Metamath Proof Explorer


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 + = ( +g𝐺 )
Assertion efmndov ( ( 𝑋𝐵𝑌𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑋𝑌 ) )

Proof

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