Theorem subrgmcl 14378

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 2232	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrgring 14369	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
3	2	3ad2ant1 1045	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Ring )
4		simp2 1025	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrgbas 14375	. . . . 5 |- ( A e. (SubRing ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1045	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2311	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1026	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2311	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2232	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2232	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	ringcl 14157	. . 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 1274	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrgrcl 14371	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
15		subrgmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 13358	. . . . 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 1045	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 6067	. 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 2316	1 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   ↾_s cress 13213   .rcmulr 13291   Ringcrg 14140  SubRingcsubrg 14362

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-subg 13887  df-mgp 14065  df-ring 14142  df-subrg 14364

This theorem is referenced by:  issubrg2  14386  subrgintm  14388  dvply2g  15631