Theorem submgmcl 41794

Description: Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)

Hypothesis

Ref	Expression
submgmcl.p	|- .+ = ( +g ` M )

Assertion

Ref	Expression
submgmcl	|- ( ( S e. (SubMgm ` M ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )

Proof of Theorem submgmcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		submgmrcl 41782	. . . . . . 7 |- ( S e. (SubMgm ` M ) -> M e. Mgm )
2		eqid 2622	. . . . . . . 8 |- ( Base ` M ) = ( Base ` M )
3		submgmcl.p	. . . . . . . 8 |- .+ = ( +g ` M )
4	2, 3	issubmgm 41789	. . . . . . 7 |- ( M e. Mgm -> ( S e. (SubMgm ` M ) <-> ( S C_ ( Base ` M ) /\ A. x e. S A. y e. S ( x .+ y ) e. S ) ) )
5	1, 4	syl 17	. . . . . 6 |- ( S e. (SubMgm ` M ) -> ( S e. (SubMgm ` M ) <-> ( S C_ ( Base ` M ) /\ A. x e. S A. y e. S ( x .+ y ) e. S ) ) )
6	5	ibi 256	. . . . 5 |- ( S e. (SubMgm ` M ) -> ( S C_ ( Base ` M ) /\ A. x e. S A. y e. S ( x .+ y ) e. S ) )
7	6	simprd 479	. . . 4 |- ( S e. (SubMgm ` M ) -> A. x e. S A. y e. S ( x .+ y ) e. S )
8		ovrspc2v 6672	. . . 4 |- ( ( ( X e. S /\ Y e. S ) /\ A. x e. S A. y e. S ( x .+ y ) e. S ) -> ( X .+ Y ) e. S )
9	7, 8	sylan2 491	. . 3 |- ( ( ( X e. S /\ Y e. S ) /\ S e. (SubMgm ` M ) ) -> ( X .+ Y ) e. S )
10	9	ancoms 469	. 2 |- ( ( S e. (SubMgm ` M ) /\ ( X e. S /\ Y e. S ) ) -> ( X .+ Y ) e. S )
11	10	3impb 1260	1 |- ( ( S e. (SubMgm ` M ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240  SubMgmcsubmgm 41778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-submgm 41780

This theorem is referenced by:  resmgmhm  41798  mgmhmima  41802