) = ( ( invr ` S ) ` x ) )
28	1, 22, 27	syl2anc 673	. . . 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 2564	. . 3 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( I ` x ) e. A )
30	29	ralrimiva 2809	. 2 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) -> A. x e. ( A \ { .0. } ) ( I ` x ) e. A )
31	3	ad2antlr 741	. . 3 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> S e. Ring )
32		eqid 2471	. . . . . . . . . 10 |- (Unit ` R ) = (Unit ` R )
33	2, 32, 18	subrguss 18101	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> (Unit ` S ) C_ (Unit ` R ) )
34	33	ad2antlr 741	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) C_ (Unit ` R ) )
35		eqid 2471	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
36	35, 32, 13	isdrng 18057	. . . . . . . . . 10 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( ( Base ` R ) \ { .0. } ) ) )
37	36	simprbi 471	. . . . . . . . 9 |- ( R e. DivRing -> (Unit ` R ) = ( ( Base ` R ) \ { .0. } ) )
38	37	ad2antrr 740	. . . . . . . 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 3454	. . . . . . 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 17966	. . . . . . . 8 |- (Unit ` S ) C_ ( Base ` S )
41	9	ad2antlr 741	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> A = ( Base ` S ) )
42	40, 41	syl5sseqr 3467	. . . . . . 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 3647	. . . . . 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 18087	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
45	44	ad2antlr 741	. . . . . . 7 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> A C_ ( Base ` R ) )
46		difin2 3696	. . . . . . 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 3455	. . . . 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 741	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> A C_ ( Base ` R ) )
50		simprl 772	. . . . . . . . . . . . . 14 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. ( A \ { .0. } ) )
51	50, 6	sylib 201	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> ( x e. A /\ x =/= .0. ) )
52	51	simpld 466	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. A )
53	49, 52	sseldd 3419	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. ( Base ` R ) )
54	51	simprd 470	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x =/= .0. )
55	35, 32, 13	drngunit 18058	. . . . . . . . . . . 12 |- ( R e. DivRing -> ( x e. (Unit ` R ) <-> ( x e. ( Base ` R ) /\ x =/= .0. ) ) )
56	55	ad2antrr 740	. . . . . . . . . . 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 936	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. (Unit ` R ) )
58		simprr 774	. . . . . . . . . 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 18104	. . . . . . . . . . 11 |- ( A e. (SubRing ` R ) -> ( x e. (Unit ` S ) <-> ( x e. (Unit ` R ) /\ x e. A /\ ( I ` x ) e. A ) ) )
60	59	ad2antlr 741	. . . . . . . . . 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 1213	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ ( x e. ( A \ { .0. } ) /\ ( I ` x ) e. A ) ) -> x e. (Unit ` S ) )
62	61	expr 626	. . . . . . . 8 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ x e. ( A \ { .0. } ) ) -> ( ( I ` x ) e. A -> x e. (Unit ` S ) ) )
63	62	ralimdva 2805	. . . . . . 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 436	. . . . . 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 3408	. . . . . 6 |- ( ( A \ { .0. } ) C_ (Unit ` S ) <-> A. x e. ( A \ { .0. } ) x e. (Unit ` S ) )
66	64, 65	sylibr 217	. . . . 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 3435	. . . 4 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> (Unit ` S ) = ( A \ { .0. } ) )
68	14	ad2antlr 741	. . . . . 6 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> .0. = ( 0g ` S ) )
69	68	sneqd 3971	. . . . 5 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> { .0. } = { ( 0g ` S ) } )
70	41, 69	difeq12d 3541	. . . 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 2505	. . 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 18057	. . 3 |- ( S e. DivRing <-> ( S e. Ring /\ (Unit ` S ) = ( ( Base ` S ) \ { ( 0g ` S ) } ) ) )
73	31, 71, 72	sylanbrc 677	. 2 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> S e. DivRing )
74	30, 73	impbida 850	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   \ cdif 3387   i^i cin 3389   C_ wss 3390   {csn 3959   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾_s cress 15200   0gc0g 15416   Ringcrg 17858  Unitcui 17945   invrcinvr 17977   DivRingcdr 18053  SubRingcsubrg 18082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-subg 16892  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-drng 18055  df-subrg 18084

This theorem is referenced by:  cnsubdrglem  19096  issdrg2  36135