Theorem subrngmcl 14056

Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14080. (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 2206	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
2	1	subrngrng 14049	. . . 4 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Rng )
3	2	3ad2ant1 1021	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( R ↾s A ) e. Rng )
4		simp2 1001	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. A )
5	1	subrngbas 14053	. . . . 5 |- ( A e. (SubRng ` R ) -> A = ( Base ` ( R ↾s A ) ) )
6	5	3ad2ant1 1021	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> A = ( Base ` ( R ↾s A ) ) )
7	4, 6	eleqtrd 2285	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> X e. ( Base ` ( R ↾s A ) ) )
8		simp3 1002	. . . 4 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. A )
9	8, 6	eleqtrd 2285	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> Y e. ( Base ` ( R ↾s A ) ) )
10		eqid 2206	. . . 4 |- ( Base ` ( R ↾s A ) ) = ( Base ` ( R ↾s A ) )
11		eqid 2206	. . . 4 |- ( .r ` ( R ↾s A ) ) = ( .r ` ( R ↾s A ) )
12	10, 11	rngcl 13791	. . 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 1250	. 2 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X ( .r ` ( R ↾s A ) ) Y ) e. ( Base ` ( R ↾s A ) ) )
14		subrngrcl 14050	. . . . 5 |- ( A e. (SubRng ` R ) -> R e. Rng )
15		subrngmcl.p	. . . . . 6 |- .x. = ( .r ` R )
16	1, 15	ressmulrg 13062	. . . . 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 1021	. . 3 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	18	oveqd 5979	. 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 2290	1 |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2177   ` cfv 5285  (class class class)co 5962   Basecbs 12917   ↾_s cress 12918   .rcmulr 12995  Rngcrng 13779  SubRngcsubrng 14044

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-iress 12925  df-plusg 13007  df-mulr 13008  df-mgm 13273  df-sgrp 13319  df-subg 13591  df-abl 13708  df-mgp 13768  df-rng 13780  df-subrng 14045

This theorem is referenced by:  issubrng2  14057  subrngintm  14059