Theorem subrngmcl 14626

Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14650. (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 2238	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrngrng 14619	. . . 4 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Rng )
3	2	3ad2ant1 1049	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Rng )
4		simp2 1029	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrngbas 14623	. . . . 5 |- ( A e. (SubRng ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1049	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2317	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1030	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2317	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2238	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2238	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	rngcl 14350	. . 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 1278	. 2 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrngrcl 14620	. . . . 5 |- ( A e. (SubRng ` R ) -> R e. Rng )
15		subrngmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 13558	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ R e. Rng ) -> .x. = ( .r ` ( R ↾s A ) ) )
17	14, 16	mpdan 425	. . . 4 |- ( A e. (SubRng ` R ) -> .x. = ( .r ` ( R ↾s A ) ) )
18	17	3ad2ant1 1049	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 6105	. 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 2322	1 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   Basecbs 13410   ↾_s cress 13411   .rcmulr 13491  Rngcrng 14338  SubRngcsubrng 14614

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-mgm 13735  df-sgrp 13776  df-subg 14032  df-abl 14180  df-mgp 14325  df-rng 14339  df-subrng 14615

This theorem is used by:  issubrng2  14627  subrngintm  14629