Theorem subrgmcl 14237

Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypothesis

Ref	Expression
subrgmcl.p	|- .x. = ( .r ` R )

Assertion

Ref	Expression
subrgmcl	|- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Proof of Theorem subrgmcl

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrgring 14228	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
3	2	3ad2ant1 1042	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Ring )
4		simp2 1022	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrgbas 14234	. . . . 5 |- ( A e. (SubRing ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1042	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2308	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1023	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2308	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2229	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2229	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	ringcl 14016	. . 3 |- ( ( ( R ↾s A ) e. Ring /\ X e. ( Base ` ( R ↾s A ) ) /\ Y e. ( Base ` ( R ↾s A ) ) ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
13	3, 7, 9, 12	syl3anc 1271	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrgrcl 14230	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
15		subrgmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 13218	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> .x. = ( .r ` ( R ↾s A ) ) )
17	14, 16	mpdan 421	. . . 4 |- ( A e. (SubRing ` R ) -> .x. = ( .r ` ( R ↾s A ) ) )
18	17	3ad2ant1 1042	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 6030	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) = ( X ( .r ` ( R ↾s A ) ) Y ) )
20	13, 19, 6	3eltr4d 2313	1 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073   .rcmulr 13151   Ringcrg 13999  SubRingcsubrg 14221

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-subg 13747  df-mgp 13924  df-ring 14001  df-subrg 14223

This theorem is referenced by:  issubrg2  14245  subrgintm  14247  dvply2g  15480