Theorem issubdrg 16890

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	subrgrng 16868	. . . . . 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 4000	. . . . . . . . 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 16874	. . . . . . . 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 2519	. . . . . 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 16872	. . . . . . . 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 2630	. . . . . 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 16837	. . . . . . 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 913	. . . . 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	rnginvcl 16768	. . . . 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 16881	. . . . 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 2524	. . 3 |- ( ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ S e. DivRing ) /\ x e. ( A \ { .0. } ) ) -> ( I ` x ) e. A )
30	29	ralrimiva 2799	. 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 16880	. . . . . . . . 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 16836	. . . . . . . . . 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 3392	. . . . . . 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 16752	. . . . . . . 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 3405	. . . . . . 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 3574	. . . . . 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 16866	. . . . . . . 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 3612	. . . . . . 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 3393	. . . . 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 3357	. . . . . . . . . . 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 16837	. . . . . . . . . . . 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 913	. . . . . . . . . 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 16883	. . . . . . . . . . 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 1171	. . . . . . . . 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 2794	. . . . . . 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 3346	. . . . . 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 3373	. . . 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 3889	. . . . 5 |- ( ( ( R e. DivRing /\ A e. (SubRing ` R ) ) /\ A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) -> { .0. } = { ( 0g ` S ) } )
70	41, 69	difeq12d 3475	. . . 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 2475	. . 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 16836	. . 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 828	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 965   = wceq 1369   e. wcel 1756   =/= wne 2606   A.wral 2715   \ cdif 3325   i^i cin 3327   C_ wss 3328   {csn 3877   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾_s cress 14175   0gc0g 14378   Ringcrg 16645  Unitcui 16731   invrcinvr 16763   DivRingcdr 16832  SubRingcsubrg 16861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-subg 15678  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-drng 16834  df-subrg 16863

This theorem is referenced by:  cnsubdrglem  17864  issdrg2  29555