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 ∈ Mnd ↔ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) ∈ 𝐵 ∧ ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) + 𝑐 ) = ( 𝑎 + ( 𝑏 + 𝑐 ) ) ) ∧ ∃ 𝑒 ∈ 𝐵 ∀ 𝑎 ∈ 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 ∈ Mnd ↔ ( 𝐺 ∈ Smgrp ∧ ∃ 𝑒 ∈ 𝐵 ∀ 𝑎 ∈ 𝐵 ( ( 𝑒 + 𝑎 ) = 𝑎 ∧ ( 𝑎 + 𝑒 ) = 𝑎 ) ) )
4		rexn0	⊢ ( ∃ 𝑒 ∈ 𝐵 ∀ 𝑎 ∈ 𝐵 ( ( 𝑒 + 𝑎 ) = 𝑎 ∧ ( 𝑎 + 𝑒 ) = 𝑎 ) → 𝐵 ≠ ∅ )
5		fvprc	⊢ ( ¬ 𝐺 ∈ V → ( Base ‘ 𝐺 ) = ∅ )
6	1 5	eqtrid	⊢ ( ¬ 𝐺 ∈ V → 𝐵 = ∅ )
7	6	necon1ai	⊢ ( 𝐵 ≠ ∅ → 𝐺 ∈ V )
8	1 2	issgrpv	⊢ ( 𝐺 ∈ V → ( 𝐺 ∈ Smgrp ↔ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) ∈ 𝐵 ∧ ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) + 𝑐 ) = ( 𝑎 + ( 𝑏 + 𝑐 ) ) ) ) )
9	4 7 8	3syl	⊢ ( ∃ 𝑒 ∈ 𝐵 ∀ 𝑎 ∈ 𝐵 ( ( 𝑒 + 𝑎 ) = 𝑎 ∧ ( 𝑎 + 𝑒 ) = 𝑎 ) → ( 𝐺 ∈ Smgrp ↔ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) ∈ 𝐵 ∧ ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) + 𝑐 ) = ( 𝑎 + ( 𝑏 + 𝑐 ) ) ) ) )
10	9	pm5.32ri	⊢ ( ( 𝐺 ∈ Smgrp ∧ ∃ 𝑒 ∈ 𝐵 ∀ 𝑎 ∈ 𝐵 ( ( 𝑒 + 𝑎 ) = 𝑎 ∧ ( 𝑎 + 𝑒 ) = 𝑎 ) ) ↔ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) ∈ 𝐵 ∧ ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) + 𝑐 ) = ( 𝑎 + ( 𝑏 + 𝑐 ) ) ) ∧ ∃ 𝑒 ∈ 𝐵 ∀ 𝑎 ∈ 𝐵 ( ( 𝑒 + 𝑎 ) = 𝑎 ∧ ( 𝑎 + 𝑒 ) = 𝑎 ) ) )
11	3 10	bitri	⊢ ( 𝐺 ∈ Mnd ↔ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) ∈ 𝐵 ∧ ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 + 𝑏 ) + 𝑐 ) = ( 𝑎 + ( 𝑏 + 𝑐 ) ) ) ∧ ∃ 𝑒 ∈ 𝐵 ∀ 𝑎 ∈ 𝐵 ( ( 𝑒 + 𝑎 ) = 𝑎 ∧ ( 𝑎 + 𝑒 ) = 𝑎 ) ) )