Theorem subrgmcl 13448

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 2187	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrgring 13439	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
3	2	3ad2ant1 1019	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Ring )
4		simp2 999	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrgbas 13445	. . . . 5 |- ( A e. (SubRing ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1019	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2266	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1000	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2266	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2187	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2187	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	ringcl 13260	. . 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 1248	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrgrcl 13441	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
15		subrgmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 12617	. . . . 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 1019	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 5905	. 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 2271	1 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Syntax hints:   -> wi 4   /\ w3a 979   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12475   ↾_s cress 12476   .rcmulr 12551   Ringcrg 13243  SubRingcsubrg 13432

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-subg 13061  df-mgp 13171  df-ring 13245  df-subrg 13434

This theorem is referenced by:  issubrg2  13456  subrgintm  13458