Theorem mgmsscl 13308

Description: If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. (Contributed by AV, 17-Feb-2024.)

Hypotheses

Ref	Expression
mgmsscl.b	|- B = ( Base ` G )
mgmsscl.s	|- S = ( Base ` H )

Assertion

Ref	Expression
mgmsscl	|- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X ( +g ` G ) Y ) e. S )

Proof of Theorem mgmsscl

Step	Hyp	Ref	Expression
1		ovres 6109	. . 3 |- ( ( X e. S /\ Y e. S ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` G ) Y ) )
2	1	3ad2ant3 1023	. 2 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` G ) Y ) )
3		simp1r 1025	. . . . 5 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> H e. Mgm )
4		simp3 1002	. . . . 5 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X e. S /\ Y e. S ) )
5		3anass 985	. . . . 5 |- ( ( H e. Mgm /\ X e. S /\ Y e. S ) <-> ( H e. Mgm /\ ( X e. S /\ Y e. S ) ) )
6	3, 4, 5	sylanbrc 417	. . . 4 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( H e. Mgm /\ X e. S /\ Y e. S ) )
7		mgmsscl.s	. . . . 5 |- S = ( Base ` H )
8		eqid 2207	. . . . 5 |- ( +g ` H ) = ( +g ` H )
9	7, 8	mgmcl 13306	. . . 4 |- ( ( H e. Mgm /\ X e. S /\ Y e. S ) -> ( X ( +g ` H ) Y ) e. S )
10	6, 9	syl 14	. . 3 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X ( +g ` H ) Y ) e. S )
11		oveq 5973	. . . . . . 7 |- ( ( ( +g ` G ) |` ( S X. S ) ) = ( +g ` H ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` H ) Y ) )
12	11	eleq1d 2276	. . . . . 6 |- ( ( ( +g ` G ) |` ( S X. S ) ) = ( +g ` H ) -> ( ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) e. S <-> ( X ( +g ` H ) Y ) e. S ) )
13	12	eqcoms 2210	. . . . 5 |- ( ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) -> ( ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) e. S <-> ( X ( +g ` H ) Y ) e. S ) )
14	13	adantl 277	. . . 4 |- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) e. S <-> ( X ( +g ` H ) Y ) e. S ) )
15	14	3ad2ant2 1022	. . 3 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) e. S <-> ( X ( +g ` H ) Y ) e. S ) )
16	10, 15	mpbird 167	. 2 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) e. S )
17	2, 16	eqeltrrd 2285	1 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X ( +g ` G ) Y ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   C_ wss 3174   X. cxp 4691   |` cres 4695   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024  Mgmcmgm 13301

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-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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-mgm 13303

This theorem is referenced by:  mndissubm  13422  grpissubg  13645