# Metamath Proof Explorer

## Theorem mgmcl

Description: Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010) (Revised by AV, 13-Jan-2020)

Ref Expression
Hypotheses mgmcl.b ${⊢}{B}={\mathrm{Base}}_{{M}}$
mgmcl.o No typesetting found for |- .o. = ( +g  M ) with typecode |-
Assertion mgmcl Could not format assertion : No typesetting found for |- ( ( M e. Mgm /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B ) with typecode |-

### Proof

Step Hyp Ref Expression
1 mgmcl.b ${⊢}{B}={\mathrm{Base}}_{{M}}$
2 mgmcl.o Could not format .o. = ( +g  M ) : No typesetting found for |- .o. = ( +g ` M ) with typecode |-
3 1 2 ismgm Could not format ( M e. Mgm -> ( M e. Mgm <-> A. x e. B A. y e. B ( x .o. y ) e. B ) ) : No typesetting found for |- ( M e. Mgm -> ( M e. Mgm <-> A. x e. B A. y e. B ( x .o. y ) e. B ) ) with typecode |-
4 3 ibi Could not format ( M e. Mgm -> A. x e. B A. y e. B ( x .o. y ) e. B ) : No typesetting found for |- ( M e. Mgm -> A. x e. B A. y e. B ( x .o. y ) e. B ) with typecode |-
5 ovrspc2v Could not format ( ( ( X e. B /\ Y e. B ) /\ A. x e. B A. y e. B ( x .o. y ) e. B ) -> ( X .o. Y ) e. B ) : No typesetting found for |- ( ( ( X e. B /\ Y e. B ) /\ A. x e. B A. y e. B ( x .o. y ) e. B ) -> ( X .o. Y ) e. B ) with typecode |-
6 5 expcom Could not format ( A. x e. B A. y e. B ( x .o. y ) e. B -> ( ( X e. B /\ Y e. B ) -> ( X .o. Y ) e. B ) ) : No typesetting found for |- ( A. x e. B A. y e. B ( x .o. y ) e. B -> ( ( X e. B /\ Y e. B ) -> ( X .o. Y ) e. B ) ) with typecode |-
7 4 6 syl Could not format ( M e. Mgm -> ( ( X e. B /\ Y e. B ) -> ( X .o. Y ) e. B ) ) : No typesetting found for |- ( M e. Mgm -> ( ( X e. B /\ Y e. B ) -> ( X .o. Y ) e. B ) ) with typecode |-
8 7 3impib Could not format ( ( M e. Mgm /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B ) : No typesetting found for |- ( ( M e. Mgm /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B ) with typecode |-