Theorem efmndov

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

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

Proof

Step	Hyp	Ref	Expression
1		efmndtset.g	⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
2		efmndplusg.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
3		efmndplusg.p	⊢ + = ( +g ‘ 𝐺 )
4		coexg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∘ 𝑌 ) ∈ V )
5		coeq1	⊢ ( 𝑓 = 𝑋 → ( 𝑓 ∘ 𝑔 ) = ( 𝑋 ∘ 𝑔 ) )
6		coeq2	⊢ ( 𝑔 = 𝑌 → ( 𝑋 ∘ 𝑔 ) = ( 𝑋 ∘ 𝑌 ) )
7	1 2 3	efmndplusg	⊢ + = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 ∘ 𝑔 ) )
8	5 6 7	ovmpog	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∘ 𝑌 ) ∈ V ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )
9	4 8	mpd3an3	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )