Theorem sdrgacs 37771

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 2622	. . . . . . . 8 |- ( invr ` R ) = ( invr ` R )
2		eqid 2622	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
3	1, 2	issdrg2 37768	. . . . . . 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 1042	. . . . . . 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 264	. . . . . 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 944	. . . . 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 18781	. . . . . . . . 9 |- ( s e. (SubRing ` R ) -> s C_ B )
9		selpw 4165	. . . . . . . . 9 |- ( s e. ~P B <-> s C_ B )
10	8, 9	sylibr 224	. . . . . . . 8 |- ( s e. (SubRing ` R ) -> s e. ~P B )
11	10	adantl 482	. . . . . . 7 |- ( ( R e. DivRing /\ s e. (SubRing ` R ) ) -> s e. ~P B )
12		iftrue 4092	. . . . . . . . . . . . . 14 |- ( x = ( 0g ` R ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) = x )
13	12	eleq1d 2686	. . . . . . . . . . . . 13 |- ( x = ( 0g ` R ) -> ( if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y <-> x e. y ) )
14	13	biimprd 238	. . . . . . . . . . . 12 |- ( x = ( 0g ` R ) -> ( x e. y -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y ) )
15		eldifsni 4320	. . . . . . . . . . . . . 14 |- ( x e. ( y \ { ( 0g ` R ) } ) -> x =/= ( 0g ` R ) )
16	15	necon2bi 2824	. . . . . . . . . . . . 13 |- ( x = ( 0g ` R ) -> -. x e. ( y \ { ( 0g ` R ) } ) )
17	16	pm2.21d 118	. . . . . . . . . . . 12 |- ( x = ( 0g ` R ) -> ( x e. ( y \ { ( 0g ` R ) } ) -> ( ( invr ` R ) ` x ) e. y ) )
18	14, 17	2thd 255	. . . . . . . . . . 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 4317	. . . . . . . . . . . . 13 |- ( x e. ( y \ { ( 0g ` R ) } ) <-> ( x e. y /\ x =/= ( 0g ` R ) ) )
20	19	rbaibr 946	. . . . . . . . . . . 12 |- ( x =/= ( 0g ` R ) -> ( x e. y <-> x e. ( y \ { ( 0g ` R ) } ) ) )
21		ifnefalse 4098	. . . . . . . . . . . . 13 |- ( x =/= ( 0g ` R ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) = ( ( invr ` R ) ` x ) )
22	21	eleq1d 2686	. . . . . . . . . . . 12 |- ( x =/= ( 0g ` R ) -> ( if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. y <-> ( ( invr ` R ) ` x ) e. y ) )
23	20, 22	imbi12d 334	. . . . . . . . . . 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 2877	. . . . . . . . . 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 2978	. . . . . . . . 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 3721	. . . . . . . . . 10 |- ( y = s -> ( y \ { ( 0g ` R ) } ) = ( s \ { ( 0g ` R ) } ) )
27		eleq2 2690	. . . . . . . . . 10 |- ( y = s -> ( ( ( invr ` R ) ` x ) e. y <-> ( ( invr ` R ) ` x ) e. s ) )
28	26, 27	raleqbidv 3152	. . . . . . . . 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 272	. . . . . . . 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 3364	. . . . . . 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 673	. . . . 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 271	. . . 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 3796	. . . 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 278	. . 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 2620	. 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 6201	. . . . 5 |- ( Base ` R ) e. _V
38	7, 37	eqeltri 2697	. . . 4 |- B e. _V
39		mreacs 16319	. . . 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 18754	. . . 4 |- ( R e. DivRing -> R e. Ring )
42	7	subrgacs 37770	. . . 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 792	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ x = ( 0g ` R ) ) -> x e. B )
45		df-ne 2795	. . . . . . 7 |- ( x =/= ( 0g ` R ) <-> -. x = ( 0g ` R ) )
46	7, 2, 1	drnginvrcl 18764	. . . . . . . 8 |- ( ( R e. DivRing /\ x e. B /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
47	46	3expa 1265	. . . . . . 7 |- ( ( ( R e. DivRing /\ x e. B ) /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
48	45, 47	sylan2br 493	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ -. x = ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
49	44, 48	ifclda 4120	. . . . 5 |- ( ( R e. DivRing /\ x e. B ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
50	49	ralrimiva 2966	. . . 4 |- ( R e. DivRing -> A. x e. B if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
51		acsfn1 16322	. . . 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 695	. . 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 16259	. . 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 1326	. 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 2701	1 |- ( R e. DivRing -> (SubDRing ` R ) e. (ACS ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   ~Pcpw 4158   {csn 4177   ` cfv 5888   Basecbs 15857   0gc0g 16100  Moorecmre 16242  ACScacs 16245   Ringcrg 18547   invrcinvr 18671   DivRingcdr 18747  SubRingcsubrg 18776  SubDRingcsdrg 37765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-subrg 18778  df-sdrg 37766

This theorem is referenced by: (None)