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Metamath Proof Explorer

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 fovcdm 
 |-  ( ( 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 biimtrdi 
 |-  ( 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 )