Theorem issubrgd 14485

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 14033	. . . 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 13795	. 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 6046	. . . 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 1230	. . . 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 2614	. 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 14274	. . 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 1206	1 |- ( ph -> D e. (SubRing ` I ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   ` cfv 5326  (class class class)co 6018   Basecbs 13100   ↾_s cress 13101   +g cplusg 13178   .rcmulr 13179   0gc0g 13357   Grpcgrp 13601   invgcminusg 13602  SubGrpcsubg 13772   1rcur 13991   Ringcrg 14028  SubRingcsubrg 14250

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-mulr 13192  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-subg 13775  df-mgp 13953  df-ur 13992  df-ring 14030  df-subrg 14252

This theorem is referenced by: (None)