Theorem efmndcl

Description: The group operation of the monoid of endofunctions on A is closed. (Contributed by AV, 27-Jan-2024)

	Ref	Expression
Hypotheses	efmndtset.g	⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
	efmndplusg.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	efmndplusg.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	efmndcl	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		efmndtset.g	⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
2		efmndplusg.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
3		efmndplusg.p	⊢ + = ( +g ‘ 𝐺 )
4	1 2 3	efmndov	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )
5	1 2	efmndbasf	⊢ ( 𝑋 ∈ 𝐵 → 𝑋 : 𝐴 ⟶ 𝐴 )
6	1 2	efmndbasf	⊢ ( 𝑌 ∈ 𝐵 → 𝑌 : 𝐴 ⟶ 𝐴 )
7		fco	⊢ ( ( 𝑋 : 𝐴 ⟶ 𝐴 ∧ 𝑌 : 𝐴 ⟶ 𝐴 ) → ( 𝑋 ∘ 𝑌 ) : 𝐴 ⟶ 𝐴 )
8	5 6 7	syl2an	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∘ 𝑌 ) : 𝐴 ⟶ 𝐴 )
9		coexg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∘ 𝑌 ) ∈ V )
10	1 2	elefmndbas2	⊢ ( ( 𝑋 ∘ 𝑌 ) ∈ V → ( ( 𝑋 ∘ 𝑌 ) ∈ 𝐵 ↔ ( 𝑋 ∘ 𝑌 ) : 𝐴 ⟶ 𝐴 ) )
11	9 10	syl	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∘ 𝑌 ) ∈ 𝐵 ↔ ( 𝑋 ∘ 𝑌 ) : 𝐴 ⟶ 𝐴 ) )
12	8 11	mpbird	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∘ 𝑌 ) ∈ 𝐵 )
13	4 12	eqeltrd	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )