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Metamath Proof Explorer

Theorem ringccat

Description: The category of unital rings is a category. (Contributed by AV, 14-Feb-2020) (Revised by AV, 9-Mar-2020)

Ref Expression
Hypothesis ringccat.c ⊢ 𝐶 = ( RingCat ‘ 𝑈 )
Assertion ringccat ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )

Proof

Step Hyp Ref Expression
1 ringccat.c ⊢ 𝐶 = ( RingCat ‘ 𝑈 )
2 id ⊢ ( 𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉 )
3 eqidd ⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Ring ) = ( 𝑈 ∩ Ring ) )
4 eqidd ⊢ ( 𝑈 ∈ 𝑉 → ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) = ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) )
5 1 2 3 4 ringcval ⊢ ( 𝑈 ∈ 𝑉 → 𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) ) )
6 eqid ⊢ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) )
7 eqid ⊢ ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 )
8 eqidd ⊢ ( 𝑈 ∈ 𝑉 → ( Ring ∩ 𝑈 ) = ( Ring ∩ 𝑈 ) )
9 incom ⊢ ( 𝑈 ∩ Ring ) = ( Ring ∩ 𝑈 )
10 9 a1i ⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Ring ) = ( Ring ∩ 𝑈 ) )
11 10 sqxpeqd ⊢ ( 𝑈 ∈ 𝑉 → ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) = ( ( Ring ∩ 𝑈 ) × ( Ring ∩ 𝑈 ) ) )
12 11 reseq2d ⊢ ( 𝑈 ∈ 𝑉 → ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) = ( RingHom ↾ ( ( Ring ∩ 𝑈 ) × ( Ring ∩ 𝑈 ) ) ) )
13 7 2 8 12 rhmsubcsetc ⊢ ( 𝑈 ∈ 𝑉 → ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) ∈ ( Subcat ‘ ( ExtStrCat ‘ 𝑈 ) ) )
14 6 13 subccat ⊢ ( 𝑈 ∈ 𝑉 → ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RingHom ↾ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) ) ∈ Cat )
15 5 14 eqeltrd ⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )