Theorem clmgmOLD

Description: Obsolete version of mgmcl as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	clmgmOLD.1	|- X = dom dom G
Assertion	clmgmOLD	|- ( ( G e. Magma /\ A e. X /\ B e. X ) -> ( A G B ) e. X )

Proof

Step	Hyp	Ref	Expression
1		clmgmOLD.1	|- X = dom dom G
2	1	ismgmOLD	|- ( G e. Magma -> ( G e. Magma <-> G : ( X X. X ) --> X ) )
3		fovrn	|- ( ( G : ( X X. X ) --> X /\ A e. X /\ B e. X ) -> ( A G B ) e. X )
4	3	3exp	|- ( G : ( X X. X ) --> X -> ( A e. X -> ( B e. X -> ( A G B ) e. X ) ) )
5	2 4	syl6bi	|- ( G e. Magma -> ( G e. Magma -> ( A e. X -> ( B e. X -> ( A G B ) e. X ) ) ) )
6	5	pm2.43i	|- ( G e. Magma -> ( A e. X -> ( B e. X -> ( A G B ) e. X ) ) )
7	6	3imp	|- ( ( G e. Magma /\ A e. X /\ B e. X ) -> ( A G B ) e. X )