Theorem mgmsscl 13193

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 6086	. . 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 2205	. . . . 5 |- ( +g ` H ) = ( +g ` H )
9	7, 8	mgmcl 13191	. . . 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 5950	. . . . . . 7 |- ( ( ( +g ` G ) |` ( S X. S ) ) = ( +g ` H ) -> ( X ( ( +g ` G ) |` ( S X. S ) ) Y ) = ( X ( +g ` H ) Y ) )
12	11	eleq1d 2274	. . . . . 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 2208	. . . . 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 2283	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 2176   C_ wss 3166   X. cxp 4673   |` cres 4677   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909  Mgmcmgm 13186

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-mgm 13188

This theorem is referenced by:  mndissubm  13307  grpissubg  13530