Theorem mgmsscl 13591

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 6196	. . 3 |- ( ( X e. S /\ Y e. S ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` G ) Y ) )
2	1	3ad2ant3 1047	. 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 1049	. . . . 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 1026	. . . . 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 1009	. . . . 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 2234	. . . . 5 |- ( +g ` H ) = ( +g ` H )
9	7, 8	mgmcl 13589	. . . 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 6058	. . . . . . 7 |- ( ( ( +g ` G ) |` ( S X. S ) ) = ( +g ` H ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` H ) Y ) )
12	11	eleq1d 2303	. . . . . 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 2237	. . . . 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 1046	. . 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 2312	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 1005   = wceq 1398   e. wcel 2205   C_ wss 3213   X. cxp 4749   |` cres 4753   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307  Mgmcmgm 13584

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 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 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-ov 6055  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-plusg 13320  df-mgm 13586

This theorem is referenced by:  mndissubm  13705  grpissubg  13928