Metamath Proof Explorer


Theorem rngcval

Description: Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020) (Revised by AV, 8-Mar-2020)

Ref Expression
Hypotheses rngcval.c 𝐶 = ( RngCat ‘ 𝑈 )
rngcval.u ( 𝜑𝑈𝑉 )
rngcval.b ( 𝜑𝐵 = ( 𝑈 ∩ Rng ) )
rngcval.h ( 𝜑𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )
Assertion rngcval ( 𝜑𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) )

Proof

Step Hyp Ref Expression
1 rngcval.c 𝐶 = ( RngCat ‘ 𝑈 )
2 rngcval.u ( 𝜑𝑈𝑉 )
3 rngcval.b ( 𝜑𝐵 = ( 𝑈 ∩ Rng ) )
4 rngcval.h ( 𝜑𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )
5 df-rngc RngCat = ( 𝑢 ∈ V ↦ ( ( ExtStrCat ‘ 𝑢 ) ↾cat ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) ) )
6 fveq2 ( 𝑢 = 𝑈 → ( ExtStrCat ‘ 𝑢 ) = ( ExtStrCat ‘ 𝑈 ) )
7 6 adantl ( ( 𝜑𝑢 = 𝑈 ) → ( ExtStrCat ‘ 𝑢 ) = ( ExtStrCat ‘ 𝑈 ) )
8 ineq1 ( 𝑢 = 𝑈 → ( 𝑢 ∩ Rng ) = ( 𝑈 ∩ Rng ) )
9 8 sqxpeqd ( 𝑢 = 𝑈 → ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) = ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) )
10 3 sqxpeqd ( 𝜑 → ( 𝐵 × 𝐵 ) = ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) )
11 10 eqcomd ( 𝜑 → ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) = ( 𝐵 × 𝐵 ) )
12 9 11 sylan9eqr ( ( 𝜑𝑢 = 𝑈 ) → ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) = ( 𝐵 × 𝐵 ) )
13 12 reseq2d ( ( 𝜑𝑢 = 𝑈 ) → ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )
14 4 eqcomd ( 𝜑 → ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) = 𝐻 )
15 14 adantr ( ( 𝜑𝑢 = 𝑈 ) → ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) = 𝐻 )
16 13 15 eqtrd ( ( 𝜑𝑢 = 𝑈 ) → ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) = 𝐻 )
17 7 16 oveq12d ( ( 𝜑𝑢 = 𝑈 ) → ( ( ExtStrCat ‘ 𝑢 ) ↾cat ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) )
18 2 elexd ( 𝜑𝑈 ∈ V )
19 ovexd ( 𝜑 → ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) ∈ V )
20 5 17 18 19 fvmptd2 ( 𝜑 → ( RngCat ‘ 𝑈 ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) )
21 1 20 syl5eq ( 𝜑𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) )