Theorem subrgmcl 14191

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 14182	. . . 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 14188	. . . . 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 13971	. . 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 14184	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
15		subrgmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 13173	. . . . 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 6017	. 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 5317  (class class class)co 6000   Basecbs 13027   ↾_s cress 13028   .rcmulr 13106   Ringcrg 13954  SubRingcsubrg 14175

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-mulr 13119  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-subg 13702  df-mgp 13879  df-ring 13956  df-subrg 14177

This theorem is referenced by:  issubrg2  14199  subrgintm  14201  dvply2g  15434