Metamath Proof Explorer


Theorem oppgoppcid

Description: The converted opposite monoid has the same identity morphism as that of the opposite category. Example 3.6(2) of Adamek p. 25. (Contributed by Zhi Wang, 22-Sep-2025)

Ref Expression
Hypotheses mndtccat.c ( 𝜑𝐶 = ( MndToCat ‘ 𝑀 ) )
mndtccat.m ( 𝜑𝑀 ∈ Mnd )
oppgoppchom.d ( 𝜑𝐷 = ( MndToCat ‘ ( oppg𝑀 ) ) )
oppgoppchom.o 𝑂 = ( oppCat ‘ 𝐶 )
oppgoppchom.x ( 𝜑𝑋 ∈ ( Base ‘ 𝐷 ) )
oppgoppchom.y ( 𝜑𝑌 ∈ ( Base ‘ 𝑂 ) )
Assertion oppgoppcid ( 𝜑 → ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) = ( ( Id ‘ 𝑂 ) ‘ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 mndtccat.c ( 𝜑𝐶 = ( MndToCat ‘ 𝑀 ) )
2 mndtccat.m ( 𝜑𝑀 ∈ Mnd )
3 oppgoppchom.d ( 𝜑𝐷 = ( MndToCat ‘ ( oppg𝑀 ) ) )
4 oppgoppchom.o 𝑂 = ( oppCat ‘ 𝐶 )
5 oppgoppchom.x ( 𝜑𝑋 ∈ ( Base ‘ 𝐷 ) )
6 oppgoppchom.y ( 𝜑𝑌 ∈ ( Base ‘ 𝑂 ) )
7 eqid ( oppg𝑀 ) = ( oppg𝑀 )
8 eqid ( 0g𝑀 ) = ( 0g𝑀 )
9 7 8 oppgid ( 0g𝑀 ) = ( 0g ‘ ( oppg𝑀 ) )
10 9 a1i ( 𝜑 → ( 0g𝑀 ) = ( 0g ‘ ( oppg𝑀 ) ) )
11 eqid ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
12 4 11 oppcbas ( Base ‘ 𝐶 ) = ( Base ‘ 𝑂 )
13 12 eqcomi ( Base ‘ 𝑂 ) = ( Base ‘ 𝐶 )
14 13 a1i ( 𝜑 → ( Base ‘ 𝑂 ) = ( Base ‘ 𝐶 ) )
15 1 2 mndtccat ( 𝜑𝐶 ∈ Cat )
16 eqid ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
17 4 16 oppcid ( 𝐶 ∈ Cat → ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) )
18 15 17 syl ( 𝜑 → ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) )
19 1 2 14 6 18 mndtcid ( 𝜑 → ( ( Id ‘ 𝑂 ) ‘ 𝑌 ) = ( 0g𝑀 ) )
20 7 oppgmnd ( 𝑀 ∈ Mnd → ( oppg𝑀 ) ∈ Mnd )
21 2 20 syl ( 𝜑 → ( oppg𝑀 ) ∈ Mnd )
22 eqidd ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) )
23 eqidd ( 𝜑 → ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 ) )
24 3 21 22 5 23 mndtcid ( 𝜑 → ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) = ( 0g ‘ ( oppg𝑀 ) ) )
25 10 19 24 3eqtr4rd ( 𝜑 → ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) = ( ( Id ‘ 𝑂 ) ‘ 𝑌 ) )