Theorem sdrgacs 29705

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 2454	. . . . . . . 8 |- ( invr ` R ) = ( invr ` R )
2		eqid 2454	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
3	1, 2	issdrg2 29702	. . . . . . 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 16988	. . . . . . . . 9 |- ( s e. (SubRing ` R ) -> s C_ B )
9		selpw 3974	. . . . . . . . 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 3904	. . . . . . . . . . . . . 14 |- ( x = ( 0g ` R ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) = x )
13	12	eleq1d 2523	. . . . . . . . . . . . 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 4108	. . . . . . . . . . . . . 14 |- ( x e. ( y \ { ( 0g ` R ) } ) -> x =/= ( 0g ` R ) )
16	15	necon2bi 2688	. . . . . . . . . . . . 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 4107	. . . . . . . . . . . . 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 3908	. . . . . . . . . . . . 13 |- ( x =/= ( 0g ` R ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) = ( ( invr ` R ) ` x ) )
22	21	eleq1d 2523	. . . . . . . . . . . 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 2764	. . . . . . . . . 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 2836	. . . . . . . . 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 3574	. . . . . . . . . 10 |- ( y = s -> ( y \ { ( 0g ` R ) } ) = ( s \ { ( 0g ` R ) } ) )
27		eleq2 2527	. . . . . . . . . 10 |- ( y = s -> ( ( ( invr ` R ) ` x ) e. y <-> ( ( invr ` R ) ` x ) e. s ) )
28	26, 27	raleqbidv 3035	. . . . . . . . 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 3223	. . . . . . 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 3646	. . . 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 2451	. 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 5808	. . . . 5 |- ( Base ` R ) e. _V
38	7, 37	eqeltri 2538	. . . 4 |- B e. _V
39		mreacs 14714	. . . 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 16961	. . . 4 |- ( R e. DivRing -> R e. Ring )
42	7	subrgacs 29704	. . . 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 2649	. . . . . . 7 |- ( x =/= ( 0g ` R ) <-> -. x = ( 0g ` R ) )
46	7, 2, 1	drnginvrcl 16971	. . . . . . . 8 |- ( ( R e. DivRing /\ x e. B /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
47	46	3expa 1188	. . . . . . 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 3928	. . . . 5 |- ( ( R e. DivRing /\ x e. B ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
50	49	ralrimiva 2829	. . . 4 |- ( R e. DivRing -> A. x e. B if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
51		acsfn1 14717	. . . 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 14655	. . 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 1219	. 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 2542	1 |- ( R e. DivRing -> (SubDRing ` R ) e. (ACS ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2647   A.wral 2798   {crab 2802   _Vcvv 3076   \ cdif 3432   i^i cin 3434   C_ wss 3435   ifcif 3898   ~Pcpw 3967   {csn 3984   ` cfv 5525   Basecbs 14291   0gc0g 14496  Moorecmre 14638  ACScacs 14641   Ringcrg 16767   invrcinvr 16885   DivRingcdr 16954  SubRingcsubrg 16983  SubDRingcsdrg 29699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-tpos 6854  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-0g 14498  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-grp 15663  df-minusg 15664  df-subg 15796  df-mgp 16713  df-ur 16725  df-rng 16769  df-oppr 16837  df-dvdsr 16855  df-unit 16856  df-invr 16886  df-drng 16956  df-subrg 16985  df-sdrg 29700

This theorem is referenced by: (None)