Theorem issubrgd 14548

Description: Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)

Hypotheses

Ref	Expression
issubrgd.s	|- ( ph -> S = ( I ↾s D ) )
issubrgd.z	|- ( ph -> .0. = ( 0g ` I ) )
issubrgd.p	|- ( ph -> .+ = ( +g ` I ) )
issubrgd.ss	|- ( ph -> D C_ ( Base ` I ) )
issubrgd.zcl	|- ( ph -> .0. e. D )
issubrgd.acl	|- ( ( ph /\ x e. D /\ y e. D ) -> ( x .+ y ) e. D )
issubrgd.ncl	|- ( ( ph /\ x e. D ) -> ( ( invg ` I ) ` x ) e. D )
issubrgd.o	|- ( ph -> .1. = ( 1r ` I ) )
issubrgd.t	|- ( ph -> .x. = ( .r ` I ) )
issubrgd.ocl	|- ( ph -> .1. e. D )
issubrgd.tcl	|- ( ( ph /\ x e. D /\ y e. D ) -> ( x .x. y ) e. D )
issubrgd.g	|- ( ph -> I e. Ring )

Assertion

Ref	Expression
issubrgd	|- ( ph -> D e. (SubRing ` I ) )

Distinct variable groups:   x, y, .0.   x, D, y   x, I, y   x, .+ , y   ph, x, y   x, S, y   x, .x. , y

Allowed substitution hints:   .1. ( x, y)

Proof of Theorem issubrgd

Step	Hyp	Ref	Expression
1		issubrgd.s	. . 3 |- ( ph -> S = ( I ↾s D ) )
2		issubrgd.z	. . 3 |- ( ph -> .0. = ( 0g ` I ) )
3		issubrgd.p	. . 3 |- ( ph -> .+ = ( +g ` I ) )
4		issubrgd.ss	. . 3 |- ( ph -> D C_ ( Base ` I ) )
5		issubrgd.zcl	. . 3 |- ( ph -> .0. e. D )
6		issubrgd.acl	. . 3 |- ( ( ph /\ x e. D /\ y e. D ) -> ( x .+ y ) e. D )
7		issubrgd.ncl	. . 3 |- ( ( ph /\ x e. D ) -> ( ( invg ` I ) ` x ) e. D )
8		issubrgd.g	. . . 4 |- ( ph -> I e. Ring )
9		ringgrp 14095	. . . 4 |- ( I e. Ring -> I e. Grp )
10	8, 9	syl 14	. . 3 |- ( ph -> I e. Grp )
11	1, 2, 3, 4, 5, 6, 7, 10	issubgrpd2 13857	. 2 |- ( ph -> D e. (SubGrp ` I ) )
12		issubrgd.o	. . 3 |- ( ph -> .1. = ( 1r ` I ) )
13		issubrgd.ocl	. . 3 |- ( ph -> .1. e. D )
14	12, 13	eqeltrrd 2309	. 2 |- ( ph -> ( 1r ` I ) e. D )
15		issubrgd.t	. . . . 5 |- ( ph -> .x. = ( .r ` I ) )
16	15	oveqdr 6056	. . . 4 |- ( ( ph /\ ( x e. D /\ y e. D ) ) -> ( x .x. y ) = ( x ( .r ` I ) y ) )
17		issubrgd.tcl	. . . . 5 |- ( ( ph /\ x e. D /\ y e. D ) -> ( x .x. y ) e. D )
18	17	3expb 1231	. . . 4 |- ( ( ph /\ ( x e. D /\ y e. D ) ) -> ( x .x. y ) e. D )
19	16, 18	eqeltrrd 2309	. . 3 |- ( ( ph /\ ( x e. D /\ y e. D ) ) -> ( x ( .r ` I ) y ) e. D )
20	19	ralrimivva 2615	. 2 |- ( ph -> A. x e. D A. y e. D ( x ( .r ` I ) y ) e. D )
21		eqid 2231	. . . 4 |- ( Base ` I ) = ( Base ` I )
22		eqid 2231	. . . 4 |- ( 1r ` I ) = ( 1r ` I )
23		eqid 2231	. . . 4 |- ( .r ` I ) = ( .r ` I )
24	21, 22, 23	issubrg2 14336	. . 3 |- ( I e. Ring -> ( D e. (SubRing ` I ) <-> ( D e. (SubGrp ` I ) /\ ( 1r ` I ) e. D /\ A. x e. D A. y e. D ( x ( .r ` I ) y ) e. D ) ) )
25	8, 24	syl 14	. 2 |- ( ph -> ( D e. (SubRing ` I ) <-> ( D e. (SubGrp ` I ) /\ ( 1r ` I ) e. D /\ A. x e. D A. y e. D ( x ( .r ` I ) y ) e. D ) ) )
26	11, 14, 20, 25	mpbir3and 1207	1 |- ( ph -> D e. (SubRing ` I ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   ` cfv 5333  (class class class)co 6028   Basecbs 13162   ↾_s cress 13163   +g cplusg 13240   .rcmulr 13241   0gc0g 13419   Grpcgrp 13663   invgcminusg 13664  SubGrpcsubg 13834   1rcur 14053   Ringcrg 14090  SubRingcsubrg 14312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-subg 13837  df-mgp 14015  df-ur 14054  df-ring 14092  df-subrg 14314

This theorem is referenced by: (None)