Theorem ismnd

Description: The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl ), whose operation is associative (so, a semigroup, see also mndass ) and has a two-sided neutral element (see mndid ). (Contributed by Mario Carneiro, 6-Jan-2015) (Revised by AV, 1-Feb-2020)

	Ref	Expression
Hypotheses	ismnd.b	|- B = ( Base ` G )
	ismnd.p	|- .+ = ( +g ` G )
Assertion	ismnd	|- ( G e. Mnd <-> ( A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )

Proof

Step	Hyp	Ref	Expression
1		ismnd.b	|- B = ( Base ` G )
2		ismnd.p	|- .+ = ( +g ` G )
3	1 2	ismnddef	|- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
4		rexn0	|- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> B =/= (/) )
5		fvprc	|- ( -. G e. _V -> ( Base ` G ) = (/) )
6	1 5	eqtrid	|- ( -. G e. _V -> B = (/) )
7	6	necon1ai	|- ( B =/= (/) -> G e. _V )
8	1 2	issgrpv	|- ( G e. _V -> ( G e. Smgrp <-> A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) ) )
9	4 7 8	3syl	|- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> ( G e. Smgrp <-> A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) ) )
10	9	pm5.32ri	|- ( ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) <-> ( A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
11	3 10	bitri	|- ( G e. Mnd <-> ( A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )