Theorem sdrgacs 29511

Description: Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)

Hypothesis

Ref	Expression
subrgacs.b	|- B = ( Base ` R )

Assertion

Ref	Expression
sdrgacs	|- ( R e. DivRing -> (SubDRing ` R ) e. (ACS ` B ) )

Proof of Theorem sdrgacs

Dummy variables x s y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2438	. . . . . . . 8 |- ( invr ` R ) = ( invr ` R )
2		eqid 2438	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
3	1, 2	issdrg2 29508	. . . . . . 7 |- ( s e. (SubDRing ` R ) <-> ( R e. DivRing /\ s e. (SubRing ` R ) /\ A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) )
4		3anass 969	. . . . . . 7 |- ( ( R e. DivRing /\ s e. (SubRing ` R ) /\ A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) <-> ( R e. DivRing /\ ( s e. (SubRing ` R ) /\ A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) ) )
5	3, 4	bitri 249	. . . . . 6 |- ( s e. (SubDRing ` R ) <-> ( R e. DivRing /\ ( s e. (SubRing ` R ) /\ A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) ) )
6	5	baib 896	. . . . 5 |- ( R e. DivRing -> ( s e. (SubDRing ` R ) <-> ( s e. (SubRing ` R ) /\ A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) ) )
7		subrgacs.b	. . . . . . . . . 10 |- B = ( Base ` R )
8	7	subrgss 16844	. . . . . . . . 9 |- ( s e. (SubRing ` R ) -> s C_ B )
9		selpw 3862	. . . . . . . . 9 |- ( s e. ~P B <-> s C_ B )
10	8, 9	sylibr 212	. . . . . . . 8 |- ( s e. (SubRing ` R ) -> s e. ~P B )
11	10	adantl 466	. . . . . . 7 |- ( ( R e. DivRing /\ s e. (SubRing ` R ) ) -> s e. ~P B )
12		iftrue 3792	. . . . . . . . . . . . . 14 |- ( x = ( 0g ` R ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) = x )
13	12	eleq1d 2504	. . . . . . . . . . . . 13 |- ( x = ( 0g ` R ) -> ( if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y <-> x e. y ) )
14	13	biimprd 223	. . . . . . . . . . . 12 |- ( x = ( 0g ` R ) -> ( x e. y -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y ) )
15		eldifsni 3996	. . . . . . . . . . . . . 14 |- ( x e. ( y \ { ( 0g ` R ) } ) -> x =/= ( 0g ` R ) )
16	15	necon2bi 2652	. . . . . . . . . . . . 13 |- ( x = ( 0g ` R ) -> -. x e. ( y \ { ( 0g ` R ) } ) )
17	16	pm2.21d 106	. . . . . . . . . . . 12 |- ( x = ( 0g ` R ) -> ( x e. ( y \ { ( 0g ` R ) } ) -> ( ( invr ` R ) ` x ) e. y ) )
18	14, 17	2thd 240	. . . . . . . . . . 11 |- ( x = ( 0g ` R ) -> ( ( x e. y -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y ) <-> ( x e. ( y \ { ( 0g ` R ) } ) -> ( ( invr ` R ) ` x ) e. y ) ) )
19		eldifsn 3995	. . . . . . . . . . . . 13 |- ( x e. ( y \ { ( 0g ` R ) } ) <-> ( x e. y /\ x =/= ( 0g ` R ) ) )
20	19	rbaibr 899	. . . . . . . . . . . 12 |- ( x =/= ( 0g ` R ) -> ( x e. y <-> x e. ( y \ { ( 0g ` R ) } ) ) )
21		ifnefalse 3796	. . . . . . . . . . . . 13 |- ( x =/= ( 0g ` R ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) = ( ( invr ` R ) ` x ) )
22	21	eleq1d 2504	. . . . . . . . . . . 12 |- ( x =/= ( 0g ` R ) -> ( if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y <-> ( ( invr ` R ) ` x ) e. y ) )
23	20, 22	imbi12d 320	. . . . . . . . . . 11 |- ( x =/= ( 0g ` R ) -> ( ( x e. y -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y ) <-> ( x e. ( y \ { ( 0g ` R ) } ) -> ( ( invr ` R ) ` x ) e. y ) ) )
24	18, 23	pm2.61ine 2682	. . . . . . . . . 10 |- ( ( x e. y -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y ) <-> ( x e. ( y \ { ( 0g ` R ) } ) -> ( ( invr ` R ) ` x ) e. y ) )
25	24	ralbii2 2738	. . . . . . . . 9 |- ( A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y <-> A. x e. ( y \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. y )
26		difeq1 3462	. . . . . . . . . 10 |- ( y = s -> ( y \ { ( 0g ` R ) } ) = ( s \ { ( 0g ` R ) } ) )
27		eleq2 2499	. . . . . . . . . 10 |- ( y = s -> ( ( ( invr ` R ) ` x ) e. y <-> ( ( invr ` R ) ` x ) e. s ) )
28	26, 27	raleqbidv 2926	. . . . . . . . 9 |- ( y = s -> ( A. x e. ( y \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. y <-> A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) )
