Theorem subrgmcl 14266

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 2231	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrgring 14257	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
3	2	3ad2ant1 1044	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Ring )
4		simp2 1024	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrgbas 14263	. . . . 5 |- ( A e. (SubRing ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1044	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2310	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1025	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2310	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2231	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2231	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	ringcl 14045	. . 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 1273	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrgrcl 14259	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
15		subrgmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 13246	. . . . 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 1044	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 6035	. 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 2315	1 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6018   Basecbs 13100   ↾_s cress 13101   .rcmulr 13179   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-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-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-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-fv 5334  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-mgm 13457  df-sgrp 13503  df-mnd 13518  df-subg 13775  df-mgp 13953  df-ring 14030  df-subrg 14252

This theorem is referenced by:  issubrg2  14274  subrgintm  14276  dvply2g  15509