Theorem sdrgacs 31394

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 31391	. . . . . . 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 975	. . . . . . 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 901	. . . . 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 17628	. . . . . . . . 9 |- ( s e. (SubRing ` R ) -> s C_ B )
9		selpw 4006	. . . . . . . . 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 464	. . . . . . 7 |- ( ( R e. DivRing /\ s e. (SubRing ` R ) ) -> s e. ~P B )
12		iftrue 3935	. . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . 14 |- ( x e. ( y \ { ( 0g ` R ) } ) -> x =/= ( 0g ` R ) )
16	15	necon2bi 2691	. . . . . . . . . . . . 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 4141	. . . . . . . . . . . . 13 |- ( x e. ( y \ { ( 0g ` R ) } ) <-> ( x e. y /\ x =/= ( 0g ` R ) ) )
20	19	rbaibr 903	. . . . . . . . . . . 12 |- ( x =/= ( 0g ` R ) -> ( x e. y <-> x e. ( y \ { ( 0g ` R ) } ) ) )
21		ifnefalse 3941	. . . . . . . . . . . . 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 318	. . . . . . . . . . 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 2767	. . . . . . . . . 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 2883	. . . . . . . . 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 3601	. . . . . . . . . 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 3065	. . . . . . . . 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 3255	. . . . . . 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 639	. . . . 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 3673	. . . 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 5858	. . . . 5 |- ( Base ` R ) e. _V
38	7, 37	eqeltri 2538	. . . 4 |- B e. _V
39		mreacs 15150	. . . 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		drngring 17601	. . . 4 |- ( R e. DivRing -> R e. Ring )
42	7	subrgacs 31393	. . . 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 753	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ x = ( 0g ` R ) ) -> x e. B )
45		df-ne 2651	. . . . . . 7 |- ( x =/= ( 0g ` R ) <-> -. x = ( 0g ` R ) )
46	7, 2, 1	drnginvrcl 17611	. . . . . . . 8 |- ( ( R e. DivRing /\ x e. B /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
47	46	3expa 1194	. . . . . . 7 |- ( ( ( R e. DivRing /\ x e. B ) /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
48	45, 47	sylan2br 474	. . . . . 6 |- ( ( ( R e. DivRing /\ x e. B ) /\ -. x = ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B )
49	44, 48	ifclda 3961	. . . . 5 |- ( ( R e. DivRing /\ x e. B ) -> if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
50	49	ralrimiva 2868	. . . 4 |- ( R e. DivRing -> A. x e. B if ( x = ( 0g ` R ) , x , ( ( invr ` R ) ` x ) ) e. B )
51		acsfn1 15153	. . . 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 661	. . 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 15091	. . 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 1226	. 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106   \ cdif 3458   i^i cin 3460   C_ wss 3461   ifcif 3929   ~Pcpw 3999   {csn 4016   ` cfv 5570   Basecbs 14719   0gc0g 14932  Moorecmre 15074  ACScacs 15077   Ringcrg 17396   invrcinvr 17518   DivRingcdr 17594  SubRingcsubrg 17623  SubDRingcsdrg 31388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-0g 14934  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-grp 16259  df-minusg 16260  df-subg 16400  df-mgp 17340  df-ur 17352  df-ring 17398  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-drng 17596  df-subrg 17625  df-sdrg 31389

This theorem is referenced by: (None)