Theorem mgmsscl 12592

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 5981	. . 3 |- ( ( X e. S /\ Y e. S ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` G ) Y ) )
2	1	3ad2ant3 1010	. 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 1012	. . . . 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 989	. . . . 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 972	. . . . 5 |- ( ( H e. Mgm /\ X e. S /\ Y e. S ) <-> ( H e. Mgm /\ ( X e. S /\ Y e. S ) ) )
6	3, 4, 5	sylanbrc 414	. . . 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 2165	. . . . 5 |- ( +g ` H ) = ( +g ` H )
9	7, 8	mgmcl 12590	. . . 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 5848	. . . . . . 7 |- ( ( ( +g ` G ) |` ( S X. S ) ) = ( +g ` H ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` H ) Y ) )
12	11	eleq1d 2235	. . . . . 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 2168	. . . . 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 275	. . . 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 1009	. . 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 166	. 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 2244	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   C_ wss 3116   X. cxp 4602   |` cres 4606   ` cfv 5188  (class class class)co 5842   Basecbs 12394   +g cplusg 12457  Mgmcmgm 12585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-ov 5845  df-inn 8858  df-2 8916  df-ndx 12397  df-slot 12398  df-base 12400  df-plusg 12470  df-mgm 12587

This theorem is referenced by: (None)