29	25, 28	syl5bb 257	. . . . . . . 8 |- ( y = s -> ( A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y <-> A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) )
30	29	elrab3 3113	. . . . . . 7 |- ( s e. ~P B -> ( s e. { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } <-> A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) )
31	11, 30	syl 16	. . . . . 6 |- ( ( R e. DivRing /\ s e. (SubRing ` R ) ) -> ( s e. { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } <-> A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) )
32	31	pm5.32da 641	. . . . 5 |- ( R e. DivRing -> ( ( s e. (SubRing ` R ) /\ s e. { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) <-> ( s e. (SubRing ` R ) /\ A. x e. ( s \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. s ) ) )
33	6, 32	bitr4d 256	. . . 4 |- ( R e. DivRing -> ( s e. (SubDRing ` R ) <-> ( s e. (SubRing ` R ) /\ s e. { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) ) )
34		elin 3534	. . . 4 |- ( s e. ( (SubRing ` R ) i^i { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) <-> ( s e. (SubRing ` R ) /\ s e. { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) )
35	33, 34	syl6bbr 263	. . 3 |- ( R e. DivRing -> ( s e. (SubDRing ` R ) <-> s e. ( (SubRing ` R ) i^i { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) ) )
36	35	eqrdv 2436	. 2 |- ( R e. DivRing -> (SubDRing ` R ) = ( (SubRing ` R ) i^i { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) )
37		fvex 5696	. . . . 5 |- ( Base ` R ) e. _V
38	7, 37	eqeltri 2508	. . . 4 |- B e. _V
39		mreacs 14588	. . . 4 |- ( B e. _V -> (ACS ` B ) e. (Moore ` ~P B ) )
40	38, 39	mp1i 12	. . 3 |- ( R e. DivRing -> (ACS ` B ) e. (Moore ` ~P B ) )
41		drngrng 16817	. . . 4 |- ( R e. DivRing -> R e. Ring )
42	7	subrgacs 29510	. . . 4 |- ( R e. Ring -> (SubRing ` R ) e. (ACS ` B ) )
43	41, 42	syl 16	. . 3 |- ( R e. DivRing -> (SubRing ` R ) e. (ACS ` B ) )
44		simplr 754	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ x = ( 0g ` R ) ) -> x e. B )
45		df-ne 2603	. . . . . . 7 |- ( x =/= ( 0g ` R ) <-> -. x = ( 0g ` R ) )
46	7, 2, 1	drnginvrcl 16827	. . . . . . . 8 |- ( ( R e. DivRing /\ x e. B /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
47	46	3expa 1187	. . . . . . 7 |- ( ( ( R e. DivRing /\ x e. B ) /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
48	45, 47	sylan2br 476	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ -. x = ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
49	44, 48	ifclda 3816	. . . . 5 |- ( ( R e. DivRing /\ x e. B ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
50	49	ralrimiva 2794	. . . 4 |- ( R e. DivRing -> A. x e. B if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
51		acsfn1 14591	. . . 4 |- ( ( B e. _V /\ A. x e. B if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B ) -> { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } e. (ACS ` B ) )
52	38, 50, 51	sylancr 663	. . 3 |- ( R e. DivRing -> { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } e. (ACS ` B ) )
53		mreincl 14529	. . 3 |- ( ( (ACS ` B ) e. (Moore ` ~P B ) /\ (SubRing ` R ) e. (ACS ` B ) /\ { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } e. (ACS ` B ) ) -> ( (SubRing ` R ) i^i { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) e. (ACS ` B ) )
54	40, 43, 52, 53	syl3anc 1218	. 2 |- ( R e. DivRing -> ( (SubRing ` R ) i^i { y e. ~P B | A. x e. y if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y } ) e. (ACS ` B ) )
55	36, 54	eqeltrd 2512	1 |- ( R e. DivRing -> (SubDRing ` R ) e. (ACS ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   {crab 2714   _Vcvv 2967   \ cdif 3320   i^i cin 3322   C_ wss 3323   ifcif 3786   ~Pcpw 3855   {csn 3872   ` cfv 5413   Basecbs 14166   0gc0g 14370  Moorecmre 14512  ACScacs 14515   Ringcrg 16633   invrcinvr 16751   DivRingcdr 16810  SubRingcsubrg 16839  SubDRingcsdrg 29505

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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-subg 15669  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-drng 16812  df-subrg 16841  df-sdrg 29506

This theorem is referenced by: (None)