Theorem subrngmcl 13765

Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 13789. (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 2196	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrngrng 13758	. . . 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 13762	. . . . 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 2275	. . 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 2275	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2196	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2196	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	rngcl 13500	. . 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 13759	. . . . 5 |- ( A e. (SubRng ` R ) -> R e. Rng )
15		subrngmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 12822	. . . . 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 5939	. 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 2280	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 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   .rcmulr 12756  Rngcrng 13488  SubRngcsubrng 13753

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-mgm 12999  df-sgrp 13045  df-subg 13300  df-abl 13417  df-mgp 13477  df-rng 13489  df-subrng 13754

This theorem is referenced by:  issubrng2  13766  subrngintm  13768