Theorem subrngmcl 14455

Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14479. (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 2234	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrngrng 14448	. . . 4 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Rng )
3	2	3ad2ant1 1045	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Rng )
4		simp2 1025	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrngbas 14452	. . . . 5 |- ( A e. (SubRng ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1045	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2313	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1026	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2313	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2234	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2234	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	rngcl 14183	. . 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 1274	. 2 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrngrcl 14449	. . . . 5 |- ( A e. (SubRng ` R ) -> R e. Rng )
15		subrngmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 13442	. . . . 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 1045	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 6075	. 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 2318	1 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   .rcmulr 13375  Rngcrng 14171  SubRngcsubrng 14443

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-mgm 13619  df-sgrp 13665  df-subg 13923  df-abl 14040  df-mgp 14160  df-rng 14172  df-subrng 14444

This theorem is referenced by:  issubrng2  14456  subrngintm  14458