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Metamath Proof Explorer

Theorem invid

Description: The inverse of the identity is the identity. (Contributed by AV, 8-Apr-2020)

Ref Expression
Hypotheses invid.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
invid.i ⊢ 𝐼 = ( Id ‘ 𝐶 )
invid.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
invid.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion invid ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 invid.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
2 invid.i ⊢ 𝐼 = ( Id ‘ 𝐶 )
3 invid.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
4 invid.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 1 2 3 4 sectid ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) )
6 eqid ⊢ ( Inv ‘ 𝐶 ) = ( Inv ‘ 𝐶 )
7 eqid ⊢ ( Sect ‘ 𝐶 ) = ( Sect ‘ 𝐶 )
8 1 6 3 4 4 7 isinv ⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ∧ ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ) ) )
9 5 5 8 mpbir2and ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) )