Theorem subrngmcl 13553

Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 13577. (Revised by AV, 14-Feb-2025.)

Hypothesis

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

Assertion

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

Proof of Theorem subrngmcl

Step	Hyp	Ref	Expression
1		eqid 2189	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrngrng 13546	. . . 4 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Rng )
3	2	3ad2ant1 1020	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Rng )
4		simp2 1000	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrngbas 13550	. . . . 5 |- ( A e. (SubRng ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1020	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2268	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1001	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2268	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2189	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2189	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	rngcl 13295	. . 3 |- ( ( ( R ↾s A ) e. Rng /\ 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 1249	. 2 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrngrcl 13547	. . . . 5 |- ( A e. (SubRng ` R ) -> R e. Rng )
15		subrngmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 12653	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ R e. Rng ) -> .x. = ( .r ` ( R ↾s A ) ) )
17	14, 16	mpdan 421	. . . 4 |- ( A e. (SubRng ` R ) -> .x. = ( .r ` ( R ↾s A ) ) )
18	17	3ad2ant1 1020	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 5912	. 2 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) = ( X ( .r ` ( R ↾s A ) ) Y ) )
20	13, 19, 6	3eltr4d 2273	1 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160   ` cfv 5235  (class class class)co 5895   Basecbs 12511   ↾_s cress 12512   .rcmulr 12587  Rngcrng 13283  SubRngcsubrng 13541

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-mgm 12829  df-sgrp 12862  df-subg 13106  df-abl 13223  df-mgp 13272  df-rng 13284  df-subrng 13542

This theorem is referenced by:  issubrng2  13554  subrngintm  13556