Theorem sdrgacs 36111

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 2461	. . . . . . . 8 |- ( invr ` R ) = ( invr ` R )
2		eqid 2461	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
3	1, 2	issdrg2 36108	. . . . . . 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 995	. . . . . . 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 257	. . . . . 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 919	. . . . 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 18057	. . . . . . . . 9 |- ( s e. (SubRing ` R ) -> s C_ B )
9		selpw 3969	. . . . . . . . 9 |- ( s e. ~P B <-> s C_ B )
10	8, 9	sylibr 217	. . . . . . . 8 |- ( s e. (SubRing ` R ) -> s e. ~P B )
11	10	adantl 472	. . . . . . 7 |- ( ( R e. DivRing /\ s e. (SubRing ` R ) ) -> s e. ~P B )
12		iftrue 3898	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 |- ( x = ( 0g ` R ) -> ( x e. y -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y ) )
15		eldifsni 4110	. . . . . . . . . . . . . 14 |- ( x e. ( y \ { ( 0g ` R ) } ) -> x =/= ( 0g ` R ) )
16	15	necon2bi 2665	. . . . . . . . . . . . 13 |- ( x = ( 0g ` R ) -> -. x e. ( y \ { ( 0g ` R ) } ) )
17	16	pm2.21d 110	. . . . . . . . . . . 12 |- ( x = ( 0g ` R ) -> ( x e. ( y \ { ( 0g ` R ) } ) -> ( ( invr ` R ) ` x ) e. y ) )
18	14, 17	2thd 248	. . . . . . . . . . 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 4109	. . . . . . . . . . . . 13 |- ( x e. ( y \ { ( 0g ` R ) } ) <-> ( x e. y /\ x =/= ( 0g ` R ) ) )
20	19	rbaibr 921	. . . . . . . . . . . 12 |- ( x =/= ( 0g ` R ) -> ( x e. y <-> x e. ( y \ { ( 0g ` R ) } ) ) )
21		ifnefalse 3904	. . . . . . . . . . . . 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 326	. . . . . . . . . . 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 2718	. . . . . . . . . 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 2828	. . . . . . . . 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 3555	. . . . . . . . . 10 |- ( y = s -> ( y \ { ( 0g ` R ) } ) = ( s \ { ( 0g ` R ) } ) )
27		eleq2 2528	. . . . . . . . . 10 |- ( y = s -> ( ( ( invr ` R ) ` x ) e. y <-> ( ( invr ` R ) ` x ) e. s ) )
28	26, 27	raleqbidv 3012	. . . . . . . . 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 265	. . . . . . . 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 3208	. . . . . . 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 17	. . . . . 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 651	. . . . 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 264	. . . 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 3628	. . . 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 271	. . 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 2459	. 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 5897	. . . . 5 |- ( Base ` R ) e. _V
38	7, 37	eqeltri 2535	. . . 4 |- B e. _V
39		mreacs 15612	. . . 4 |- ( B e. _V -> (ACS ` B ) e. (Moore ` ~P B ) )
40	38, 39	mp1i 13	. . 3 |- ( R e. DivRing -> (ACS ` B ) e. (Moore ` ~P B ) )
41		drngring 18030	. . . 4 |- ( R e. DivRing -> R e. Ring )
42	7	subrgacs 36110	. . . 4 |- ( R e. Ring -> (SubRing ` R ) e. (ACS ` B ) )
43	41, 42	syl 17	. . 3 |- ( R e. DivRing -> (SubRing ` R ) e. (ACS ` B ) )
44		simplr 767	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ x = ( 0g ` R ) ) -> x e. B )
45		df-ne 2634	. . . . . . 7 |- ( x =/= ( 0g ` R ) <-> -. x = ( 0g ` R ) )
46	7, 2, 1	drnginvrcl 18040	. . . . . . . 8 |- ( ( R e. DivRing /\ x e. B /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
47	46	3expa 1215	. . . . . . 7 |- ( ( ( R e. DivRing /\ x e. B ) /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
48	45, 47	sylan2br 483	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ -. x = ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
49	44, 48	ifclda 3924	. . . . 5 |- ( ( R e. DivRing /\ x e. B ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
50	49	ralrimiva 2813	. . . 4 |- ( R e. DivRing -> A. x e. B if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
51		acsfn1 15615	. . . 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 674	. . 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 15553	. . 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 1276	. 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 2539	1 |- ( R e. DivRing -> (SubDRing ` R ) e. (ACS ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   A.wral 2748   {crab 2752   _Vcvv 3056   \ cdif 3412   i^i cin 3414   C_ wss 3415   ifcif 3892   ~Pcpw 3962   {csn 3979   ` cfv 5600   Basecbs 15169   0gc0g 15386  Moorecmre 15536  ACScacs 15539   Ringcrg 17828   invrcinvr 17947   DivRingcdr 18023  SubRingcsubrg 18052  SubDRingcsdrg 36105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-tpos 6998  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-0g 15388  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-grp 16721  df-minusg 16722  df-subg 16862  df-mgp 17772  df-ur 17784  df-ring 17830  df-oppr 17899  df-dvdsr 17917  df-unit 17918  df-invr 17948  df-drng 18025  df-subrg 18054  df-sdrg 36106

This theorem is referenced by: (None)