Theorem mgmsscl 13658

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 6219	. . 3 |- ( ( X e. S /\ Y e. S ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` G ) Y ) )
2	1	3ad2ant3 1051	. 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 1053	. . . . 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 1030	. . . . 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 1013	. . . . 5 |- ( ( H e. Mgm /\ X e. S /\ Y e. S ) <-> ( H e. Mgm /\ ( X e. S /\ Y e. S ) ) )
6	3, 4, 5	sylanbrc 421	. . . 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 2238	. . . . 5 |- ( +g ` H ) = ( +g ` H )
9	7, 8	mgmcl 13656	. . . 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 6081	. . . . . . 7 |- ( ( ( +g ` G ) |` ( S X. S ) ) = ( +g ` H ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` H ) Y ) )
12	11	eleq1d 2307	. . . . . 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 2241	. . . . 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 1050	. . 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 2316	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 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   X. cxp 4767   |` cres 4771   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408  Mgmcmgm 13651

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-mgm 13653

This theorem is referenced by:  mndissubm  13759  grpissubg  13974